Conservative Structured Noncommutative Multidimensional Linear Systems

Operator Theory: Advances and Applications, Vol. 161, 179–223 c 2005 Birkhauser
¨ Verlag Basel/Switzerland

Conservative Structured Noncommutative Multidimensional Linear Systems Joseph A. Ball, Gilbert Groenewald and Tanit Malakorn Abstract. We introduce a class of conservative structured multidimensional linear systems with evolution along a free semigroup. The system matrix for such a system is unitary and the associated transfer function is a formal power series in noncommuting indeterminates. A formal power series T (z1 , . . . , zd ) in the noncommuting indeterminates z1 , . . . , zd arising in this way satisfies a noncommutative von Neumann inequality, i.e., substitution of a d-tuple of noncommuting operators δ = (δ1 , . . . , δd ) on a fixed separable Hilbert space which is contractive in the appropriate sense yields a contraction operator T (δ) = T (δ1 , . . . , δd ). We also obtain the converse realization theorem: any formal power series satisfying such a von Neumann inequality can be realized as the transfer function of such a conservative structured multidimensional linear system. Mathematics Subject Classification (2000). Primary 47A56; Secondary 13F25, 47A60, 93B28. Keywords. Formal power series, noncommuting indeterminates, energy balance, Hahn-Banach separation argument, noncommutative Schur-Agler class.

Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Structured noncommutative multidimensional linear systems: basic definitions and properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 Adjoint systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Dissipative and conservative structured multidimensional linear systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Conservative SNMLS-realization of formal power series in the class SAG (U, Y) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

180 183 191 193 199 220

The first author was partially supported by US National Science Foundation under Grant Number DMS-9987636; The second author is supported by the National Research Foundation of South Africa under Grant Number 2053733; The third author was supported by a grant from Naresuan University, Thailand.

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1. Introduction This paper concerns extensions of the classical theory of conservative discrete-time linear systems to the setting of conservative structured multidimensional linear systems with evolution along a finitely generated free semigroup (words in a finite set of letters). By way of introduction we first review the relevant points of the classical theory. By a (classical) conservative discrete-time input/state/output (i/s/o) linear system, we mean a system of equations of the form  x(n + 1) = Ax(n) + Bu(n) Σ = Σ(U ) : (1.1) y(n) = Cx(n) + Du(n) such that the so-called connection matrix or colligation C D C D C D A B H H U= : → C D U Y

(1.2)

is unitary. Here we assume that x(n) takes values in the state space H, u(n) takes values in the input space U and y(n) takes values in the output space Y where H, U and Y are all assumed to be Hilbert spaces. The unitary property of the colligation U leads to the energy balance relation x(n + 1)2 − x(n)2 = u(n)2 − y(n)2 .

(1.3)

Summing over all n with 0 ≤ n ≤ N leads to x(N + 1)2 − x(0)2 =

N  2

3 u(n)2 − y(n)2 .

n=0

In particular, if we assume that x(0) = 0 and let N → ∞ we get ∞ ∞   y(n)2 ≤ u(n)2 . n=0

(1.4)

n=0

Application of the Z-transform {x(n)}n∈Z+ → x (z) :=



x(n)z n

n∈Z+

to the system equations (1.1) leads to the frequency-domain formulas x (z) = (I − zA)−1 x(0) + z(I − zA)−1 B u(z)

(1.5)

(z) y(z) = C(I − zA)−1 x(0) + TΣ (z) · u

(1.6)

where

TΣ (z) = D + zC(I − zA)−1 B. In particular, if we assume x(0) = 0 we get the input-output relation y(z) = TΣ (z) · u (z).

From (1.4) and the Plancherel theorem we then see that H 2 (D,Y) ≤  uH 2 (D,U ) T TΣ · u

(1.7)

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∞ for all u  ∈ H 2 (D, U) (the Hardy space of U-valued functions u (z) = n=0 u(n)z n on the unit disk D with norm-square summable Taylor coefficients (u2H 2 (D,U ) = ∞ 2 TΣ is in the operator-valued n=0 u(n) < ∞). As a result it follows that n Schur class S(U, Y) consisting of functions S(z) = ∞ n=0 Sn z analytic on D with values equal to contraction operators from ∞U into Y. Conversely, it is well known that any Schur class function S(z) = n=0 Sn z n ∈ S(U, Y) can be realized as the transfer function of a conservative linear system, i.e., any S ∈ S(U, Y) can be written in the form S(z) = D + zC(I − zA)−1 B for some unitary colligation 2 3 H A B ] : [H] → U = [C Y . Moreover any such S satisfies a von Neumann inequality: D U if K is another Hilbert space and T ∈ L(K)has T  < 1, then S(T ) ≤ 1 where ∞ S(T ) ∈ L(U ⊗K, Y ⊗K) is given by S(T ) = n=0 Sn ⊗T n . The following theorem is a convenient summary of the various equivalent characterizations of the operatorvalued Schur class S(U, Y). Theorem 1.1. Let z → S(z) be a L(U, Y)-valued function defined on the unit disk D. Then the following conditions are equivalent: 1. S ∈ S(U, Y), i.e., S is analytic on D and S(z) ≤ 1 for all z ∈ D. 2. S is analytic on D and S(T ) ≤ 1 for any operator T on some Hilbert space K with T  < 1. 3. There exists a Hilbert space H and an operator-valued function z → H(z) ∈ L(H , Y) so that I − S(z)S(w)∗ = H(z)H(w)∗ for all z, w ∈ D. 1 − zw 4. S(z) can be realized as the transfer function of a conservative discrete-time i/s/o linear system, i.e., there is a unitary colligation U of the form (1.2) so that S(z) = D + zC(IIH − zA)−1 B. For more information on the Schur class and its applications in both operator theory and engineering, we refer the reader to [42, 25, 26, 46, 38, 47]. Recent work has generalized these ideas to multivariable settings in several ways. We mention [1, 2, 17, 15, 16] for extensions to the polydisk Dn ⊂ Cn setting, [33, 52, 10, 30, 3, 8, 18, 45, 37, 4] for extensions to the unit ball Bn ⊂ Cn (where additional refinements concerning Nevanlinna-Pick-type interpolation and lifting theorems are also explored), and the recent work [56, 7, 6, 12] which suggests how a unification of these two settings can be achieved. In the present paper we generalize these ideas to other types of conservative structured multidimensional linear systems. This paper can be considered as a sequel to our paper [13] where we introduced and studied a general class of systems called structured noncommutative multidimensional linear systems (SNMLSs). These systems have evolution along a free semigroup rather than along an integer lattice as is usually taken in work in multidimensional linear system theory, and the transfer function is a formal power series in noncommuting indeterminates rather than an analytic function of several complex variables. In [13] it is assumed

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that the input space, state space and output space were all finite-dimensional linear spaces, and analogues of the standard results in finite-dimensional linear system theory (such as controllability, observability, Kalman decomposition, state space similarity theorem, Hankel operators and realization theory) were developed. Here we use the same notion of SNMLS as introduced in [13] but take the input space, state space and output space all to be Hilbert spaces and introduce a notion of conservative SNMLS for which the system and its adjoint satisfy an energy balance relation. The main result is Theorem 5.3 which can be viewed as a far-reaching generalization of Theorem 1.1. In this generalization, the unit disk is replaced by a tuple of (not necessarily commuting) operators δ = (δ1 , . . . , δd ) on some Hilbert  space K in an appropriate noncommutative Cartan domain ( di=1 Ii ⊗ δi  < 1 for an appropriate collection of n∞× m matrices I1 , . . . , Id ), and analytic operatorvalued functions z → T (z) = n=0 Tn z n on the unit disk are replaced by formal power series  Tw z w (1.8) T (z) = w∈F Fd

in a set of noncommuting formal indeterminates z = (z1 , . . . , zd ), where the coefficients Tw are operators from U to Y. Here Fd is the free semigroup generated by the set of letters {1, . . . , d}; thus elements of Fd are words w of the form w = iN · · · i1 where ik ∈ {1, . . . , d} for each k = 1, . . . , N . We also consider the empty word ∅ as an element of Fd which serves as the unit element for Fd : ∅ · w = w · ∅ = w for all w ∈ Fd . Given a formal power series T (z) as in (1.8) and an operator-tuple δ = (δ1 , . . . , δd ) we may define T (δ) ∈ L(U ⊗ K, Y ⊗ K) by  Tw ⊗ δ w (1.9) T (δ) = w∈F Fd

whenever the series converges, where δ w = δiN · · · δi1 ∈ L(K) if w = iN · · · i1 and δik ∈ L(K) for k = 1, . . . , N. Theorem 5.3 characterizes formal power series T (z) for which a noncommutative von Neumann inequality T (δ) ≤ 1 holds for all operator tuples δ = (δ1 , . . . , δd ) in d a suitable noncommutative Cartan domain  i=1 Ii ⊗ δi  < 1 in terms analogous to those in Theorem 1.1. One can view the result as a noncommutative analogue of the recent work of Ambrozie-Timotin [7], Ball-Bolotnikov [12] and AmbrozieEschmeier [6] on extensions of the so-called Schur-Agler class to more general domains in Cd . In this more general setting there is no analogue of condition (1) in Theorem 1.1. In the classical case, the implication (2) = =⇒ (3) follows in a rather straightforward way as a consequence of the fact that the Schur class can be identified with the space of contractive multipliers on the Hardy space over the unit disk. This type of argument applies in our setting only in special cases (see Remark 5.11); the general case requires a rather involved separation argument of HahnBanach type first used in this context by Agler for the (commutative) polydisk setting (see [1]). The analogue of implication (3) = =⇒ (4) in Theorem 1.1 follows the now standard “lurking isometry” argument which now has appeared in many

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contexts (see [11] for a survey), while the implication (4) =⇒ = (1) is elementary but in our setting requires some care (see Theorem 4.2). This functional calculus of formal power series considered as functions of noncommuting operator-tuples has been used in the context of robust control and the theory of structured singular values (µ-analysis) – see [22, 23, 24, 44, 57]; we explore these connections further in our paper [14]. We mention that results on formal power series (including polynomials in noncommuting indeterminates) closely related to our Theorem 5.3 below have appeared in the recent work of Helton, McCullough and Putinar [39, 40, 41] on representations of polynomials in noncommuting indeterminates as sums of squares as well as in related work of Kalyuzhny˘-Verbovetzki˘ ˘ ˘ı and Vinnikov [43]. This work has motivation from somewhat different connections with system theory. We indicate more precise connections between this work and our Theorem 5.3 in Remark 5.15 below. In a different direction, the paper of Alpay and Kalyuzhny˘-Verbovetzki˘ ˘ ˘ı [5] introduces the notion of a rational, inner formal power series and develops a realization theory for these (see Remarks 5.2 and 5.5 below). The paper is organized as follows. Following the present Introduction, in Section 2 we review the needed material from [13] on structured noncommutative multidimensional linear systems (SNMLSs). In Section 3 we define the adjoint of a SNMLS (having all signal spaces equal to Hilbert spaces). This gives the natural setting for the definition of a conservative SNMLS in Section 4. Section 5 contains the main Theorem 5.3 on the identification of the structured noncommutative Schur-Agler class with the set of formal power series capable of being realized as the transfer function of a conservative SNMLS.

2. Structured noncommutative multidimensional linear systems: basic definitions and properties We present an infinite-dimensional Hilbert-space version of the structured noncommutative multidimensional linear systems (SNMLS) introduced in [13]. As in graph theory, a graph G consists of a set of vertices V = V (G) and edges E = E(G) connecting vertices. We assume throughout that the sets V and E are both finite, i.e., that G is a finite graph. We are interested only in what we call admissible graphs, i.e., a bipartite graph such that each connected component is a complete bipartite graph. This means simply that: ˙ into the set of 1. the set of vertices V has a disjoint partitioning V = S ∪R source vertices S and range vertices R, K K 2. S and R in turn have disjoint partitionings S = ∪˙ k=1 Sk and R = ∪˙ k=1 Rk into nonempty subsets S1 , . . . , SK and R1 , . . . , RK such that, for each sk ∈ Sk and rk ∈ Rk (with the same value of k) there is a unique edge e = esk ,rk connecting sk to rk (s(e) = sk , r(e) = rk ), and 3. every edge of G is of this form.

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If v is a vertex of G (so either v ∈ S or v ∈ R) we denote by [v] the path-connected component p (i.e., the complete bipartite graph p = Gk with set of source vertices equal to Sk and set of range vertices equal to Rk for some k = 1, . . . , K) containing v. Thus, given two distinct vertices v1 , v2 ∈ S ∪ R, there is a path of G connecting v1 to v2 if and only if [v1 ] = [v2 ] and this path has length 2 if both v1 and v2 are either in S or in R and has length 1 otherwise. In case s ∈ S and r ∈ R are such that [s] = [r], we shall use the notation es,r for the unique edge having s as source vertex and r as range vertex: es,r ∈ E determined by s(es,r ) = s, r(es,r ) = r.

(2.1)

Note that es,r is well defined only for s ∈ S and r ∈ R with [s] = [r]. We define a structured noncommutative multidimensional linear system (SNMLS) to be a collection Σ = (G, H, U ) where G is an admissible graph, H = {Hp : p ∈ P } is a collection of (separable) Hilbert spaces (called auxiliary state spaces) indexed by the path-connected components P of the graph G, and where U is a connection matrix (sometimes also called colligation) of the form C D C D C D C D A B [Ar,s ] [Br ] ⊕s∈S H[s] ⊕r∈R H[r] U= = : → (2.2) C D [Cs ] D U Y where U and Y are additional (separable) Hilbert spaces (to be interpreted as the input space and the output space respectively). This definition differs from that in [13] in that here we take the auxiliary state spaces Hp , the input space U and the output space Y to be separable (possibly infinite-dimensional) Hilbert spaces rather than finite-dimensional linear spaces. With any SNMLS we associate an input/state/output linear system with evolution along a free semigroup as follows. We denote by FE the free semigroup generated by the edge set E. An element of FE is then a word w of the form w = eN · · · e1 where each ek is an edge of G for k = 1, . . . , N . We denote the empty word (consisting of no letters) by ∅. The semigroup operation is concatenation: if w = eN · · · e1 and w = eN  · · · e1 , then ww is defined to be ww = eN · · · e1 eN  · · · e1 . Note that the empty word ∅ acts as the identity element for this semigroup. On occasion we shall have use of the notation we−1 for a word w ∈ FE and an edge e ∈ E; by this notation we mean

if w = w e, w −1 we = (2.3) undefined otherwise. with a similar convention for e−1 w. If Σ = (G, H, U ) is an SNMLS, we associate the system equations (with evolution along FE ) ⎧ ⎨ xs(e) (ew) = Σs∈S Ar(e),s xs (w) + Br(e) u(w) xs (ew) = 0 if s = s(e) (2.4) Σ: ⎩ y(w) = Σs∈S Cs xs (w) + Du(w).

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Here the state vector x(w) at position w (for w ∈ FE ) has the form of a column vector x(w) = cols∈S xs (w) with column entries indexed by the source vertices s ∈ S and with column entry xs (w) taking values in the auxiliary state space H[s] (and thus x(w) takes values in the state space ⊕s∈S H[s] ), while u(w) ∈ U denotes the input at position w and y(w) ∈ Y denotes the output at position w. Just as in the classical case, if we specify an initial condition x(∅) ∈ ⊕s∈S H[s] and feed in an input string {u(w)}w∈F FE , then equations (2.4) enables us to recursively compute x(w) for all w ∈ FE \ {∅} and y(w) for all w ∈ FE . The solution of these recursions can be made more explicit as follows. Note first of all that a consequence of the system equations is that x(ew) ∈ Hs(e) := cols∈S [δ s,s(e) H[s(e)] ] for all e ∈ E and w ∈ FE (where δ s,s is the Kronecker delta function). Given x(∅) and {u(w)}w∈F (E), we can solve the system equations (2.4) or (2.7) uniquely for {x(w)}w∈F FE \{∅} and {y(w)}w∈F as follows: FE  xs(eN ) (eN · · · e1 ) = Ar(eN ),s(eN −1 ) Ar(eN −1 ),s(eN −2 ) · · · Ar(e1 ),s xs (∅) s∈S

+

N 

Ar(eN ),s(eN −1 ) · · · Ar(er+1 ),s(er ) Br(er ) u(er−1 · · · e1 )

(2.5)

r=1

where we interpret u(er−1 · · · e1 ) to be u(∅) where r = 1, and xs (eN eN −1 · · · e1 ) = 0 if s = s(eN ). Also,  Cs(eN ) Ar(eN ),s(eN −1 ) Ar(eN −1 ),s(eN −2 ) · · · Ar(e1 ),s xs (∅) y(eN · · · e1 ) = s∈S

+

N 

Cs(eN ) Ar(eN ),s(eN −1 ) · · · Ar(er+1 ),s(er ) Br(er ) u(er−1 · · · e1 )

r=1

+ Du(eN · · · e1 ).

(2.6)

This formula must be interpreted appropriately for special cases. As examples, for the particular cases r = 1 and r = N we have the interpretations Ar(eN ),s(eN −1 ) · · · Ar(er+1 ),s(er ) Br(er ) u(er−1 · · · e1 )|r=1 = Ar(eN ),s(eN −1 ) · · · Ar(e2 ),s(e1 ) Br(e1 ) u(∅), Ar(eN ),s(eN −1 ) · · · Ar(er+1 ),s(er ) Br(er ) u(er−1 · · · e1 )|r=N = Br(eN ) u(eN −1 · · · e1 ). The system equations (2.4) can be written more compactly in operatortheoretic form as  x(ew) = IΣ;e Ax(w) + IΣ;e Bu(w) (2.7) Σ: y(w) = Cx(w) + Du(w)

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where IΣ;e : ⊕r∈R H[r] → ⊕s∈S H[s] is given via matrix entries

IH[s(e)] = IH[r(e)] if s = s(e)and r = r(e), [IIΣ;e ]s,r = 0 otherwise. A consequence of the system equations (2.8) is the identity D D C C colr∈R xs[r] (es[r] ,r w) cols∈S xs (w) =U u(w) y(w)

(2.8)

(2.9)

for any choice of source-vertex cross-section p → sp . Here we say that a map p → sp from the set of path-connected components P of G into the set of source vertices S of G is a source-vertex cross-section if, for each path-connected component p ∈ P , the path-connected component of G containing sp ∈ S is equal to p: sp ∈ S for each p ∈ P and [sp ] = p.

(2.10)

More precisely, the system of equations (2.9) is equivalent to (2.7) in the sense that the function w → (u(w), x(w), y(w)) satisfies (2.7) if and only if the function w → (u(w), x(w), y(w)) satisfies (2.9) for every choice of source-vertex crosssection map p → sp ∈ S (see (2.10)). From the fact that (2.9) holds for any choice of source-vertex cross-section p → sp we deduce that the state vector w → x(w) of any system trajectory w → (u(w), x(w), y(w)) satisfies the compatibility condition xs (es,r w) is independent of s ∈ [r] for each fixed r ∈ R and w ∈ FE ,

(2.11)

as can also be seen directly from the system equations (2.4). Note that IΣ;e is already determined by the first two pieces G and H of Σ = (G, H, U ). On occasion we shall need these objects in situations where we have a graph G and a collection of Hilbert spaces H = {Hp : p ∈ P } without the presence of any particular connection matrix U . In this situation we shall use the in place of IΣ;e . We shall also have occasion notation IG,H;e   to need the operator pencil ZΣ (z) = e∈E IΣ,e ze , also written as ZG,H (z) = e∈E IG,H;e ze when U is absent or suppressed. Also just as in the classical case, it is convenient to introduce “frequencydomain” notation for explicit representation of system trajectories. For any linear space H, we define the formal noncommutative Z-transform of a sequence of Hvalued functions as a formal power series in several noncommuting indeterminates z = (ze : e ∈ E) as follows:   h(w)z w , (2.12) {h(w)}w∈F FE → h(z) = w∈F FE

where z ∅ = 1, z w = zeN zeN −1 · · · ze1 if w = eN eN −1 · · · e1 . Thus 



z w · z w = z ww ,

z w · ze = z we for w, w ∈ FE and e ∈ E.

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On occasion we shall have need of multiplication on the right or left by ze−1 ; we use the convention

−1 if we−1 ∈ FE is defined; z we w −1 (2.13) z ze = 0 if we−1 is undefined. where we use the convention (2.3) for the meaning of we−1 . We use the obvious analogous convention to define ze−1 z w . As derived in [13], application of the formal noncommutative Z-transform to the system equations (2.4) and solving gives a frequency-domain formula for the state and output trajectory: x (z) = (I − ZΣ (z)A)−1 x(∅) + (I − ZΣ (z)A)−1 ZΣ (z)B u(z) u(z) y(z) = C(I − ZΣ (z)A)−1 x(∅) + TΣ (z) where we have set ZΣ (z) =



IΣ;e ze

(2.14) (2.15)

e∈E

and where the formal power series given by TΣ (z) = D + C(I − ZΣ (z)A)−1 ZΣ (z)B = T∅ +

∞ 



(2.16)

Cs(eN ) Ar(eN ),s(eN −1 ) · · · Ar(e2 ),s(e1 ) Br(e1 ) zeN zeN −1 · · · ze2 ze1

N =1 e1 ,...,eN ∈E

(2.17) is the transfer function of the SNMLS Σ. As explained in [13], there are three particular examples worth special mention; we refer to these as (1) noncommutative Fornasini-Marchesini systems, (2) noncommutative Givone-Roesser systems, and (3) noncommutative full-structured multidimensional linear systems. These special cases are defined as follows. Example 2.1. Noncommutative Fornasini-Marchesini systems. We let GF M be the admissible graph with source-vertex set S F M consisting of a single element S F M = {1} and with range-vertex set RF M and set of edges E F M both equal to the finite set {1, . . . , d}, with edge j having source vertex 1 and range vertex j: sF M (j) = 1 and rF M (j) = j for j = 1, . . . , d. Suppose now that Σ = (GF M , H, U F M ) is a SNMLS with structure graph GF M . As GF M has only one path-connected component (P F M = {p1 }, the collection of Hilbert spaces H = {Hp : p ∈ P } collapses to a single Hilbert space H. The connection matrix U F M then has the form ⎡ ⎤ A1 B1 D D ⎢ . C C d .. ⎥ CHD ⊕j=1 H A B ⎢ . . ⎥ =⎢ . UFM = → ⎥: C D U Y ⎣Ad Bd ⎦ C

D

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and the system equations (2.4) have ⎧ x(1w) ⎪ ⎪ ⎪ ⎨ .. . ΣF M : ⎪ ⎪ x(dw) ⎪ ⎩ y(w) or in more compact form  x(jw) ΣF M : y(w) ···

A1 x(w) + B1 u(w) .. .

= =

Ad x(w) + Bd u(w) Cx(w) + Du(w).

(2.18)

(2.19)

3 0 where IH occurs in the jth column.

···

IH

=

= IΣF M ,j Ax(w) + IΣF M ,j Bu(w) = Cx(w) + Du(w)

where we set

2 IΣF M ,j = 0

the form

The transfer function TΣF M (z) then has the form TΣF M (z) = D + C(I − ZΣF M (z)A)−1 ZΣF M (z)B = D + C(I − z1 A1 − · · · − zd Ad )−1 (z1 B1 + · · · + zd Bd ) =D+

d  

CAv Bj z v zj

(2.20)

v∈F Fd j=1

where the structure matrix ZΣF M (z) is given by ZΣF M (z) =

d 

2 IΣF M ,j zj = z1 IH

···

3 zd IH .

j=1

We consider these as noncommutative Fornasini-Marchesini systems (with evolution along the free semigroup generated by {1, . . . , d}) to be a noncommutative analogue of the commutative multidimensional systems (with evolution along an integer lattice rather than a tree or free semigroup) introduced and studied by Fornasini and Marchesini (see, e.g., [34]). Example 2.2. Noncommutative Givone-Roesser systems. We let GGR be the graph with source-vertex set S GR , range vertex set RGR and set of edges E GR all equal to the finite set {1, . . . , d} with edge j having source vertex j and range vertex j: sF M (j) = j and rF M (j) = j for j = 1, . . . , d. Suppose now that Σ = (GGR , H, U GR ) is a SNMLS with structure graph GGR . As GGR has d path-connected components (P GR = {p1 , . . . , pd }), the collection of Hilbert spaces H can be labeled as H = {Hj : j = 1, . . . , d}. The connection matrix U GR then has the form ⎡ ⎤ A11 · · · A1d B1 D ⎢ . D C C d .. .. ⎥ C⊕d H D A B ⎢ . i=1 i → ⊕j=1 Hj . . ⎥ U GR = =⎢ . ⎥: C D U Y ⎣Ad1 · · · Add Bd ⎦ C1

···

Cd

D

Conservative Noncommutative Systems and the system ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ GR Σ : ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

189

equations (2.4) have the form x1 (1w) = A11 x1 (w) + · · · + A1d xd (w) + B1 u(w) .. .. . . xd (dw) = Ad1 x1 (w) + · · · + Add xd (w) + Bd u(w) xi (iw) = 0 if i = i, y(w) = C1 x1 (w) + · · · + Cd xd (w) + Du(w).

or in more compact form  x(jw) GR Σ : y(w)

= IΣGR ,j Ax(w) + IΣGR ,j Bu(w) = Cx(w) + Du(w) ⎡ 0 ⎢ ⎢ ⎢ =⎢ ⎢ ⎢ ⎣

where we set

(2.21)

(2.22)

⎤

⎥ ⎥ ⎥ ⎥ IHj IΣGR ,j ⎥ ⎥ .. ⎦ . 0 where the nonzero entry occurs in the jth diagonal slot. The transfer function TΣGR (z) then has the form ..

.

(2.23) TΣGR (z) = D + C(I − ZΣGR (z)A)−1 ZΣGR (z)B ⎛⎡ ⎤ ⎡ ⎤⎞−1 ⎡ ⎤ IH1 z1 B1 z1 A11 · · · z1 A1d 2 3 ⎜⎢ ⎥ ⎢ .. .. ⎥⎟ ⎢ .. ⎥ .. = D + C1 · · · Cd ⎝⎣ ⎦−⎣ . . . ⎦⎠ ⎣ . ⎦ IHd zd Bd zd Ad1 · · · zd Add ∞   CiN AiN ,iN −1 AiN −1 ,iN −2 · · · Ai2 ,i1 Bi1 ziN ziN −1 · · · zi2 zi1 =D+ N =1 i1 ,...,iN ∈{1,...,d}

(2.24) where the structure matrix ZΣGR (z) is given by ⎡ z1 IH1 d  ⎢ IΣGR ,j zj = ⎣ ZΣGR (z) =

⎤ ..

⎥ ⎦.

.

j=1

zd IHd

We consider these noncommutative Givone-Roesser systems to be a noncommutative analogue of the commutative multidimensional systems introduced and studied by Givone and Roesser (see, e.g., [35, 36]). Example 2.3. Noncommutative full-structured multidimensional systems. We take Gfull to be the complete bipartite graph on source-vertex set S full = {1, . . . , n} and range-vertex set Rfull = {1, . . . , m}. Thus we may label the edge set E full as E full = {(i, j) : i = 1, . . . , n; j = 1, . . . , m} with sfull(i, j) = i,

rfull (i, j) = j.

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We let Fn,m denote the free semigroup generated by the set E full = {1, . . . , n} × {1, . . . , m}. Thus elements of Fn,m are words w of the form (iN , jN )(iN −1 , jN −1 ) · · · (i1 , j1 ) where ik ∈ {1, . . . , n} for all k = 1, . . . , N and jk ∈ {1, . . . , m} for all k = 1, . . . , N . Suppose that Σfull = (Gfull , H, U full) is a SNMLS with structure graph equal to Gfull . As Gfull has only one connected component in this case, the collection of Hilbert spaces H = {Hp : p ∈ P full } collapses to a single Hilbert space denoted as H. The connection matrix U full then has the form ⎡ ⎤ A11 · · · A1n B1 D ⎢ . C C m D .. .. ⎥ C⊕n HD ⊕j=1 H A B ⎢ .. ⎥ full i=1 . . =⎢ = → U ⎥: C D U Y ⎣Am1 · · · Amn Bm ⎦ C1 · · · Cn D and the associated system equations have the form ⎧ ⎪x1 ((1, j) · w) = Aj1 x1 (w) + · · · + Ajn xn (w) + Bj u(w) for j = 1, . . . , m, ⎪ ⎪ ⎪ .. ⎪ ⎪ . ⎨ Σfull :

⎪xn ((n, j) · w) ⎪ ⎪ ⎪ ⎪xi ((i, j) · w) ⎪ ⎩ y(w)

= Aj1 x1 (w) + · · · + Ajn xn (w) + Bj u(w) for j = 1, . . . , m, = 0 if i = i, = C1 x1 (w) + · · · + Cn xn (w) + Du(w). (2.25) Note that, as is consistent with (2.11), xi ((i, j) · w) is independent of i for each fixed j ∈ {1, . . . , m} and w ∈ Fn,m . The transfer function TΣfull then has the form TΣfull (z) = D + C(I − ZΣfull (z)A)−1 ZΣfull (z)B (2.26) 2 3 = D + C1 · · · Cn m ⎛⎡ ⎤ ⎡ m ⎤⎞−1 ⎡ m ⎤ ··· IH j=1 z1j Aj1 j=1 z1j Ajn j=1 z1j Bj ⎜⎢ ⎥ ⎢ ⎥⎟ ⎢ ⎥ .. .. .. .. · ⎝⎣ ⎦−⎣ ⎦⎠ ⎣ ⎦ . m . m . m . IH ··· j=1 znj Aj1 j=1 znj Ajn j=1 znj Bj =D+

∞ 





CiN AjN ,iN −1 AjN −1 ,iN −2 · · · Aj2 ,i1 Bi1

N =1 i1 ,...,iN ∈{1,...,n} j1 ,...,jN ∈{1,...,m}

· ziN ,jN ziN −1 ,jN −1 · · · zi2 ,j2 zi1 ,j1 and where ZΣfull (z) is given by

(2.27)

z1,1 IH ⎢ .. ZΣfull (z) = ⎣ .

···

⎤ z1,m IH .. ⎥ . . ⎦

zn,1 IH

···

zn,m IH

⎡

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3. Adjoint systems It turns out that the adjoint system for a SNMLS Σ has a somewhat different form. Let us say that the collection Σ∗ = (G, H∗ , U∗ ) is a SNMLS of adjoint form if 1. G is an admissible finite graph, 2. H∗ = {H∗p : p ∈ P } is a collection of Hilbert spaces (the auxiliary state spaces for Σ∗ ) indexed by the set P of path-connected components of G, and 3. the connection matrix U∗ for Σ∗ has the form C D C D C D C D A∗ B∗ [A∗s,r ] [B∗s ] ⊕r∈R H∗[r] ⊕H∗[s] U∗ = = : → (3.1) [C∗r ] C∗ D∗ D∗ U∗ Y∗ where U∗ (the input space for Σ∗ ) and Y∗ (the output space for Σ∗ ) are Hilbert spaces. The system equations associated with an SNMLS of adjoint form Σ∗ involve also a choice of source-vertex cross-section p → sp as in (2.10) and are given by   x∗s (w) = A x (e w) + B∗s u∗ (w) r∈R ∗s,r s[r] s[r] ,r (3.2) Σ∗ : y∗ (w) = r∈R C∗r x∗s[r] (es[r] ,r (w) + D∗ u∗ (w). The state vector x∗ (w) = cols∈S x∗s (w) takes values in the state space ⊕s∈S H∗[s] with components x∗s (w) in the auxiliary state space H∗[s] for each s ∈ S and is required to satisfy the compatibility condition x∗s (es,r w) = x∗s (es ,r w) for all s, s ∈ S with [s] = [s ] and for all r ∈ R and w ∈ FE .

(3.3) (3.4)

The adjoint input signal u∗ (w) takes values in U∗ and the adjoint output signal y∗ (w) takes values in Y∗ . Given a positive integer N , suppose that we are given an input signal {u∗ (w)}w : |w|≤N on the finite horizon {w ∈ Fd : |w| ≤ N } along with a finalization of the state {x∗ (w)}w : |w|=N +1 . We can then apply the recursions in (3.2) to compute x∗ (w) and y∗ (w) for all w ∈ Fd with |w| ≤ N . The compatibility condition (3.4) implies that the resulting solution x∗ (w) and y∗ (w) is independent of the choice of source-vertex cross-section p → sp . In general we say that a triple of functions w → (u∗ (w), x∗ (w), y∗ (w)) is a trajectory of the system of adjoint form Σ∗ if x∗ satisfies the compatibility condition (3.4) and (u∗ , x∗ , y∗ ) satisfy the adjoint system equations (3.2) for some (and hence for any) choice of source-vertex cross-section p → sp . Given a SNMLS Σ = (G, H, U ), we define the adjoint system Σ∗ of Σ to be the SNMLS of adjoint form given by Σ∗ = (G, H, U ∗ ).

(3.5)

From the definition (3.2) we see that the system equations associated with Σ∗ therefore have the form   x∗s (w) = A∗ x∗s[r] (es[r] ,r w) + Cs∗ u∗ (w) ∗ r∈R r,s Σ : (3.6) ∗ ∗ y∗ (w) = r∈R Br x∗s[r] (es[r] ,r w) + D u∗ (w).

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where the adjoint state vector x∗ (w) = cols∈S x∗s (w) taking values in ⊕s∈S H[s] , adjoint input signal u∗ (w) taking values in Y and adjoint output signal y∗ (w) taking values in U. The defining condition of the adjoint system is given by the following Proposition. In the following statement, by a local trajectory of the system Σ at the word w we mean a function w → (u(w ), x(w ) = ⊕s∈S xs (w ), y(w )) defined at least for w = w and w = ew for each e ∈ E which satisfies the system equations (2.4) at position w. Similarly, by a local trajectory of Σ∗ at w we mean a function w → (u∗ (w ), x∗ (w ) = ⊕s∈S x∗s (w ), y∗ (w )) defined at least for w = w and w = ew for each e ∈ E which satisfies the compatibility condition (3.4) and the adjoint system equations (3.6) at w. With these notions we avoid the issue of whether a local trajectory (of Σ or Σ∗ ) necessarily extends to a global trajectory. Proposition 3.1. Suppose that we are given a SNMLS Σ = (G, H, U ) with adjoint system Σ∗ = (G, H, U ∗ ). 1. The adjoint pairing relation  xs[r] (es[r] ,r w), x∗s[r] (es[r] ,r w)H[r] + y(w), u∗ (w)Y r∈R

=



xs (w), x∗s (w)H[s] + u(w), y∗ (w)U

(3.7)

s∈S

holds for any trajectory (u, x, y) of Σ and any trajectory (u∗ , x∗ , y∗ ) of Σ∗ . 2. Conversely, if a given function w → (u(w), x(w), y(w)) ∈ U × (⊕s∈S H[s] ) × Y satisfies the adjoint pairing relation (3.7) with respect to every local trajectory (u∗ (w), x∗ (w), y∗ (w)) of Σ∗ at each w ∈ FE , then (u, x, y) is a trajectory of Σ. 3. Conversely, if a given function w → (u∗ (w), x∗ (w), y∗ (w)) ∈ Y × (⊕s∈S H[s] ) × U satisfies the adjoint pairing relation (3.7) with respect to every local trajectory (u(w), x(w), y(w)) of Σ at w for each w ∈ FE , then (u∗ , x∗ , y∗ ) is a trajectory of Σ∗ . Proof. Note that the system equations (2.9) for Σ can be written in vector form as D C D C colr∈R xs[r] (es[r] ,r w) cols∈S xs (w) . (3.8) =U u(w) y(w) Similarly, in vector form, the adjoint system equations (3.6) are C D D C cols∈S x∗s (w) ∗ colr∈R x∗s[r] (es[r] ,r w) =U y∗ (w) u∗ (w)

(3.9)

Conservative Noncommutative Systems and the adjoint pairing relation is %C D& D C colr∈R xs[r] (es[r] ,r w) colr∈R x∗s[r] (es[r] ,r w) , y(w) u∗ (w) (⊕r∈R H[r] )⊕Y %C D C D& cols∈S xs (w) cols∈S x∗s (w) = . , u(w) y∗ (w) (⊕ H )⊕U s∈S

193

(3.10)

[s]

If (u, x, y) is a trajectory of Σ and (u∗ , x∗ , y∗ ) is a trajectory of Σ∗ , then substitution of (3.8) and (3.9) into (3.10) shows that (3.10) holds for (u, x, y) and (u∗ , x∗ , y∗ ) by definition of the adjoint U ∗ of U . More precisely, if (u, x, y) is a trajectory such that (3.7) holds for any local trajectory (u∗ , x∗ , y∗ ) of Σ∗ at w, then we see that D C D& %C colr∈R x∗s[r] (es[r] ,r w) colr∈R xs[r] (es[r] ,r w) , y(w) u∗ (w) (⊕r∈R H[r] )⊕Y D& D %C C x (e w) col cols∈S xs (w) r∈R ∗s[r] s[r] ,r ∗ ,U = . u(w) u∗ (w) (⊕s∈S H[s] )⊕U 4 col 5 r∈R xs[r] (es[r] ,r w) As can be taken to be an arbitrary element of (⊕r∈R H[r] ) ⊕ U y(w) and the source-vertex cross-section p → sp is also arbitrary, it follows that (u, x, y) satisfies (3.8) at w. As the choice of w ∈ FE is arbitrary, we conclude that (u, x, y) is a trajectory of Σ. A similar argument shows that (u∗ , x∗ , y∗ ) is a trajectory of Σ∗ if (u∗ , x∗ , y∗ ) satisfies (3.7) against every local trajectory (u, x, y) of Σ at each w ∈ FE , and Proposition 3.1 now follows.


4. Dissipative and conservative structured multidimensional linear systems In case U is contractive (U  ≤ 1), we say that Σ is a dissipative SNMLS. In this case the trajectories of Σ have the following energy dissipation property:  xs[r] (es[r] ,r w)2 − x(w)2 ≤ u(w)2 − y(w)2 (4.1) r∈R

for every choice of source-vertex cross-section p → sp . We say that the SNMLS is isometric in case the connection matrix U is isometric. In this case the dissipation inequality (4.1) is replaced with the energy balance relation:  xs[r] (es[r] ,r w)2 − x(w)2 = u(w)2 − y(w)2 (4.2) r∈R

for every choice of source-vertex cross-section p → sp . An interesting special case is the case where there is a unique source-vertex cross-section. This happens exactly when each path-connected component p of the admissible graph G contains exactly one source vertex sp ; this occurs, e.g., for the case of noncommutative Fornasini-Marchesini systems (see Example 2.1) and for

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noncommutative Givone-Roesser systems (see Example 2.2). In this case, each edge e has the form es[r] ,r and hence can be indexed more simply by r ∈ R: es[r] ,r → er . Then the property that xs (ew) = 0 if s = s(e) translates to xs (er w) = 0 if s = s[r] . With the use of this fact we see that, when we sum (4.1) over all words w of length at most some N , the left side of the inequality telescopes and we arrive at  2  3 x(w)2 − x(∅)2 ≤ u(w)2 − y(w)2 . (4.3) w : |w|=N +1

w : |w|≤N

This can be rearranged as  y(w)2 ≤



w : |w|≤N

w : |w|≤N

≤



y(w)2 +



x(w)2

w : |w|=N +1

u(w) + x(∅)2 , 2

(4.4)

w : |w|≤N

and hence, letting N → ∞ gives   y(w)2 ≤ u(w)2 + x(∅)2 . w∈F FE

(4.5)

w∈F FE

In particular, if we impose zero initial condition x(∅) = 0 and take formal Ztransform, from the fact that y(z) = TΣ (z) · u (z) (see (2.14)) we arrive at u(z)2L2 (F u(z)2L2 (F (z) ∈ L2 (F FE , U), T TΣ (z) FE ,Y) ≤  FE ,U ) for all u

(4.6)

FE , U) into L2 (F FE , Y) i.e., multiplication by TΣ is a contraction operator from L2 (F in case there is a unique source-vertex cross-section p → sp for G. We shall have further discussion of this point in Remark 5.14 below. Given a SNMLS Σ = (G, H, U ), we say that Σ is a conservative SNMLS if the connection matrix D C D C D C D C [Ar,s ] [Br ] ⊕s∈S H[s] ⊕r∈R H[r] A B = U= : → C D [Cs ] D U Y is unitary. In particular U is isometric, so system trajectories satisfy the energy balance relation (4.2). Just as in the classical case, for a system-theoretic interpretation of the meaning of the adjoint U ∗ of U also being isometric, we need to introduce the adjoint system Σ∗ . Recall the definition of the adjoint Σ∗ of a SNMLS Σ = (G, H, U ) given by (3.5). Theorem 4.1. Suppose that Σ = (G, H, U ) is a SNMLS. Then Σ is conservative (i.e., U is unitary) if and only if either one of the following conditions holds: 1. The function (u, x, y) : FE → U × ⊕s∈S H[s] × Y is a local trajectory of Σ at w if and only if the function (y, x, u) : FE → Y × ⊕s∈S H[s] × U is a local trajectory of Σ∗ at w.

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2. A local trajectory (u, x, y) of Σ at w satisfies the energy balance relation  xs[r] (es[r] ,r w)2 − x(w)2 = u(w)2 − y(w)2 (4.7) r∈R

and any local trajectory (u∗ , x∗ , y∗ ) of Σ∗ at w satisfies the adjoint energy balance relation   x∗s[r] (es[r] ,r w)2 + u∗ (w)2 = x∗s (w)2 + y∗ (w)2 . (4.8) r∈R

s∈S

for all source-vertex cross-sections p → sp . In particular, if Σ = (G, H, U ) is a conservative SNMLS, then 1. (u, x, y) is a trajectory of Σ if and only if (y, x, u) is a trajectory of Σ∗ , 2. any trajectory (u, x, y) of Σ satisfies (4.7), and 3. any trajectory (u∗ , x∗ , y∗ ) of Σ∗ satisfies (4.8). Proof. From the block forms (3.8) and (3.9) of the system equations for Σ and Σ∗ , we see that the equivalence between (u, x, y) being a local trajectory for Σ and (y, x, u) being a local trajectory for Σ∗ is in turn equivalent to U ∗ = U −1 , i.e., to U being unitary. Again from the system equations (3.8) and (3.9), we see that (4.7) holding for all local trajectories just means that U is isometric while (4.8) holding for all local trajectories of Σ∗ just means that U ∗ is isometric. This essentially completes the proof of Theorem 4.1.
A useful property of dissipative (and hence in particular of conservative) SNMLSs is the possibility of interpreting the transfer function as a function acting on tuples of noncommuting contraction operators as we now explain. In general, suppose that  G is an admissible graph and that we are given a formal power v series T (z) = v∈F FE Tv z in noncommuting variables z = (ze : e ∈ E) indexed by the edge set E of the graph G, with coefficients Tv equal to bounded operators acting between Hilbert spaces U and Y. Suppose that we are also given a collection δ = (δe : e ∈ E) of bounded, linear operators (not necessarily commuting) on some separable infinite-dimensional Hilbert space K also indexed by the edge set E of G. We define an operator T (δ) : U ⊗ K → Y ⊗ K by  T (δ) := lim Tv ⊗ δ v N →∞

v∈F FE : |v|≤N

∅

where δ = IK and δ v = δeN · · · δe1 if v = eN · · · e1 .

(4.9)

whenever the limit (say, in the norm or the strong operator topology) exists. In general there is no reason for the limit in (4.9) to exist. However, for the case that T (z) = TΣ (z) is the transfer function of a conservative SNMLS Σ = (G, H, U ), T (δ) always makes sense for a natural class of operator-tuples δ = (δe : e ∈ E). To state the result we first need to agree on some notation. Suppose that Σ is a conservative SNMLS with system structure matrix ZΣ (z) =  I e∈E Σ,e ze as in (2.15). For δ = (δe : e ∈ E) a finite collection of (not necessarily

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commuting) bounded linear operators on some Hilbert space K indexed by the edge set E, define an operator   ZΣ (δ) : ⊕r∈R H[r] ⊗ K → ⊕s∈S H[s] ⊗ K  by ZΣ (δ) = e∈E IΣ,e · δe where IΣ,e · δe is given in terms of its matrix entries

IH[s] ⊗ δe = IH[r] ⊗ δe if s = s(e) and r = r(e), [IIΣ,e · δe ]s,r = (4.10) 0 otherwise. Note that the definition of IΣ,e and of ZΣ (z) uses only the first two pieces G and H of the SNMLS Σ = (G, H, U ). In case Hp is taken to be the complex numbers C for each path-connected component p ∈ P , we denote the associated coefficient matrices IΣ,e and the structure matrix ZΣ (z) simply as IG,e and ZG (z). Thus IG,e : ⊕r∈R C → ⊕s∈S C with matrix entries

1 if s = s(e) and r = r(e), [IIG,e ]s,r = 0 otherwise  and ZG (z) = e∈E IG,e ze . We then define a class BG L(K) of tuples δ = (δe : e ∈ E) of bounded, linear operators on the Hilbert space K (the G-unit ball of L(K)nE ) (where nE denotes the number of edges in the graph G) by BG L(K) = {δ = (δe : e ∈ E) : δe ∈ L(K) for e ∈ E and ZG (δ) < 1}.

(4.11)

It is easy to see that ZG (δ) = ZΣ (δ) whenever Σ = (G, H, U ) is a SNMLS with structure graph G; thus ZΣ (δ) < 1 for all δ ∈ BG L(K). Theorem 4.2. Suppose that T (z) = D + C(I − ZΣ (z)A)−1 ZΣ (z)B is the transfer function of a dissipative SNMLS Σ = (G, H, U ) and that K is some other separable Hilbert space. Then for any collection δ = (δe : e ∈ E) of operators in BG L(K), T (δ) as defined in (4.9) is a well-defined contraction operator (with the limit of the partial sums in (4.9) existing in the operator-norm topology) from U ⊗ K to Y ⊗ K (T (δ) ≤ 1), and can alternatively be expressed as T (δ) = (D ⊗ IK ) + (C ⊗ IK ) (I − ZΣ (δ)(A ⊗ IK ))−1 ZΣ (δ)(B ⊗ IK ). Proof. A general fact is that, if C  A  U = C

(4.12)

D C D C  D B H H : → D U Y

is contractive and ∆ : H → H is a strict contraction, then the upper feedback connection Fu [U  , ∆] : U  → Y  defined implicitly by Fu [U  , ∆] : u → y  if there exist h ∈ H and h ∈ H C  D C D C  D A B  h h so that =  and ∆h = h C  D  u y

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is well defined, moreover is contractive (F Fu [U  , ∆] ≤ 1), and is given explicitly by the linear-fractional formula Fu [U  , ∆] = D + C  (I − ∆A )−1 ∆B  .

(4.13)

This fact can be found in any of a number of places where linear-fractional transformations are discussed, e.g., in [57] where there is a comprehensive treatment for the control-theory context, or in Section 3 of [7] where there is a concise summary of what we are using here. Now suppose that Σ = (G, H, U ) is a dissipative SNMLS and suppose that δ = (δe : e ∈ E) is an operator-tuple in BG L(K). We shall use a different font δ s,s for the Kronecker delta function

1 if s = s , δ s,s = (4.14) 0 otherwise for which we shall have use on occasion in the sequel. We apply the linear-fractional construction (4.13) to the case C

D C D C D A ⊗ IK B ⊗ IK (⊕s∈S H[s] ) ⊗ K (⊕r∈R H[r] ) ⊗ K U = : → , C ⊗ IK D ⊗ IK U ⊗K Y ⊗K   ∆ = ZΣ (δ) : ⊕r∈R H[r] ⊗ K → ⊕s∈S H[s] ⊗ K. 

Note that U  is then contractive since by assumption U is contractive and that ∆ < 1 since δ ∈ BG L(K). Hence it follows that Fu [U  , ZΣ (δ)] = (D ⊗ IK ) + (C ⊗ IK ) (I − ZΣ (δ)(A ⊗ IK ))

−1

ZΣ (δ)(B ⊗ IK )

is a well-defined contraction operator from U ⊗ K into Y ⊗ K. It remains to show that Fu [U  , ZΣ (δ)] = TΣ (δ). Verification of this identity draws upon repeated use of the product rule for tensor products (A ⊗ B)(C ⊗ D) = (AC) ⊗ (BD) as we now show. Since ZΣ (δ)(A ⊗ IK ) ≤ ZΣ (δ)A < 1, it follows that the inverse of I − ZΣ (δ)(A ⊗ IK ) is given by the Neumann expansion (I − ZΣ (δ)(A ⊗ IK ))

−1

=

∞ 

N

[ZΣ (δ)(A ⊗ IK )] .

N =0

From this we see that the (s, s ) matrix entry of (I − ZΣ (δ)(A ⊗ IK ))−1 : ⊕s∈S H[s] → ⊕s∈S H[s]

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is given by [(I − ZΣ (δ)(A ⊗ IK ))−1 ]s,s = δ s,s IH[s] +

∞ 



N =1 eN ,...,e1 ∈E : s(eN )=s

(Ar(eN ),s(eN −1 ) ⊗ δeN ) · · · (Ar(e2 ),s(e1 ) ⊗ δe2 )(Ar(e1 ),s ⊗ δe1 ) = δ s,s IH[s] +

∞ 



(Ar(eN ),s(eN −1 ) · · · Ar(e2 ),s(e1 ) Ar(e1 ),s )⊗

N =1 eN ,...,e1 ∈E : s(eN )=s

⊗ (δeN · · · δe2 δe1 )

(4.15)

 Note next that C ⊗ IK : ⊕s∈S H[s] ⊗ K → Y ⊗ K has row matrix representation (4.16) C ⊗ IK = rows∈S [Cs ⊗ IK ]  while ZΣ (δ)(B ⊗ IK ) : U ⊗ K → ⊕s ∈S H[s ] ⊗ K has column matrix representation ⎡



ZΣ (δ)(B ⊗ IK ) = cols ∈S ⎣

⎤ Br(e) ⊗ δe ⎦ .

(4.17)

e : s(e)=s

Using (4.15), (4.16) and (4.17), we then compute −1

(C ⊗ IK ) (I − ZΣ (δ)(A ⊗ IK )) ZΣ (δ)(B ⊗ IK )   3 2 = (Cs ⊗ IK ) (I − ZΣ (δ)(A ⊗ IK ))−1 s,s (Br(e) ⊗ δe ) s,s ∈S e∈E : s(e)=s

= X1 + X2 . where we have set   (Cs ⊗ IK )(Br(e) ⊗ δe ) X1 =

(4.18)

s∈S e : s(e)=s

X2 =





∞ 



(Cs ⊗ IK )·

s,s ∈S e : s(e)=s N =1 eN ,...,e1 ∈E : s(eN )=s

· (Ar(eN ),s(eN −1 ) · · · Ar(e1 ),s ⊗ δeN · · · δe2 δe1 )(Br(e) ⊗ δe ). The first term X1 simplifies to  X1 =



(4.19)

(Cs Br(e) ) ⊗ δe

s∈S e∈E : s(e)=s

=



e∈E

T e ⊗ δe

(4.20)

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while the second term X2 can be simplified to X2 =





∞ 



(Cs Ar(eN ),s(eN −1 ) · · ·

s,s ∈S e : s(e)=s N =1 eN ,...,e1 ∈E : s(eN )=s

=



· · · Ar(e2 ),s(e1 ) Ar(e1 ),s Br(e) ) ⊗ (δeN · · · δe2 δe1 δe ) Tv ⊗ δ v .

(4.21)

v∈E : |v|≥2

Combining (4.20) and (4.21) along with the identity T∅ = D immediately gives us the identity (4.12) as wanted. This completes the proof of Theorem 4.2.


5. Conservative SNMLS-realization of formal power series in the class SAG (U, Y) Let G be a fixed admissible graph with source-vertex set S, range-vertex set R and 4.2 suggests that we consider the class of all formal power edge set E. Theorem  v series T (z) = v∈F T FE v z having the property in the conclusion of Theorem 4.2. We view this class as a noncommutative analogue of the Schur-Agler class studied in a series of papers (see, e.g., [1, 17, 15, 6, 7, 12]). Definition 5.1. We say that T (z) is in the noncommutative Schur-Agler class Hilbert space K and SAG (U, Y) (for a given admissible graph G) if, for each  v each δ = (δe : e ∈ E) ∈ BG L(K), the limit T (δ) = limN →∞ v∈F Fd : |v|≤N Tv ⊗ δ exists (in the operator-norm topology) and defines an operator T (δ) : U ⊗ K → Y ⊗ K which is contractive T (δ) ≤ 1.

(5.1)

Remark 5.2. Alpay and Kalyuzhny˘-Verbovetzki˘ ˘ ˘ı in [5] have shown that a given formal power series  T (z) = Tv z v ∈ L(U, Y)z v∈F FE

belongs to the noncommutative Schur-Agler class SAG (U, Y) if and only if (5.1) holds for each δ = (δe : e ∈ E) ∈ BG L(CN ) for each finite N = 1, 2, 3, . . . . The proof there is done explicitly only for the case where each component of G consists of a single source vertex and a single range vertex (the Givone-Roesser case); we expect that this result continues to hold for the case of a general admissible graph G. Our next goal is a converse to Theorem 4.2 (see Theorem 5.3 below). For the statement we shall need some additional notation and terminology. We let z  = (ze : e ∈ E) be a second system of noncommuting indeterminates; while ze ze = ze ze and ze ze  = ze  ze unless e = e , we will use the convention that

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ze ze  = ze  ze for all e, e ∈ E. We also shall need the convention (2.13) to give meaning to expressions of the form 

For H(z) =





−1

−1 

ze−1 z v z v ze−1 = (z v ze−1 ) · (ze−1 z v ) = z ve z e

v

.

Hv z v , we will use the convention that  ∗     ∗ v H(z) = Hv z := Hv∗ z v = Hv∗ z v . v∈F FE

v∈F FE

v∈F FE

v∈F FE

 v v  In general let us say that a formal power series K(z, z ) = v,v ∈F FE [K]v,v  z z with coefficients [K]v,v equal to operators on a Hilbert space X (so K(z, z ) ∈ L(X )z, z  ) is positive-definite provided that  [K]v,v yv , yv X ≥ 0 (5.2) v,v  ∈F FE

for all choices of yv ∈ X with yv = 0 for all but finitely many v ∈ FE . By the standard results concerning reproducing kernel Hilbert spaces ([9]), it is known that condition (5.2) is equivalent to the existence of an auxiliary Hilbert space H and operators Hv ∈ L(H , X ) for each v ∈ FE so that [K]v,v = Hv Hv∗ . Equivalently we therefore have: K(z, z ) ∈ L(X )z, z   is positive-definite if and only if there exists an auxiliary Hilbert space H and a formal power series H(z) ∈ L(H , X )z so that K(z, z ) = H(z)H(z  )∗ . We shall be particularly interested in this concept for the case where X = ⊕s∈S Y. We therefore consider a formal power series K(z, z ) of the form K(z, z  ) = [Ks,s (z, z  )]s,s ∈S ∈ L(⊕s∈S Y)z, z  . Such a K(z, z ) therefore is positive-definite if   [Ks,s ]v,v ys ,v , ys,v Y ≥ 0

(5.3)

s,s ∈S v,v  ∈F FE

for all choices of ys,v ∈ Y for s ∈ S and v ∈ FE with ys,v = 0 for all but finitely many such s, v, or equivalently, if and only if there exist an auxiliary Hilbert space H and formal power series Hs (z) ∈ L(H , Y) for each s ∈ S so that Ks,s (z, z  ) = Hs (z)Hs (z  )∗ .  v Theorem 5.3. Let T (z) = v∈F FE Tv z be a formal power series in noncommuting indeterminates z = (ze : e ∈ E) indexed by the edge set E of the admissible graph G with coefficients Tv ∈ L(U, Y) for two Hilbert spaces U and Y. Then the following conditions are equivalent: 1. T (z) is in the noncommutative Schur-Agler class SAG (U, Y). 2. There exists a positive-definite formal power series K(z, z ) = [Ks,s (z, z  )]s,s ∈S ∈ L(⊕s∈S Y)z, z  

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so that IY − T (z)T (z )∗   = Ks,s (z, z  ) − s∈S

r∈R

 s,s ∈S :

[s]=[s ]=[r]

ze s ,r Ks,s (z, z  )zes,r .

(5.4)

3. There exists a collection of Hilbert spaces H = {Hp : p ∈ P } (where P is the set of path-connected  components of the admissible graph G) and a formal v  power series H(z) = v∈F FE Hv z with coefficients Hv ∈ L(⊕s∈S H[s] , Y) so that we have the noncommutative Agler decomposition I − T (z)T (z )∗ = H(z) (I − ZG,H (z)ZG,H (z  )∗ ) H(z  )∗ (5.5)  where we have set ZG,H (z) = e∈E IG,H ;e ze with coefficients IG,H ;e equal   to operators acting from ⊕r∈R H[r] to ⊕s∈S H[s] determined from matrix entries [IIG,H ;e ]s,r given by

IH[s] = IH[r] if s = s(e) and r = r(e), (5.6) [IIG,H ;e ]s,r = 0 otherwise. 4. There is a conservative SNMLS Σ = (G, H, U ) with structure graph equal to the given admissible graph G so that T (z) = TΣ (z), i.e., so that T (z) = D + C(I − ZΣ (z)A)−1 ZΣ (z)B 5 4 5 4 ⊕r∈R H[r] ⊕s∈S H[s] B]: → is unitary and where ZΣ (z) = D Y U

A where U = [ C  e∈E IΣ,e ze with IΣ,e as in (2.8).

Remark 5.4. We give the name Agler decomposition to an identity of the form (5.4) or (5.5) since representations of this type to our knowledge originate in the work of Agler (see [1]) in the context of the commutative polydisk. Remark 5.5. We note that the paper [5] of Alpay and Kalyuzhny˘-Verbovetzki˘ ˘ gives a uniqueness result for conservative realizations of rational inner formal power series in the Givone-Roesser case. We leave a systematic development of the uniqueness theory for realizations as in part (4) of Theorem 5.3 to another occasion. Proof. The proof breaks up into several implications which need to be shown: (2) ⇐⇒ (3): Suppose that K(z, z  ) = [Ks,s (z, z  )]s,s ∈S ∈ L(⊕s∈S Y)z, z   is positive definite. Then, by the remarks preceding the statement of the Theorem, Ks,s (z, z  ) has a factorization Ks,s (z, z  ) = Hs (z)Hs (z  )∗  for a formal power series Hs (z) ∈ L(H[s] , Y) for s ∈ S.

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 Then, if we set H(z) = rows∈S Hs (z) ∈ L(⊕s∈S H[s] , Y)z, then (5.4) assumes the form

I − T (z)T (z )∗   = Hs (z)Hs (z  )∗ − s∈S



r∈R s,s ∈S : [s]=[s ]=[r]

Hs (z) · (1 − zes,r ze s ,r )IIH · Hs (z  )∗

 = H(z) I⊕s∈S H[s] − ZG,H (z)ZG,H (z  )∗ H(z  )∗ . from which (5.5) follows. Conversely, if  H(z) = rows∈S Hs (z) ∈ L(⊕s∈S H[s] , Y)z

is as in (5.5), we may embed each Hp into a common Hilbert space H and without loss of generality assume that Hp = H for each p ∈ P . We then set Ks,s (z, z  ) = Hs (z)Hs (z  )∗ ∈ L(Y)z, z   and K(z, z  ) = [Ks,s (z, z  )]s,s ∈S ∈ L(⊕s∈S Y)z, z  . Then by the factored form of K(z, z ) = (cols∈S Hs (z)) (cols ∈S Hs (z  ))

∗

we see that K(z, z  ) is positive-definite. Reversal of the steps above then shows that K(z, z ) satisfies (5.4). In this way we see that (2) is equivalent to (3) in Theorem 5.3. (4) =⇒ = (1): Since any conservative system is also dissipative, this follows from Theorem 4.2. (1) =⇒ = (2) or (3): As a first case we assume that dim Y < ∞. Let X denote the  v v  linear space L(Y)z, z   of all formal power series ϕ(z, z  ) = v,v ∈F FE ϕv,v  z z in the sets of noncommuting indeterminates z = (ze : e ∈ E) and z  = (ze : e ∈ E) (but where ze ze  = ze  ze for each e, e ∈ E) with coefficients ϕv,v in the space of bounded linear operators L(Y) on the Hilbert space Y. We define a sequence of increasing seminorms  · N on L(Y)z, z   according to the rule ϕ(z, z  )N =

sup v,v  ∈F FE : |v|,|v  |≤N

ϕv,v .

(5.7)

Then X is a locally convex topological vector space in the topology induced by these seminorms and this Let C be the set of all formal  topology is vmetrizable. v   power series ϕ(z, z  ) = v,v ∈F ϕ z z in L(Y)z, z   such that FE v,v ϕ(z, z  ) = H(z)(I − ZG,H (z)ZG,H (z  )∗ )H(z  )∗

(5.8)

for some collection of Hilbert spaces H = {Hp : p ∈ P } indexed by the pathconnected components P of G and for some formal power series  Hv z v H(z) = v∈F FE

 with coefficients Hv ∈ where ZG,H (z) = e∈E IG,H ;e ze is defined as in (5.6). From the equivalence (2) ⇐⇒ (3), we see that an equivalent  L(⊕s∈S H[s] , Y),

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condition for membership of ϕ in C is the existence of a positive-definite formal power series K(z, z ) = [Ks,s (z, z  )]s,s ∈S ∈ L(⊕s∈S Y)z, z   so that    ϕ(z, z  ) = Ks,s (z, z  ) − ze s ,r Ks,s (z, z  )zes,r . (5.9) r∈R s,s ∈S : [s]=[s ]=[r]

s∈S

When working with the decomposition (5.8), we may assume without loss of generality that Hp is a fixed separable infinite-dimensional Hilbert space independent of the choice of ϕ ∈ C and of the particular representation (5.8) for a given ϕ ∈ C. It is easily checked that C is closed under sums and multiplication by nonnegative scalars, i.e., that C is a cone in X . We need to establish a few preliminary facts concerning C. Lemma 5.6. Any positive-definite formal power series ϕ(z, z  ) in L(Y)z, z   is also in C. Proof. As ϕ is a positive kernel, we know that we can factor ϕ as ϕ(z, z  ) = H(z)H(z  )∗ for some H(z) ∈ L(K, Y)z for some auxiliary Hilbert space K. We must produce  , Y)z so that a formal power series H  (z) ∈ L(⊕s∈S H[s] H(z)H(z  )∗ = H  (z)[II⊕s∈S H[s] − ZG,H (z)ZG,H (z  )∗ ]H  (z  )∗ . Let s0 ∈ S be any fixed choice of particular source vertex. Without loss of generality we may assume H is presented in the form H = 2 (F FE0 , K) 

We take H (z) ∈

where

 L(⊕s∈S H[s] , Y)z

E0 = {e ∈ E : s(e) = s0 }.

to be of the form

 , K)z H  (z) = H(z)K  (z) where K  (z) = rows∈S Ks (z) with Ks (z) ∈ L(H[s]

where Ks (z) is given by



Ks (z)

=

v rowv∈F FE0 z IK 0

if s = s0 , otherwise.

Then we check

4 5 H  (z) I⊕s∈S H[s] − ZG,H (z)ZG,H (z  )∗ H  (z  )∗ A  B  = H(z)Ks 0 (z) 1− ze ze IH Ks 0 (z  )∗ H(z  )∗ ⎡⎛ = H(z) ⎣⎝

e∈E0



z v z v IK −

v∈F F E0



⎞

⎤

z v z v ⎠ IK ⎦ H(z  )∗

v∈F FE0 \{∅}

 ∗

= H(z)H(z )

as wanted, and Lemma 5.6 follows.




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J.A. Ball, G. Groenewald and T. Malakorn

We shall need to approximate the cone C by the cone Cε (where ε > 0) defined as the set of all ϕ ∈ L(Y)z, z   having a representation  ϕ(z, z  ) = H(z) I − (1 + ε)2 ZG,H (z)ZG,H (z  )∗ H(z  )∗  + γe (z)(1 − ε2 ze ze )γe (z  )∗ (5.10) e∈E  , Y)z and some γe (z) ∈ L(H , Y)z for e ∈ E. for some H(z) ∈ L(⊕s∈S H[s] Equivalently, just as in the proof of (2) ⇐⇒ (3) (Step 1 above), we see that, in terms of positive-definite formal power series, Cε can be defined as the set of all ϕ ∈ L(Y)z, z   having a representation    ϕ(z, z  ) = Ks,s (z, z  ) − (1 + ε)2 ze s ,r Ks,s (z, z  )zes,r s∈S

+





Γe (z, z ) − ε

2

e∈E

r∈R s,s ∈S : [s]=[s ]=[r] ze Γe (z, z  )ze e∈E



(5.11)

for some positive-definite formal power series K(z, z ) = [Ks,s (z, z  )]s,s ∈S ∈ L(⊕s∈S Y)z, z   and some positive-definite formal power series Γe (z, z  ) in L(Y)z, z   for each e ∈ E. Lemma 5.7. Assume that ϕ ∈ L(Y)z, z   is in the cone Cε for all ε > 0 sufficiently small. Then ϕ ∈ C, i.e., ϕ has a representation (5.8) or equivalently (5.9). Proof. The assumption is that, for all ε > 0 sufficiently small, there is a positiveKε,s,s (z, z  )]s,s ∈S in L(⊕s∈S Y)z, z   definite formal power series Kε (z, z  ) = [K and a positive-definite formal power series Γε,e (z, z  ) in L(Y)z, z   so that (5.11) holds (with K (z, z  ) in place of K(z, z  ) and Γε,e in place of Γe ) for each e ∈ E. In particular for the (∅, ∅)-coefficient we get   [K Kε,s,s ]∅,∅ + [Γε,e ]∅,∅ . ϕ∅,∅ = s∈S

e∈E

Hence [K Kε,s,s ]∅,∅ and [Γε,e ]∅,∅ are all uniformly bounded as tends to 0. By using Kε,s,s ]∅,∅  is uniformly bounded the positive-definiteness of Kε (z, z  ), we see that [K  as ε tends to zero for all s, s ∈ S as well. More generally, computation of the (v, v  ) coefficient of ϕ from (5.11) yields    −1 ϕv,v = [K Kε,s,s ]v,v − (1 + ε)2 [K Kε,s,s ]ve−1  s,r ,e  v s∈S

+

 e∈E

r∈R s,s ∈S : [s]=[s ]=[r]

[Γε,e ]v,v − ε2



[Γε,e ]ve−1 ,e−1 v .

s ,r

(5.12)

e∈E

Here we are using (2.3) to define words of the form ve−1 or e−1 v with the convention that the coefficient is taken to be equal to zero if any of its indices is an undefined word. Inductively assume that there is a uniform bound on [K Kε,s,s ]w,w  for all words w, w ∈ FE having length at most N . From (5.12) we can then see that [K Kε,s,s ]v,v  is uniformly bounded for all v ∈ FE with |v| = N + 1. Using the

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205

positive-definiteness of K(z, z  ), we then see that this leads to a uniform bound for [K Kε,s,s ]v,v  for all v, v  ∈ FE of length at most N + 1 as ε tends to zero. A similar inductive argument gives that [Γε,e ]v,v  is uniformly bounded as ε tends to zero for all words v, v  with length |v|, |v  | at most some N < ∞. Since we are assuming that Y is finite-dimensional, it follows that bounded subsets of L(Y) are precompact in the operator-norm topology. By this fact combined with a Cantor diagonalization procedure, there exists a sequence of numbers εn > 0 tending to zero such that the limits lim [K Kεn ,s,s ]v,v = [Ks,s ]v,v ,

n→∞

lim [Γεn ,e ]v,v = [Γe ]v,v

n→∞

all exist in the operator-norm topology of L(Y). We then take limits in (5.12) to deduce that     −1 ϕv,v = [Ks,s ]v,v − [Ks,s ]ve−1 [Γe ]v,v (5.13)  + s,r ,e  v and hence ϕ(z, z  ) =



s ,r

r∈R s,s ∈S : [s]=[s ]=[r]

s∈S

Ks,s (z, z  ) −





r∈R s,s ∈S : [s]=[s ]=[r]

s∈S

with Ks,s (z, z  ) and Γe (z, z  ) given by   Ks,s (z, z  ) = [Ks,s ]v,v z v z v ,

e∈E

ze s ,r Ks,s (z, z  )zes,r +



Γe (z, z  )

e∈E

(5.14) Γe (z, z  ) =

v,v  ∈F FE





[Γe ]v,v z v z v .

v,v  ∈F FE

Kεn ,s,s (z, z  )]s,s ∈S and Γεn ,e (z, z  ) are positive-definite for each As Kεn (z, z  ) = [K fixed n, we know that    [K Kεn ,s,s ]v,v ys ,v , ys,v Y ≥ 0, [Γεn ,e ]v,v gv , gv Y ≥ 0 s,s ∈S v,v  ∈F FE

v,v  ∈F FE

(5.15) for all finitely supported Y-valued functions (s, v) →  ys,v and s → gv . We may then take the limits as n → ∞ in (5.15) to get    [Ks,s ]v,v ys ,v , ys,v Y ≥ 0, [Γe ]v,v gv , gv Y ≥ 0 (5.16) s,s ∈S v,v  ∈F FE

v,v  ∈F FE 

from which we see that K(z, z ) = [Ks,s (z, z  )]s,s ∈S and Γe (z, z  ) for e ∈ E are positive-definite formal power series as well. By Lemma 5.6, for each e ∈ E the formal power series Γe (z, z  ) is therefore in the cone C. As the difference in the first two terms on the right-hand side of (5.14) is clearly in C by the characterization (5.9) for C and C is closed under addition, it follows that ϕ ∈ C as asserted. Lemma 5.7 now follows.
Lemma 5.8. If ϕ(z, z  ) ∈ L(Y)z, z   is a positive-definite formal power series and if ε > 0, then: 1. ϕ ∈ Cε , and 2. for each e ∈ E, the kernel ϕ(z,

z  ) := ϕ(z, z  ) − ε2 ze ϕ(z, z  )ze is also in Cε .

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Proof. As ϕ(z, z  ) is positive-definite, we have a factorization ϕ(z, z  ) = H(z)H(z  )∗ for some H(z) ∈ L(K, Y)z. To show that ϕ(z, z  ) ∈ Cε it suffices to produce a representation (5.10) for ϕ with γε,e (z) = 0 for each e ∈ E. As in the proof of Lemma 5.6, to produce a representation of this latter form it suffices to produce  , Y)z with a formal power series H  (z) ∈ L(⊕s∈S H[s] ϕ(z, z  ) = H(z)H(z  )∗ = H  (z)(1 − (1 + ε)2 ZG,H (z)ZG,H (z  )∗ )H  (z  )∗ . (5.17) FE0 , K) where E0 = For this purpose we assume that H is presented as H = 2 (F {e ∈ E : s(e) = s0 } where s0 is some fixed source vertex s0 ∈ S. We then take H  (z) to be of the form H  (z) = H(z)K  (z) where  , K)z K  (z) = rows∈S Ks (z) ∈ L(⊕s∈S H[s]

is given by

Ks (z)

=

|v| v rowv∈F FE0 (1 + ε) z IK 0

if s = s0 , otherwise.

Then a direct computation as in the proof of Lemma 5.6 gives H  (z)(II⊕s∈S H[s] − (1 + ε)2 ZG,H (z)ZG,H (z  )∗ )H  (z  )∗ A  B   2  = H(z)Ks0 (z) 1 − (1 + ε) ze ze IH Ks 0 (z  )∗ H(z  )∗ ⎡⎛ = H(z) ⎣⎝

e∈E0



(1 + ε)2|v| z v z v IK −

v∈F F E0  ∗



⎞

⎤

(1 + ε)2|v| z v z v ⎠ IK ⎦ H(z  )∗

v∈F FE0 \{∅} 

= H(z)H(z ) = ϕ(z, z ), as wanted, and part (1) of Lemma 5.8 follows. For the second assertion, use the characterization (5.11) for membership in Cε with Ks,s (z, z  ) = 0 for all s, s ∈ S and with Γe (z, z  ) = 0 for e = e and Γe (z, z  ) = ϕ(z, z  ).
Lemma 5.9. For each ε > 0, the cone Cε is closed as a subspace of X = L(Y)z, z   with the locally convex topology induced by the sequence of seminorms  · N given by (5.7) for N = 1, 2, . . . . Proof. Suppose that {ϕn }n=1,2,... is a sequence of elements of Cε converging to ϕ ∈ X in the locally convex topology of X . By the characterization (5.11) we have the existence of positive-definite formal power series Kn;s,s (z, z  )]s,s ∈S ∈ L(⊕s∈S Y)z, z  , Γn,e (z, z  ) ∈ L(Y)z, z   Kn (z, z  ) = [K

Conservative Noncommutative Systems so that the representation   ϕn (z, z  ) = Kn;s,s (z, z  ) − (1 + ε)2 s∈S



+



r∈R s,s ∈S : [s]=[s ]=[r]



Γn,e (z, z  ) − ε2

e∈E

207

ze s ,r Kn;s,s (z, z  )zes,r

ze Γn,e (z, z  )ze

(5.18)

e∈E

holds for each n = 1, 2, . . . . In terms of coefficients we then have    −1 [ϕn ]v,v = [K Kn;s,s ]v,v − (1 + ε)2 [K Kn;s,s ]ve−1 s,r ,e  s∈S

+



r∈R s,s ∈S : [s]=[s ]=[r]

[Γn,e ]v,v − ε2

e∈E



s ,r

[Γn,e ]ve−1 ,e−1 v .

v

(5.19)

e∈E

By assumption [ϕn ]v,v converges in the operator-norm of L(Y) to [ϕ]v,v as n → ∞. An inductive argument on the length of words combined with the positivedefiniteness of Kn (z, z  ) and Γn,e (z, z  ) as in the proof of Lemma 5.7 can now be used to show that [K Kn;s,s ]v,v  and [Γn,e ]v,v  remain uniformly bounded as n → ∞ for each v, v  ∈ FE and e ∈ E. Since we are assuming that dim Y < ∞, a compactness argument together with a Cantor diagonalization argument (as in the proof of Lemma 5.7) can be used to show that there exists a subsequence n1 < n2 < n3 < . . . such that the limits lim [K Knk ;s,s ]v,v = [Ks,s ]v,v ,

k→∞

lim [Γnk ,e ]v,v = [Γe ]v,v

k→∞

all exist in L(Y)-norm for each s, s ∈ S, e ∈ E, and v, v  ∈ FE . We may then take limits in (5.19) to conclude that    −1 [Ks,s ]v,v − (1 + ε)2 [Ks,s ]ve−1 [ϕ]v,v =  s,r ,e  v s∈S

+



r∈R s,s ∈S : [s]=[s ]=[r]

[Γe ]v,v − ε2

e∈E



[Γe ]ve−1 ,e−1 v . 

K(z, z  ) = [Ks,s (z, z  )]s,s ∈S with Ks,s (z, z  ) = Γe (z, z ) =



(5.20)

e∈E

If we then set



s ,r



[Ks,s (z, z  )]v,v z v z v ,

v,v  ∈F FE v v 

[Γe ]v,v z z

,

v,v  ∈F FE

we conclude that   ϕ(z, z  ) = Ks,s (z, z  ) − (1 + ε)2 s∈S

+

 e∈E



r∈R s,s ∈S : [s]=[s ]=[r]

Γe (z, z  ) − ε2



ze Γe (z, z  )ze .

ze s ,r Ks,s (z, z  )zes,r (5.21)

e∈E

Furthermore, as Ks,s (z, z  ) is the coefficientwise limit of Kn,s,s (z, z  ) where each [K Kn;s,s ]s,s ∈S is positive definite and Γe (z, z  ) is the coefficientwise limit of

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Γn,e (z, z  ) which is positive definite, it follows as in the proof of Lemma 5.7 that K(z, z ) and Γe (z, z  ) for each e ∈ E are positive definite. The identity (5.21) then shows that ϕ(z, z  ) satisfies the criterion (5.11) for membership in Cε as wanted, and Lemma 5.9 follows.
We are now ready to commence the proof of (1) =⇒ = (2) in Theorem 5.3 for the case where dim Y < ∞. Suppose that we are given a formal power series T (z) which is in the Schur-Agler class SAG (U, Y). The issue is to show that ϕT (z, z  ) := IY − T (z)T (z )∗ is in C. By Lemma 5.7, it suffices to show that ϕT is in Cε for all ε > 0 small enough. Recall the notation X for the topological linear space L(Y)z, z   with the locally convex topology of norm-convergence of power-series coefficients. By the Hahn-Banach separation principle (apply the contrapositive version of part (b) of Theorem 3.4 in [54] with X = X , A = {ϕT }, and B = Cε ), it suffices to show: for fixed ε > 0 and for any continuous linear functional L : X → C such that L(ϕ) ≥ 0 for all ϕ ∈ Cε , it follows that L(ϕT ) ≥ 0. Here  denotes “real part”. Fix ε > 0 and let L be any continuous linear functional on X with L|Cε ≥ 0. Define L1 : X → C by 1 L(ϕ) + L(ϕ) ˘ (5.22) L1 (ϕ) = 2 where we have set ϕ(z, ˘ w) = ϕ(w, z)∗ . Note that L1 (ϕ) = L(ϕ) in case ϕ ˘ = ϕ. We define a sesquilinear form ·, ·L on the space H0 := L(Y, C)z   according to the formula f, gL = L1 (g(z)∗ f (z  )).

(5.23) 

 ∗

Note that any formal power series ϕ of the form ϕ(z, z ) = f (z)f (z ) has the property that ϕ˘ = ϕ; by part (1) of Lemma 5.8, any such ϕ is in Cε . We conclude that f, f L = L(f (z)∗ f (z  )) ≥ 0 for all f ∈ H0 . We may thus identify elements of zero norm and then take a completion in the L-norm to get a Hilbert space HL . We next seek to define operators δe for each e ∈ E on HL so that δe∗ is given by δe∗ : f (z) → ze f (z) for f ∈ H0 . (5.24) ∗ 2   By part (2) of Lemma 5.8 we know that the kernel f (z) (1 − ε ze ze )f (z ) belongs to Cε , and hence  f 2HL − ε2 δe∗ f 2HL = L f (z)∗ (1 − ε2 ze ze )f (z  ) ≥ 0 for f ∈ H0 . Hence δe extends to a bounded operator on all of HL with δe  = δe∗  ≤ 1/ε for each e ∈ E. It is then easy to see that the operator ZG,HL (δ)∗ : (⊕s∈S HL ) → (⊕r∈R HL ) is given by multiplication by ZG,HL (z  )∗ on the left: ZG,HL (δ)∗ : f (z  ) → ZG,HL (z  )∗ f (z  ) for f ∈ ⊕s∈S H0 .

Conservative Noncommutative Systems

209

Note that an element f ∈ ⊕s∈S H0 can be viewed as an element of the space L(Y, ⊕s∈S C)z  . The (⊕s∈S HL )-norm of an element f = ⊕s∈S fs ∈ ⊕s∈S H0 can be computed as follows:   f 2⊕s∈S HL = ffs 2HL = L (ffs (z)∗ fs (z  )) = L (f (z)∗ f (z  )) . s∈S

s∈S

Similarly, ZG,HL (δ)∗ f 2⊕r∈R HL = L (f (z)∗ ZG,HL (z)ZG,HL (z  )∗ f (z  )) . We may then compute f 2⊕s∈S HL − (1 + )2 ZG,HL (δ)∗ f 2⊕r∈R HL  = L f (z)∗ (II⊕s∈S C − (1 + ε)2 ZG,HL (z)ZG,HL (z  )∗ )f (z  ) .

(5.25)

Clearly, ϕ(z, z  ) given by ϕ(z, z  ) := f (z)∗ (II⊕s∈S C − (1 + ε)2 ZG,HL (z)ZG,HL (z  )∗ )f (z  ) is in the cone Cε : simply take γe (z) = 0 for all e ∈ E in the defining representation (5.10) for elements of Cε . From (5.25) and the assumption that L is nonnegative on Cε , we therefore deduce that ZG,HL (δ) = ZG,HL (δ)∗  ≤

1 < 1. 1+ε

From our assumption that T (z) ∈ SAG (U, Y), we deduce that T (δ) ≤ 1. If we are in the scalar-valued case U = Y = C, then we see from the form (5.24) for the action of δe∗ and from the continuity of L that T (δ)∗ is given by T (δ)∗ : f (z  ) → T (z  )∗ f (z  ) with T (δ)∗ f 2 = L (f (z)∗ T (z)T (z )∗ f (z  )) . As T (δ)∗  ≤ 1 we therefore have 0 ≤ 12HL − T (δ)∗ (1)2HL = L (IIY − T (z)T (z )∗ ) = L(ϕT (z, z  )) as wanted. The general case is a little more intricate. For Φ ∈ L(U, Y) and v ∈ FE , the  on an element f (z  ) ⊗ y of HL ⊗ Y. We tensor product operator δ ∗v ⊗ Φ∗ acts  v assume that the formal power series f = v∈F FE fv z consists only of its constant  term (so fv = 0 for v = ∅ and f (z ) =  where  ∈ L(Y, C) is a linear functional on  Y). We compute the (HL ⊗ U)-inner product of (δ ∗v ⊗ Φ∗ )( ⊗ y) against another

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such object (δ ∗v



⊗ Φ∗ )( ⊗ y  ) as follows:

(δ ∗v



⊗ Φ∗ )( ⊗ y  ), 



(δ ∗v ⊗ Φ∗ )( ⊗ y)HL ⊗U 

= z v  ⊗ Φ∗ y  , z v  ⊗ Φ∗ yHL ⊗U 



= z v  , z v HL · Φ∗ y  , Φ∗ yU   = L1 z v z v ∗ · ΦΦ∗ y  , yY ·    = L1 ∗ y ∗ (Φz v )(Φ∗ z v )y   .

(5.26)

Here we have viewed the vector y ∈ Y as the operator y : α → αy from C to Y with adjoint operator y ∗ : Y → C given by y ∗ : y  → y  , yY ∈ C. In this way, the inner product ΦΦ∗ y  , yY , when viewed as an operator on C, can be written as the operator composition ΦΦ∗ y  , yY = y ∗ ΦΦ∗ y  : C → C. By linearity we can generalize (5.26) to G (δ)∗ ( ⊗ y  ), G(δ)∗ ( ⊗ y)HL ⊗U = L1 (∗ y ∗ G(z)G (z  )∗ y   )

(5.27)

for any polynomials G(z  ), G (z  ) in the noncommuting indeterminates z  with coefficients in L(U, Y) (G, G ∈ L(U, Y)z  ). More generally, by the assumed continuity of L on X , (5.27) continues to hold if G and G are formal power series in L(U, Y)z   for which G(δ) and G (δ) are defined. We now apply (5.27) to the case where G = G = T ∈ SAG (U, Y) and where  ∗ ( ) = y  = yj and ∗ = y = yi , where y1 , y2 , . . . , yM is an orthonormal basis for Y, to get  T (δ)∗ (yj∗ ⊗ yj ), T (δ)∗ (yi∗ ⊗ yi )HL ⊗U = L1 yi yi∗ T (z)T (z )∗ yj yj∗ . Summing over i, j = 1, . . . , M then gives < ⎛ ⎞