A Parametrization of External Spectral Factors

A PARAMETRIZATION OF EXTERNAL SPECTRAL FACTORS L. BARATCHART∗ and A. GOMBANI∗∗1 ∗

INRIA, Route des Lucioles, Sophia–Antipolis, 06560 Valbonne, FRANCE LADSEB-CNR, Corso Stati Uniti 4, 35020 Padova, ITALY

∗∗

Abstract: We study the geometric structure of the spectral factors of a given spectral density Φ. We show that these factors can be associated to a set of invariant subspaces and we exhibit the manifold structure of this set, providing also an explicit parametrization for it, in the special case of coinciding algebraic and geometric multiplicity of the zeros of the maximum-phase spectral factor. We also make some connection with the set of solutions to the Riccati Inequality. 1991 Mathematics Subject Classification: 93E03 Keywords: spectral factors, minimal realization, state space, Riccati inequality

1

Introduction

ifold structure of this set, in the special case when algebraic and geometric multiplicity of the zeros of The characterization of all the minimum degree the maximum-phase spectral factor coincide, and spectral factors W of a given p × p spectral density we indicate how the corresponding set P of soluΦ of rank m0 is a problem which has been widely tions to (1) can be obtained as a quotient modulo a studied in connection with the lattice of symmet- group of (m−m0 )×(m−m0 ) all-pass functions. For ric solutions to the Riccati inequality associated to simplicity we only consider strictly proper factors, Φ. In particular, Anderson and Faurre gave a pre- which actually means that D in (1) is 0. cise characterization of the set of solutions (see e.g. [6]) Denote by Z the positive real part of Φ and by (A, G, C, L) a minimal realization. Then all spec- 2 Preliminaries and notation tral factors are given by the solutions P to We work with the row vectors Hardy spaces of the     P − AP A0 G − AP C 0 B 0 0 disk ID. We define [8] L2m (TT) to be the set of the = [B , D ] G0 − CP A0 L − CP C 0 D square integrable m-dimensional row vector valued 2 (1) 2 (H m ) to be functions on the unit circle TT, and Hm for suitable matrices B, D which are determined up the subspace of L2m (TT) whose non negative (strictly to a unitary transformation and can therefore alnegative) Fourier coefficients vanish. The functions ways be chosen of full rank. Then the associated 2 2 spectral factor is W (z) = D + C(z − A)−1 B (again in Hm (H m ) are defined on TT, but they can be extended to analytic functions on the open compleup to a unitary transformation). This characterization, though, does not provide ment IE of the closure of the disk ID (to the disk much insight in the geometry of the set of spec- ID), by taking the Cauchy integral: tral factors; moreover, it is not easy to characterize the elements on the boundary (that is the P for which rank([B 0 , D0 ]) drops). For the case of internal realizations (dim(W ) = p × m0 ) Lindquist and Picci have characterized the minimal stable spectral factors in terms of the coinner divisors of the minimum-phase spectral factor W− . More recently the problem has been studied by Lindquist, Michaletzky and Picci [9] for external realizations. We are not aware, though, of a geometric characterization of the set of all the spectral factors of Φ of a given dimension p × m. We exhibit here the man-

Z f (e−iω ) 1 f (z) := deiω 2π TT eiω − z −1



Z 1 g(eiω ) iω g(z) := de 2π TT eiω − z

f ∈ Hp2

g∈

2 Hp



T

z −1 ) , The transposed of the function (f ∗ (z)) := f (¯ T where denotes transposition, represents the ex2 tension in H p of f (eiω ). The inner product on Hp2 thus becomes:

1 Supported in part by CNR, CNRS and the SCIENCEERNSI project

hf, gi = 1

1 2πi

R

dz ∗ TT g(z)f (z) z

2 A p × m matrix function F with rows in zHm is said to be rigid if p ≥ m and the product F ∗ (eiω )F (eiω ) = Ip a.e. (Ip is the identity) or if p ≤ m and the product F (eiω )F ∗ (eiω ) = Ip a.e. It is inner (or stable all-pass) if p = m. The orthogonal projection onto a subspace X or onto the span of a vector b will be denoted by PX and Pb respectively; in particular, projections onto 2 2 Hm and H m will be denoted by P− and P+ , respectively. If B is an inner function in with columns in 2 2 2 zHm , we define H(B) := Hm Hm B. The shift 2 −1 on Hm is defined as U f := z f . The adjoint is U ∗ = P− Mz , and clearly U is an isometric opera2 tor, i.e. U ∗ U = I. A space X in Hm is invariant, if U X ⊂ X. Invariant subspaces are characterized by the Beurling Theorem (see e.g. [8]):

2 2 Theorem 2.2 Let X = Hm Hm Q be a coinvari2 ant subspace in Hm : then there exists a controllable pair A, B such that the rows of (z − A)−1 B form a basis x1 , x2 , ..., xn in X; the inner product matrix P defined as pij := hxi , xj i satisfies the equation:

P = AP A∗ + BB ∗ Moreover, Q(z) = D +C(z −A)−1 B where C, D are the unique solution to AP C ∗ + BD∗ = 0 CP C ∗ + DD∗ = I Q(1) = D + C(I − A)−1 B

(2) (3) (4)

Remark: the usual formulation is for an observable pair A, C and the Lyapunov equation has A∗ on the other side. The choice of the point 1 for the normalization condition (4) is quite arbitrary. We define now minimum-phase functions. An el2 ement f ∈ Hm is generating an invariant subspace X, if X is the minimal invariant subspace contain2 ing f . An invariant subspace in Hm has multiplicity m0 if it can be generated by no less than m0 vectors. It is maximal if there is no subspace Y ⊃ X of the same multiplicity m0 . A matrix function is 2 if its rows generate minimum-phase, or outer in Hm a maximal subspace. Let Φ be a rational spectral density, i.e. a symmetric p × p rational function of rank m0 which is nonegative definite on the circle TT. It is well known that for any m > m0 there are infinitely many factorizations W of dimension p × m, i.e. Φ = W W ∗ , but there exists an essentially unique factorization 2 . Φ = W− W−∗ with W− minimum-phase in Hm 0 Similarly there is a unique factorization (still up ∗ to a unitary transformation) Φ = W + W + where

Theorem 2.1 A subspace X is invariant for U in 2 2 Hm if and only if X = Hm Q for some inner function Q, which is uniquely determined up to a constant unitary factor. The function Q becomes unique if a further condition is imposed, for example Q(eiω0 ) = T0 , where −π ≤ ω0 < π is in the domain of analiticity of Q (or more generally if Q has radial limit in eiω0 ) and T0 is unitary. In our setting, since all the inner functions we will consider will be rational, it is not a restriction to make the following: Assumption 2.1 We will assume from now on that a coinvariant subspace Z = H(Q) is characterized by the unique inner function satisfying Q(1) = I. A subspace Y is said to be coinvariant , if U ∗ Y ⊂ Y . We also say that X ⊂ Z is invariant in Z if PZ U X ⊂ X. Coinvariance in Z is defined 2 analogously. A subspace X ⊂ Hm is called semiinvariant if there exists a subspace Y , such that Z := X ∨ Y is coinvariant, and X is invariant in Z. It can be shown (see [3]) that, although the subspace Y is not unique, there is only one such space which is orthogonal to X, and it will be coinvariant. It is also easy to see this fact directly: define Z := span{(U ∗ )n X; n ≥ 0}. If X is, according to the definition, invariant for U in Z, its orthogonal complement in Z will be invariant for the adjoint U ∗ of U , i.e. it is coinvariant. Therefore we have several corollaries to Beurling Theorem:

2

W + ∈ H m0 , with W + conjugate minimum-phase. Both this factorizations have dimension p × m0 . Moreover, it can be shown (see [7]) that there exists an essentially unique inner function K+ of dimension m0 × m0 of minimal degree such that the rows of W+ := W + K+ (5)

2 are in Hm ; the matrix function W+ is called the 0 maximum-phase spectral factor. The factorization (5) is also known as Douglas-Shapiro-Shield factor2 ization (in this case of W + in H m0 ). The unitary transormation up to which the spec1. A space Y is coinvariant if and only if there tral factor is determined, can be fixed by selecting exists an (essentially unique) inner function Q 2 a particular basis in Hm for each m; the basis we such that Y = H(Q) 2 2 choose is such that Hm0 and Hm−m are imbedded 0 2 2. A space Z is seminvariant if and only if there in Hm by the following maps: exist two (essentially unique) inner functions  2  2 2 Hm 7→ Hm , 0 ⊂ Hm Q1 , Q2 such that Z = H(Q2 )Q1 0 0

  When these spaces are finite dimensional, the 2 2 2 Hm−m 7→ 0, Hm−m ⊂ Hm 0 0 issue of how to represent the spaces H(Q) is adWe still have to fix a basis in the two subspaces dressed by the following well known result (see e.g. 2 2 2 Hm and Hm−m . We can assume that in Hm a [5]): 0 0 2

particular factorization W+ of Φ is given; we will see shortly that we can impose conditions so that all the other spectral factors can be chosen uniquely. We recall that a scalar zero of a rational transfer function W is a complex number z such that the matrix   ζI − A B −C D

But Q has minimal degree, and this implies that the inner function Qi is constant. The value at z0 determines this constant. As we said above, we will assume z0 = 1; thus Q(1) = I. Putting things together we have the following well known corollary (see [11]) to Theorem 2.3: Corollary 2.1 Let W be a spectral factor of minimal McMillan degree. Then there exist unique in¯ W , of dimension m × m, ner functions QW and Q ¯ W ∗ = [W− , 0]. such that W QW = [W+ , 0] and W Q Moreover,   ¯ W = Q+ QW Q (6) ˜˜ Q Z

associated to any minimal realization of W has a rank drop in z. It is well known that if zi is un2 stable, this implies that there exists a bi ∈ Hm 1 such W bi 1−zzi is analytic in the complement of the closed unit disk. In the Hilbert space setting, this is to say that 1 2 is orthogonal to W in Hm . In particzi = bi z−z i 1 ular, zi = bi z−zi is an unstable zero of W+ if bi is a row vector with components C I m0 , zi is a scalar zero of W+ and the rows of W+ are orthogonal to zi . This is not a new definition of zero; it is simply an adaptation to the Hilbert space setting of the standard definitions existing in the literature (see e.g. [12]). In particular, the set of unstable zeros generates the equivalent of the zero module for W+ . Going back to a general W , we can try to extend a definition of unstable zero along the above lines. The relation between minimal spectral factors has been characterized by Lindquist and Picci

where Q+ is independent of the choice of W , Z˜ = 2 ˜ ˜ is the inner function asPHm−m0 H(QW ) and Q Z ˜ sociated to Z The function Q+ is also relating the minmumphase and maximum -phase factors, by the well known relation W+ = W− Q+ The function Q+ will play an important role in the following.

Theorem 2.3 Let W be a spectral factor of miniMain Results mal McMillan degree. Then there exist (essentially 3 ˆ ˆ ¯ unique) rigid functions QW and QW , of dimen- As announced, we will assume for simplicity that sion m × m0 and m0 × m respectively, such that W is strictly proper, and that zW is regular at inˆ¯ ∗ = W . ˆ W = W+ and W Q WQ finity. The results are much simpler to state, and it − W is not difficult to extend them to the general case, ˆ¯ are used to extend ˆ W and Q These functions Q W as will be done in [2]. Therefore, in this section, by the definition of zeros to W as functions of those state space for a given spectral factor W we mean of W+ . Clearly W might not have any zero in the a semiinvariant subspace X in H 2 such that the m usual sense. The idea then is to complete the rigid rows of W belong to X. It can be shown that in ˆ¯ (which might not have any ˆ W and Q functions Q general (in the external case) for each spectral facW ¯ W (which always tor there exists a unique minimal state space (in zeros) to inner function QW and Q have zeros); it is shown in [11] that this extension the sense that it has dimension as small as possible: exists. We also show uniqueness under mild condi- it can be shown [11] that this dimension is always tions. n). By minimal spectral factor we therefore mean ˆ be an m × m0 rigid function of a factor of McMillan degree n. Denote by X+ the Lemma 2.1 Let Q ˜ state space of the maximum-phase spectral factor, degree n: then there exists a unique rigid function Q ˆ Q] ˜ is inner, has minimal degree i.e. such that Q = [Q, X+ = span{P− U n W+ } and takes a specified value at a fixed point z0 in its 2 Let Z be coinvariant in Hm and Z ⊥ W+ , where domain of analiticity. with the abuse of notation we intend that Z is ˜ be a minimal spectral factor of Im − Proof: let Q orthogonal to the rows of W+ . We set XZ := ˆQ ˆ ∗ ; then Q = [Q, ˆ Q] ˜ is inner; but Q ˜ has an outerQ (X+ ∨ Z) Z. ˜ − Qi , and inner factorization Q Lemma 3.1 XZ is a semiinvariant minimal sub˜1Q ˜ ∗1 = Im − Q ˆQ ˆ∗ = Q ˜−Q ˜ ∗− Q space containing W+ 2 Proof: X+ ∨ Z is coinvariant in Hm , since both X+ and Z are. Since Z is coinvariant, it is also coinvariant in X+ ∨ Z. Thus its orthogonal complement XZ is semiinvariant in X+ ∨ Z. Since Z ⊥ W+ ,

˜ − is also an extension. Therefore, In other words, Q   Im0 ˆ ˜ ˆ ˜ Q = [Q, Q] = [Q, Q− ] Qi 3

◦

W+ ∈ XZ , and in view of the dimension, XZ is minimal. Denote now by Z+ the maximal coinvariant sub2 space in Hm orthogonal to W+ . Since we know 0 from Corollary 2.1 that there exists a unique inner function Q+ such that W+ = W− Q+ , it is clear that Z+ = H(Q+ ) We need to clarify the link between W and the space Z. We will assume, for the time being, that Z 2 is a minimal set of zeros, i.e. dimZ = dimPHm0 Z

Theorem 3.2 The space Z m is homeomorphic to CI r×(m−m0 ) Proof: from the above lemma, for a given choice of basis in Z+ , we can associate to each ele˜ such that ment Z ∈ Zm a unique matrix B ◦m ˜ ∈ Bz . [Bz , B] From [1], the map from +

{(Az , Bz ) controllable; Bz ∈ CI r×m } to the set of Irm inner functions of degree r, endowed with the L2 norm on the rows, is smooth, and therefore so is its restriction to the open subset (in the induced ˆz , B ˜ z ] ∈ Bz } to the topology) {(Az , Bz ); Bz = [B image. Therefore, our task reduces to prove that the map φ from the image of IB z to Zm defined as φ(Q) = H(Q) is continuous. By construction the map is bijective. Let first 2 kQn − Qk → 0; then, for x ∈ Hm , kx[PH(Qn ) − PH(Q) ]k is bounded by the quantity kx(Q−1 n − ¯2 ¯2 −1 H −1 H m m Q )P Qn k + kxQ P (Qn − Q)k which converges to zero as kQn − Qk does. Conversely, let kPH(Qn ) − PH(Q) k → 0; we want show that in the limit also kQn − Qk is vanishing. 2 Let x ∈ Hm Then

Theorem 3.1 There a one to one correspondence between minimal spectral factors W and minimal 2 coinvariant subspaces Z such that PHm0 Z ⊂ Z . +

Proof: given W = [W+ ,0](QW )∗ , we have set by definition ZW = H(QW ). Conversely, given Z such 2 that PHm0 Z ⊂ Z+ , we know from Corollary 2.1 that there exists a unique inner function Q such that Z = H(Q) and Q(1) = I. Moreover, the elements of H(Q) are orthogonal to the rows of [W+ , 0], and therefore Q divides [W+ , 0] on the right. The desired factor is then given by W = [W+ ,0]Q∗ . How do we characterize all the W ? From the above theorem we need to characterize all the Z 2 such that PHm0 Z ⊂ Z+ and is invariant in Z+ .

kxQ[PH(Qn ) − PH(Q) k] ¯2

−1 = kxQ[(Q−1 )PHm Qn n −Q

Lemma 3.2 Let Z = H(Q) be a coinvariant sub¯2 2 +Q−1 PHm (Qn − Q)]k 2 space in Hm of dimension r such that PHm0 Z = ¯2 H mQ k = kxQQ−1 Z+ , and let Az+ , Bz+ , Cz+ , Dz+ be a controllable ren n P ¯2 −1 H alization for Q+ ; then there exists a unique matrix = kxQQn P m k ˜ such that z := (z − A)−1 [Bz , B] ˜ is a basis for B + 2 ˜ ∈ IRr×(m−m0 ) yields that is, the operator kQQ−1 PH¯ m Z. Conversely, any matrix B k converges to zero n H(Q ) H(Q) n a space satisfying the above condition. strongly as kP −P k → 0. Since this is means that the strictly anProof: let AQ , BQ , CQ , DQ be a minimal realiza- a Hankel operator, this −1 tistable part of QQ vanishes. In the same mann tion of Q. From Lemma 2.2 we know that a basis ¯2 −1 H −1 ner we can conclude that kQ Q P m k is strongly n for H(Q) is (z − A) B; it is also obvious that we ˆQ , B ˜Q ] where the dimensions convergent to zero, and therefore the stricly stable can partition BQ = [B is also vanishing. But this means part of QQ−1 n are m0 and m − m0 , and that −1 converges to a constant, and since the that QQ n 2 functions are inner and take the value I in 1, this Z+ = PHm0 H(Q) constant is necessarily the identity, as wanted. ˆQ ]i ; i = 1, ..., r} = span{[(z − AQ )−1 B ¯◦ The structure of Z m is quite complicated to deˆQ and (Az , Bz ) are related by a simso, (AQ , B + + scribe; nevertheless this is not so crucial, since we ilarity transformation, which is unique in view of ˜ is also uniquely deter- are actually interested in equivalence classes of this controllability. Therefore, B set, which have a better behaviour. mined. The converse is also trivially following from We define the following equivalence relation ∼ on controllability of (Az+ , Bz+ ) ¯◦ We denote by Hm the set of linear spaces with Z m : Z1 ∼ Z2 if both 2 rows in Hm , endowed with the gap topology: 2 2 Z1 (Z1 ∩ Hm−m ) = Z2 (Z2 ∩ Hm−m ) (8) 0 0 M, N ∈ Hm (7) d(M, N ) := kPM − PN k and set, for a given basis z+ = (z − Az+ )−1 Bz+ The interest of this relation is explained in the folof Z corresponding to an arbitrary realization lowing simple results +

(Az+ , Bz+ ) of Q+ , ◦m Bz

and

◦

:= {B ∈ C I

m

¯◦ Lemma 3.3 Let Z1 , Z2 ∈ Z m , and let Q1 , Q2 the associated inner matrices. Then Z1 is equivalent Z2 if and only if there exists an (m − m0) × (m − m0 ) Im0 0 all-pass matrix Q such that Q1 = Q2 0 Q

˜ ; B = [Bz+ , B] 2

H Z m := {Z ∈ Hm : P m0 Z = Z+ } 4

2 Proof: by construction, Z1 ∩Hm−m is coinvariant, 0 and it has associated an inner function    of the form  Im0 0 Im0 0 0 , and Q factors as Q . 1 1 0 Qp1 Qp1  0 Im0 0 Similarly, Q2 factors as Q02 . Then the 0 Qp2 relation (8) in terms of inner functions writes     Im0 0 Im0 0 0 0 H(Q1 ) = H(Q2 ) 0 Qp1 0 Qp2

∆ = diag{δ1 , ..., δr } with δi in the one point compactification of the unit disk S1 := {z; |z| < 1∪{1}} (we are working over the complex field) and ˜bi ∈ Sm−m0 −1 and we consider the following map φ from r to Z¯m : Sm−m 0 ˜ φ(∆, B)

:= span{(z − Az+ )−1

(9) ∗ 1/2

2

[∆ Bz+ , ∆(I − (∆∆ )

˜ )B]}

The map is clearly continuous, since (Az , Bz ) can be completed smoothly. It is bijective: since for any 2 Z ∈ Z¯m of dimension s ≤ r, dim PHm0 Z = s, Z is ∗ generated by eigenvectors of U|Z , which are of the form (z − aik )−1 [βˆi , β˜i ], with β˜i = 0 if βˆi = 0 Now, comparing with (9), it is seen that

which implies the conclusion. Notice that all the matrices we are considering are are normalized, i.e. Q(1) = I; ¯◦ Clearly for each equivalence class of Z m there 2 exists a unique element Z¯ such that Z∩Hm−m = 0; 0 2 H m0 ¯ ¯ so dim Z = dim P Z. We take this element as the representative of the equivalence class in Z¯m .

|δi | =

kβˆi k kβˆi k + kβ˜i k

Corollary 3.1 Let Z1 = H(Q1 ) and Z2 = H(Q2 ). and that for 0 < |δi | < 1 ˆ1 = Q ˆ2 Then Z1 ∼ Z2 if and only if Q ¯◦ We now define Z¯m := Z m / ∼.

arg δi =

arg βˆik 2 arg ˆbk i

ˆ m be the set of m × m0 rigid Corollary 3.2 let Q ˆ such that W = W+ Q ˆ ∗ has rows in H 2 where ˆbki is a nonzero component of ˆbi ; then ˜bi is function Q m ˜ for determined, for 0 < |δi | < 1 by and W (1) = W+ (1). Then the completion Q ˆ Q] ˜ is minimal inner is such that Z = which Q = [Q, δi (1 − |δi |)β˜i = ˜bi ˆ m to H(Q) ∈ Z¯m and the completion map from Q Z¯m is continuous. and is zero otherwise. So the map φ is onto, and since the solution is obviously unique, it is also one ˆ have the same Proof: note first that Q and Q to one. Continuity of φ−1 is obvious for |δi | < 1, i.e. ˆ B]) ˜ is a realizaMcMillan degree. In fact, if (A, [B, ˆ is not controllable, we can when k˜bi k remains bounded. In the case k˜bi k → ∞, tion of Q such that (A, B) −1 ˜ ¯ find a change of basis for which A is lower triangular the basis for Z contains the vector (z −aik ) [0, βi ], so that we can substitute zeros in the i − th row ˆ has a zero row in correspondence of Jordan, and B ˜ the uncontrollable mode ai ; if ˜bi is the correspond- for βi . But the element δi in the inverse image is also converging to zero, and so the map φ is a ˜ we know that H(Q) 3 (z −ai )−1 [0, ˜bi ], ing row of B, homeomorphism. ˜ ai and therefore, defining Bbi (z) := P[0,bi ] 1−z¯ z−ai it is We now come to the main result of the paper, ∗ 2 QBbi ∈ Hm , contradicting minimality of the exten- which is to represent the set P as the quotient of a sion. smooth manifold. In our simplified setting, the set ˆ n } is a Cauchy sequence convergTherefore, if {Q of solutions P of (1) is restricted those for which ˆ the completion Q has always degree less ing to Q, D = 0; each P ∈ P determines then a set of sothan or equal to the limit of the degrees of the se- lutions B to (1) and an equivalence class W of P P quence. If it is equal, there is nothing to show: the spectral factors W = C(z − A)−1 B, B ∈ B where P topology coincides with that of inner functions. If the equivalence relation is multiplication by a uni¯ there is a degree drop, it means that the limit Q tary matrix on the right. Define of{Qn } in the usual topology can be factored as I 0 Zm := Z¯m /Gm Q , and therefore Q is the representative 0 Q1 ¯ where we have set of the equivalence class of Q.   I 0 Theorem 3.3 Let Az+ be diagonalizable. Then the Gm := 0 U (m − m0 ) set Z¯m is a manifold diffeomorphic to the product r Sm−m of r (m − m0 ) − dimensional spheres. 0 and set Proof: we prove continuity, and refer to [2] for Pm := {P ∈ P : rank(P − AP A∗ ) ≤ m} smoothness, since the construction of the charts is quite complex. Let (Az+ , Bz+ ) be a realization of Theorem 3.4 for each m there is a homeomorQ+ such that Az+ is diagonal and the rows of Bz+ phism r ˜ where have norm 1; we represent Sm−m as (∆, B) πm : Pm 7→ Zm 0 5

such that ◦

[5] H. Dym. J-contractive matrix functions, reproducing kernel spaces and interpolation, volume 71 of CBMS lecture notes. American mathematical society, Rhodes island, 1989.

◦

πm :P m 7→ Z m is a diffeomorphism i) the points such that rank(P − AP A∗ ) = s < m correspond to the points of Zm whose centralizer has dimension m − s ii) the extremal points for which rank(P −AP A∗ ) = m0 (internal realizations) correspond to fixed points of Gm

[6] Faurre P., Clerget M., Germain F., Op´erateurs Rationnels Positifs, Dunod, 1979 [7] P. A. Fuhrmann , Linear Systems and Operators in Hilbert Space, McGraw-Hill, 1981

[8] Hoffman, K. (1962) Banach Spaces of Analytic Functions. Prentice-Hall, Englewood Cliffs. Proof: (sketch: see [2] for details) to each P ∈ P there corresponds an equivalence class WP of spec[9] Lindquist A., Michaletzky G., Picci G. Zeros tral factors. It has been shown that Z¯m is the of spectral factors, the geometry of splitting manifold of inner functions satisfying the condition subspaces, and the algebraic Riccati inequalQ(1) = I and yielding the factorization W Q = ity, to appear on SIAM Journal of Control and [W+ , 0]; let T ∈ Gm : then also W T T ∗ QT = [W+ , 0] Optimization and satisfies the normalizing condition; therefore all the conjugates of Q will yield W in the same [10] Lindquist A., Pavon M., On the structure of equivalence class. These are the only ones, since state space models for discrete-time stochastic [W+ , 0]T = [W+ , 0] and T unitary forces T ∈ Gm . vector processes, IEEE Trans. Automatic ConIn conclusion, πm (P ) associates to an element P trol, AC-29, p.418-432, 1984 the set of spectral factors in WP ∩ Z¯m , and these sets are disjoint orbits of G. This shows that π is [11] Lindquist A., Picci G. A geometric approach to modeling and estimation of linear stochastic a map. Invertibility follows from direct computasystems, Journal of Math. Systems, Estimation tion: if Z ∈ Zm , then Z = H(Q) and W Q∗ is a and Control, vol. 1, pp. 241–333, 1991. stable spectral factor admitting a minimal realization W = zC(z − A)1 B with (A, C) independent of W , and P is then the controllability gramian of [12] Wyman B. F. Sain M.K., Module theoretic zero structures for system matrices, SIAM (A, B). It is continuous, since the set of solutions to Journal on Control and Optimization, 25, 86the positive real equations depend bicontinuosly on 99 the coefficients. To see that it is a diffeomorphism state space formulas are needed and we refer to [2]. About i) and ii), it is clear that if W = C(z −Az )B, with rank B = s, then its centralizer has dimension m − s; in particular, for internal realizations, this dimension is m − m0 and therefore the centralizer is the whole Gm . Notice that, in particular, the the lattice of symmetric solutions to (1) P = Pn+p is homeomorphic to Zm+p .

References [1] D. Alpay, L. Baratchart, A. Gombani, On the differential structure of matrix-valued rational inner functions, to appear on Journal of Integral Equations and Operator Theory [2] L. Baratchart, A. Gombani, State space formulas for the geometry of external spectral factors, under preparation [3] J. Ball, I. Gohberg, and L. Rodman Interpolation of rational matrix functions. Birkh¨auser Verlag, Basel, 1990. [4] P.E.Caines, Linear Stochastic Systems, Wiley, 1988 6