Spectral Properties for Matrix Algebras

SPECTRAL PROPERTIES FOR MATRIX ALGEBRAS ´ CARMEN FERNANDEZ, ANTONIO GALBIS, AND JOACHIM TOFT

Abstract. We consider Banach algebras of infinite matrices defined in terms of a weight measuring the off-diagonal decay of the matrix entries. If a given matrix A is invertible as an operator on `2 we analyze the decay of its inverse matrix entries in the case where the matrix algebra is not inverse closed in B(`2 ), the Banach algebra of bounded operators on `2 . To this end we consider a condition on sequences of weights which extends the notion of GRScondition. Finally we focus on the behavior of inverses of pseudodifferential operators whose Weyl symbols belong to weighted modulation spaces and the weights lack the GRS condition.

0. Introduction In science and technology it is common to reformulate questions into analysis of matrices, because they are usually convenient to implement into computer programmings. This in turn usually leads to efficient numerical computations and approximations. An important question here concerns invertibility properties. In particular, the question whether a given class of matrices (on `2 = `2 (Zd )) is inverse closed in B(`2 ). Recall that a Banach algebra of matrices on `2 is called inverse closed in B(`2 ) if whenever an element in the algebra is invertible in B(`2 ), its inverse is also in the algebra. (For other definitions, see Section 1.) In the paper we especially consider invertibility questions for the Banach algebra Cv = Cv (Zd ), the set of all matrices (a(j, k))j,k∈Zd which satisfy X kAkCv ≡ sup |a(j, j − k)|v(k) < ∞, A = (a(j, k))j,k∈Zd . (0.1) k∈Zd

j∈Zd

Here v is a weight on Rd which is symmetric and submultiplicative. ∞,1 We remark that there are strong links between Cv and Opt (M(v) ), the set of all pseudo-differential operators Opt (a) with the symbol a ∞,1 in the modulation space (or the weighted Sj¨ostrand class) M(v) = ∞,1 M(v) (R2d ). 2010 Mathematics Subject Classification. 15A09, 35S05, 46H05, 47G30. Key words and phrases. Off-diagonal decay of matrices; inverse closedness; GRScondition; spectral invariance; pseudodifferential operators; modulation spaces. 1

There are several situations when matrices in Cv or pseudo-differential ∞,1 operators in Opt (M(v) ) appear. For example, matrices in Cv appear when considering discrete slow time-varying systems, and when discretizing non-stationary filters in time-frequency analysis and signal processing (see [12] and the references therein). In the latter situation, ∞,1 the link between Opt (M(v) ) and the matrices in Cv is essential. Another example can be found in [20], where Strohmer when modeling mobile wireless channels, considers pseudodifferential operators in the ∞,1 class Opt (M(v) ). The set Cv is contained in B(`2 ). Moreover, Cv is inverse closed in B(`2 ), if and only if v satisfies the GRS-condition (Gelfand-RaikovShilov condition), i. e., lim v(`x)1/` = 1,

`→∞

when x ∈ Rd

(0.2)

(cf. [12, Corollary 5.31]). A similar fact is true when Cv and `2 are ∞,1 replaced by Opt (M(v) ) and L2 , respectively. We note that the set of weights on Rd which satisfy the GRS-condition is a monoid with respect to the pointwise multiplication, and contains important weights of the form s

x 7→ C(1 + |x|)t

and x 7→ et|x| ,

when C ≥ 1, t ≥ 0 and 0 ≤ s < 1. Exponential growth in some direction is the reason why GRS-condition can fail. In the limit case where v(x) = et|x| some information about the entries of the inverse of a matrix A ∈ Cv was obtained by Jaffard [17] (see also Baskakov [3]), despite the fact that Cv is not inverse closed in B(`2 ). In fact, it was proved in [17] that the existence of A−1 ∈ B(`2 ) implies X sup |c(j, j − k)|es|k| < ∞, A−1 = (c(j, k))j,k∈Zd , k

j

for some 0 < s < t. In several situations, the lack of inverse closedness in B(`2 ), e. g. when the GRS-condition is violated, can have bad impact when solving problems. For example, in the analysis of wireless channels in [20], Strohmer avoids exponential weights, because of the absence of fulfilling the GRS-conditions of such weights. In Section 1 we consider a condition on sequences of weights which extends the notion of GRS-condition, and which also covers the case considered by Jaffard in [17]. Moreover in Section 2 we prove that Cv possess a weaker form of inverse closedness in B(`2 ) under this sequence version of GRS-condition. 2

More precisely, we consider decreasing sequences (vn )n∈N of (submultiplicative) weights vn on Rd satisfying   1/` inf lim vn (`x) = 1, when x ∈ Rd . (0.3) n

`→∞

In Theorem 2.1 we then prove that [ Cvn

(0.4)

n

is equal to an (uncountable) intersection of algebras Cv , where each v satisfies the GRS condition, by using the projective description technique due to Bierstedt, Meise and Summers (cf. [6]). Since any Cv is inverse closed in B(`2 ), it follows that the union (0.4) is inverse closed in B(`2 ). In particular, if A ∈ Cvn for some n is invertible on `2 , then its inverse A−1 belongs to Cvm for some m. If vn = v is independent of n, then (0.3) agrees with the GRScondition. Hence Theorem 2.1 in this case means that Cv is inverse closed in B(`2 ) when v satisfies the GRS-condition. If instead vn (k) = et|x|/n , for some t > 0, then Theorem 2.1 in this case gives Jaffard’s result. Our approach is completely different and gives additional information that clarifies why Jaffard’s result holds. Moreover, we also show that off-diagonal decay of the inverse matrix is as good as possible in those directions for which GRS-condition does not fail. See Corollary 2.5. For a submultiplicative weight ω on R2d , it is convenient to let Θ be the extension operator to weights on R4d , i. e. we put (Θω)(X, Y ) = ω(Y ) for X, Y ∈ R2d . ∞,1 According to Gr¨ochenig and Rzeszotnik in [14], Opt (M(Θv) (R2d )) is inverse closed in B(L2 (Rd )) if and only if v satisfies GRS-condition. In Section 3 we apply our results to analyze the behavior of inverses ∞,1 of pseudo-differential operators in Opt (M(Θv) (R2d )), where the GRS conditions might be violated for the weight v. 1. Preliminaries In this section we introduce convenient matrix classes, and recall some basic facts for weights, Gelfand-Shilov spaces, modulation spaces and pseudo-differential operators. 1.1. Weights. First we consider weight functions. Let X denote either X = Rd or X = Zd . A weight on X is a positive and locally bounded function on X. The weight v on X is called submultiplicative, if it is even and satisfies v(x + y) ≤ v(x)v(y),

for x, y ∈ X.

For two positive and locally bounded functions f and g on X, f . g means that there exists C > 0 such that f (x) ≤ Cg(x) for all x ∈ X. 3

It is well-known that v(x) . et|x| for some t > 0, when v is submultiplicative. In the sequel, v and vj always denote submultiplicative weights on Rd or on Zd . Let ω be a weight on Rd . Then ω is called v-moderate if ω(x + y) . ω(x)v(y),

for x, y ∈ Rd .

(1.1)

A weight ω which is v-moderate for some v is also called moderate. If ω is v-moderate, then v(x)−1 . ω(x) . v(x). If in addition v satisfies the GRS-condition (cf. (0.2)), then v is called a GRS-weight. More generally, the sequence (vn )n∈N of submultiplicative weights on Rd is called a GRS-sequence of weights, if (0.3) is fulfilled. By letting vn = v when v is a GRS-weight, it follows that the set of GRS-weights can be considered as a subset of the set of GRS-sequences of weights. Finally, if ω is a weight on Rd , which is v-moderate for some v, then `p(ω) = `p(ω) (Zd ) denotes the set of all sequences a on Zd such that j 7→ aj ω(j) belongs to `p (Zd ). 1.2. Matrices. Next we consider matrices indexed on Zd . Let v be submultiplicative on Rd . We recall that Cv = Cv (Zd ) consists of all matrices A = (ai,j )(i,j)∈Zd ×Zd such that (0.1) holds. In particular, the off-diagonal decay of the entries of the matrix A is controlled by the weight v. We note that Cv is a Banach ∗-algebra with respect to matrix multiplication and taking the adjoint matrix as involution. As Cv is a subalgebra of the Banach algebra B(`2 ) it makes sense to consider when Cv is inverse closed in B(`2 ). This is the case if and only if v satisfies the GRS-condition. Remark 1.1. Any weight on Rd is also a weight on Zd . On the other hand, let ω be a weight on Rd which is v-moderate for some v, and let ω0 be the weight on Rd , given by ω0 (x) = ω(k),

when k ∈ Zd , x ∈ k + Q.

(1.2)

Here Q is the unit cube such that 2Q = [−1, 1)d . The condition (1.1) now implies that ω0 is equivalent to ω, in the sense ω0 . ω . ω0 . ∞,1 Here we note that the classes `p(ω) , Cv , M(v) and other similar weighted spaces, do not change if the weights ω and v are replaced by other equivalent ones. Hence, when investigating such weighted spaces it suffices to consider weights ω0 on Rd , given by (1.2), where ω is a weight on Zd .

4

1.3. Gelfand-Shilov spaces. Our discussion on modulation spaces later on is formulated in the framework of Gelfand-Shilov spaces and their distribution spaces. In order to recall the definition of the latter spaces, we let 0 < h, s ∈ R be fixed, and let Ss,h (Rd ) be the set of all f ∈ C ∞ (Rd ) such that kf kSs,h ≡ sup

|xβ ∂ α f (x)| h|α|+|β| α!s β!s

is finite. Here the supremum should be taken over all α, β ∈ Nd0 and x ∈ Rd . We have that Ss,h (Rd ) is a Banach space which increases with h and s and is contained in S (Rd ), the set of all Schwartz functions on Rd . The Gelfand-Shilov space Ss (Rd ) is the inductive limit of Ss,h (Rd ). In particular, [ Ss (Rd ) = Ss,h (Rd ), (1.3) h>0

and the Gelfand-Shilov distribution space Ss0 (Rd ) is the projective limit 0 of Ss,h (Rd ). This implies that \ 0 Ss,h (Rd ). (1.3)0 Ss0 (Rd ) = h>0

We remark that in [9, 18, 19] it is proved that Ss0 (Rd ) is the dual of Ss (Rd ) (also in topological sense). Evidently, Ss (Rd ) increases with s, and Ss0 (Rd ) decreases with s. If s < 1/2, then Ss (Rd ) is trivial. Otherwise Ss (Rd ) is embedded and dense in S (Rd ), and S 0 (Rd ) is contained in Ss0 (Rd ). We let the Fourier transform on S 0 (Rd ) be the linear and continuous map which takes the form Z −d/2 b (F f )(ξ) = f (ξ) ≡ (2π) f (x)e−ihx,ξi dx Rd

when f ∈ L1 (Rd ). For every s ≥ 1/2, the Fourier transform is continuous and bijective on Ss (Rd ), and extends uniquely to a continuous and bijective map on Ss0 (Rd ). 1.4. Modulation spaces. Let 1/2 ≤ s0 < s, and let φ ∈ Ss0 (Rd ) \ {0} be fixed. Then the short-time Fourier transform Vφ f of f ∈ Ss0 (Rd ) with respect to the window function φ is the element in Ss0 (R2d ) ∩ C ∞ (R2d ), defined by the formula (Vφ f )(x, ξ) := (2π)−d/2 (f, φ( · − x)eih · ,ξi ), where ( · , · ) is the (unique) extension of the L2 -form on Ss0 (Rd ). If in addition f is locally integrable, then Vφ f is given by Z −d/2 (Vφ f )(x, ξ) = (2π) f (y)φ(y − x)e−ihy,ξi dy. Rd

5

Let φ ∈ S1/2 (Rd ), p, q ∈ [1, ∞], v be submultiplicative on R2d , and let p,q ω be a v-moderate weight on R2d . Then the modulation space M(ω) (Rd ) 0 is the Banach space which consists of all f ∈ S1/2 (Rd ) such that !1/q  Z Z q/p

kf k

p,q M(ω)

p

|Vφ f (x, ξ)ω(x, ξ)| dx

:= Rd

dξ

(1.4)

Rd

is finite (with obvious modifications when p = ∞ or q = ∞). We rep,q mark that the definition of M(ω) (Rd ) is independent of the choice of the window φ ∈ S1/2 (Rd ) \ {0}, and different φ gives rise to equivalent norms. See e. g. [11, Proposition 11.4.2] or [21, Proposition 1.11]. Furp,q thermore, if s < 1 and ω is a moderate weight, then M(ω) contains Ss 0 and is contained in Ss (see e. g. [21]). For distributions defined on the phase space R2d , we may define modulation spaces in terms of the symplectic short-time Fourier transform, which is given by (VΦ a)(X, Y ) := π −d (a, Φ( · − X)e2iσ( · ,Y ) ), when Φ ∈ Ss0 (R2d ) \ {0} is fixed, a ∈ Ss0 (R2d ) and X, Y ∈ R2d . Here σ is the symplectic form, given by σ(X, Y ) = hy, ξi − hx, ηi when X = (x, ξ) ∈ R2d and Y = (y, η) ∈ R2d . Again we note that if in addition a is locally integrable, then Z −d (VΦ a)(X, Y ) = π a(Z)Φ(Z − X)e2iσ(Y,Z) dZ. R2d

If ω is a v-moderate weight on R4d for some submultiplicative weight 2d v on R4d , then the modulation space Mp,q (ω) (R ) is now defined in the same way, after the short-time Fourier transforms have been replaced by corresponding symplectic transforms in the definition of the norms. We note that (VΦ a)(X, Y ) = 2d (VΦ a)((X, Y )), when (X, Y ) = (x, ξ, −2η, 2y), X = (x, ξ) and Y = (y, η), which implies that p,q 2d 2d Mp,q (1.5) (ω◦) (R ) = M(ω) (R ), with equivalent norms. 1.5. Pseudo-differential operators. Next we recall some properties in pseudo-differential calculus. Let s ≥ 1/2, a ∈ Ss (R2d ), and t ∈ R be fixed. Then the pseudo-differential operator Opt (a) is a linear and continuous operator on Ss (Rd ), given by Z Z −d (Opt (a)f )(x) = (2π) a((1 − t)x + ty, ξ)f (y)eihx−y,ξi dydξ. Rd

Rd

(1.6) 6

For t = 0, then Op0 (a) agrees with Kohn-Nirenberg or normal representation a(x, D). If instead t = 1/2, then Opt (a) is the Weyl quantization, and is denoted by Opw (a). For general a ∈ Ss0 (R2d ), the pseudo-differential operator Opt (a) is defined as the continuous operator from Ss (Rd ) to Ss0 (Rd ) with distribution kernel given by Ka,t (x, y) = (2π)−d/2 (F2−1 a)((1 − t)x + ty, x − y),

(1.7)

that is, hOpt (a)f, gi = hKa,t , g ⊗ f i,

f, g ∈ Ss (Rd ).

Here h · , · i is the dual form between Ss and Ss0 , and F2 F is the partial Fourier transform of F (x, y) ∈ Ss0 (R2d ) with respect to the y variable. This definition of Opt (a) makes sense, since both of the mappings F2

and F (x, y) 7→ F ((1 − t)x + ty, y − x)

(1.8)

are homeomorphisms on Ss0 (R2d ). In particular, the map a 7→ Ka,t is a homeomorphism on Ss0 (R2d ). For future references we recall the link between the Weyl quantization and the (cross)-Wigner distribution Wφ,ψ of φ, ψ ∈ Ss0 (Rd ), which belongs to Ss0 (R2d ), and is defined by the formula
Wφ,ψ (x, ξ) = F φ(x + · /2)ψ(x − · /2) (ξ). We note that Wφ,ψ is given by Z −d/2 Wφ,ψ (x, ξ) = (2π) φ(x + y/2)ψ(x − y/2)e−ihy,ξi dy, Rd

when φ, ψ ∈ L2 (Rd ). By straight-forward computations we have (Opw (a)ψ, φ) = (a, Wφ,ψ ), for admissible a, φ and ψ. The set of pseudo-differential operators Opt (a) with the symbols a in the spaces Ss (R2d ),

S (R2d ),

S 0 (R2d ) or Ss0 (R2d ),

(1.9)

is independent of the choice of t ∈ R. In fact, by [16, Section 18.5] and its analysis, we have Opt1 (a1 ) = Opt2 (a2 )

a2 = ei(t2 −t1 )hDξ ,Dx i a1 ,

⇐⇒

(1.10)

where the map ei(t2 −t1 )hDξ ,Dx i is a continuous bijection on each one of the spaces in (1.9). There are several established continuity results for pseudo-differential operators. We are especially interested in the following special case of [21, Theorem 6.15]. Here PE (Rd ) is the set of all moderate weights on Rd . 7

Theorem 1.2. Let t ∈ R and p, q ∈ [1, ∞]. Also let ω ∈ PE (R4d ) and ω1 , ω2 ∈ PE (R2d ) be such that ω2 (x − ty, ξ + (1 − t)η) . ω(x, ξ, η, y). ω1 (x + (1 − t)y, ξ − tη) ∞,1 0 If a ∈ M(ω) (R2d ), then Opt (a) from S1/2 (Rd ) to S1/2 (Rd ) extends p,q p,q uniquely to a continuous map from M(ω (Rd ) to M(ω (Rd ). 1) 2)

We note that the previous result agrees with [11, Theorem 14.5.2] when the weights are trivial (ω = ωj = 1) and t = 0. Remark 1.3. Theorem 1.2 attains the following convenient form in the case of the Weyl quantization. Let p, q ∈ [1, ∞]. Also let ω ∈ PE (R4d ) and ω1 , ω2 ∈ PE (R2d ) be such that ω2 (X − Y ) . ω(X, Y ). ω1 (X + Y ) w d 2d d 0 If a ∈ M∞,1 (ω) (R ), then Op (a) from S1/2 (R ) to S1/2 (R ) extends p,q p,q uniquely to a continuous map from M(ω (Rd ) to M(ω (Rd ). 1) 2)

Finally we also need the following result which is an immediate consequence of [21, Proposition 6.14] and (1.10). Here recall that Θ is the extension operator (Θω)(X, Y ) = ω(Y ). Proposition 1.4. Let ω ∈ PE (R2d ). Then the operator class ∞,1 Opt (M(Θω) )

is independent of t ∈ R. 2. Off-Diagonal Decay Matrices In this section we investigate off-diagonal decay of inverses of matrices in Cv in absence of GRS-condition. We consider unions of the form (0.4) where (vn )n∈N is a GRS sequence of weights on Rd (which may contain weights which fail to meet the GRS-condition). We prove that this union can be represented as an (uncountable) intersection of algebras Cv , in such a way that each v satisfies the GRS-condition. Since any Cv is inverse closed in B(`2 ), it follows that the union is also inverse closed in B(`2 ). Theorem 2.1. Let (vn )n∈N be a decreasing sequence of submultiplicative weights on Rd such that (0.3) holds. Then, there is a family V of submultiplicative weights satisfying GRS-condition such that [ \ Cvn = Cv . (2.1) n

v∈V

In particular, if A ∈ Cvn for some n ∈ N and A is invertible on `2 (Zd ), then A−1 ∈ Cvm for some m ∈ N. 8

Proof. By Remark 1.1 we may assume that (vn )n∈N is a family of weights on Zd (instead of Rd ). Let W be the family of positive even functions on Zd which are dominated by vn , for every n. That is,   w(k) d 0. Proof. Let vn := v 1/n when n ∈ N. As 1 ≤ vn (`x)1/` ≤ v(x)1/n , the previous theorem can be applied.



It is well known that a weight fails the GRS condition when it grows exponentially in some directions. When these directions can be isolated, we can say more about the behavior of the inverses. Corollary 2.4. Let v be submultiplicative on Rd1 × Rd2 of the form v(x, y) = u(x)·et|y| , where u is a weight on Rd1 satisfying GRS-condition and t > 0. If A ∈ Cv is invertible on `2 (Zd ) then A−1 ∈ Cw where w(x, y) = u(x) · es|y| for some s > 0. Proof. It suffices to apply Theorem 2.1 to the sequence of weights vn (x, y) = u(x) · et|y|/n .  The previous corollary can be extended and reformulated as follows. Corollary 2.5. Let vj on Rdj , j = 1, 2, be submultiplicative weights such that v1 satisfies the GRS-condition. If A ∈ Cv1 ⊗v2 is invertible on `2 (Zd ) then A−1 ∈ Cv1 ⊗v2t , for some t > 0. In [7] (see also [13, Remark 1]) it is shown that for any exponential weight vt (k) = et|k| there exists a banded matrix A which is invertible as a bounded T operator on `2 (Z) and yet A−1 ∈ / Cvt . As A is banded, we have A ∈ s>0 Cvs . This shows that the off-diagonal exponential decay of the inverse matrix strongly depends on the matrix itself and not only on the off-diagonal decay of the entries of A. Recall that given an algebra X with unit e (no topology is considered) the spectrum of an element x in X, denoted by σX (x), is the 10

set of those complex numbers λ for which the element λe − x is not invertible. X is called a locally m-convex algebra if it is endowed with a locally convex topology defined by a system of submultiplicative seminorms (qi )i∈I . Normed algebras as well as countable inductive limits of normed algebras are locally m-convex algebras (see [1, Theorem 2.2] or [8] for an easier proof). In the commutative case, this was shown in [2, Proposition 12]. In particular given (vn )n∈N a decreasing sequence of submultiplicative weights on Rd the spaces k 1 :=

[

`1(vn ) (Zd )

n∈N

and [

Cvn

n

equipped with the corresponding inductive limit topologies are locally m-convex algebras. Given a locally m-convex algebra X, its spectrum, denoted by Spec(X), is the set of all non-trivial, multiplicative, continuous and linear functionals ϕ : X → C. If x ∈ X is invertible then ϕ(x) 6= 0 for each ϕ ∈ Spec(X). By [2, Theorem 1] the converse holds if X is a countable inductive limit of commutative Banach algebras. Consequently, in this case {ϕ(x) : ϕ ∈ Spec(X)} = σX (x). In particular, given a ∈ k1 , σk1 (a) = {ϕ(a) : ϕ ∈ Spec(k1 )} . T 1 On the other hand, it is clear that Spec(k1 ) = ∞ n=1 Spec(`(vn ) ). For m = 1, . . . , d, Jm ∈ Zd is the vector with all coordinates zero except the m-th coordinate equal to 1. In our next result, we will use that for each n, the canonical basis {ej : j ∈ Zd } in `1(vn ) can be obtained from the eJm , m = 1,  . . . ,d, and their inverses, by convolution. Therefore, every ϕ ∈ Spec `1(vn ) is completely determined by ξ = (ϕ(eJ1 ), . . . , ϕ(eJd )) ∈ Cd . We recall that, given two algebras X ⊂ Y with a common unit, X is inverse closed in Y if and only if σX (x) = σY (x) for all x ∈SX. In particular, under the assumptions of Theorem 2.1, S every A ∈ n Cvn 2 has the same spectrum in the algebras B (` ) and n Cvn . Proposition 2.6. Let (vn )n∈N be a decreasing sequence of weights on Rd that do not satisfy GRS-condition but (0.3). Then there is A ∈ Cv1 such that σB(`2 ) (A) 6= σCvn (A) for each n ∈ N. 11

Proof. We may assume that vn are weights on Zd . Let a ∈ `1(v1 ) (Zd ) be a sequence with all the coordinates positive and let A be the matrix of the convolution operator b 7→Sa ∗ b on `2 (Zd ). The spectrum of A in the locally m-convex algebra n Cvn coincides with the S spectrum of a in the commutative locally m-convex algebra k1 := n `1(vn ) which, according to [2, Theorem 1], is given by ) ( ∞ \ σk1 (a) = ϕ(a) : ϕ ∈ Spec(`1(vn ) ) . n=1

Spec(`1(vn ) )

For ϕ ∈ we have ϕ(a) = j∈Zd aj ξ j where ξ ∈ Cd is given by ξm = ϕ(eJm ), m = 1, . . . , d. It easily follows that P

−1 rn,m ≤ |ξm | ≤ rn,m

where rn,m = lim vn (`Jm )1/` . `→∞

From the condition (0.3) and Theorem 2.1 we conclude that   X  j d σB(`2 ) (A) = σk1 (a) = a ξ :ξ∈T .  d j  j∈Z

On the other hand, for a fixed n ∈ N, there is k ∈ Zd such that lim vn (`k)1/` > 1. From

`→∞

vn (`k) ≤

d Y

vn (`km Jm )

m=1 1/`

we get rn,m = lim vn (`Jm )

> 1 for some m = 1, . . . , d. For simplicity  −1 we assume m = 1 and denote Un = z ∈ C : rn,1 < |z| < rn,1 . Then for every z ∈ Un we define the element of Spec(`1(vn ) ) given by X ϕz (b) = bj z j1 , b ∈ `1(vn ) , `→∞

j∈Zd

that is, ϕz is the unique element in the spectrum of `1(vn ) satisfying ϕz (eJ1 ) = z and ϕz (eJm ) = 1 for m 6= 1. Consequently, h(Un ) ⊂ σ`1(v ) (a) = σCvn (A) n

being h the holomorphic function ! h(z) = ϕz (a) =

X j∈Zd

aj z j1 =

X

X

k∈Z

s∈Zd−1

ak,s z k , z ∈ Un .

Therefore σCvn (A) has non empty interior, from where it follows σB(`2 ) (A) 6= σCvn (A) for any n ∈ N.  According to Theorem 2.1, for every λ ∈ / σB(`2 ) (A) there is m ∈ N such that λ ∈ / σCvm (A). The previous proposition shows that, in general, m depends on λ. 12

Remark 2.7. Arguing as in the proof of Proposition 2.6, if condition (0.3) is violated then we may assume that lim vn (`J1 )1/` > ρ > 1

`→∞

for all n ∈ N. Then, proceeding as in [13, Remark 1], there exists a banded matrix S A which is invertible as a bounded operator on `2 (Zd ) and yet A−1 ∈ / n∈N Cvn . Hence the condition (0.3) in Theorem 2.1 is also necessary. The next result will be needed to prove the main result of Section 3. Lemma 2.8. Let (vn )n∈N be a decreasing sequence of submultiplicative weights on Rd such that (0.3) holds. Let A ∈ Cvn for some n ∈ N, and let K ⊂ C be compact with K ∩ σB(`2 ) (A) = ∅. Then there exists m ∈ N such that (zI − A)−1 ∈ Cvm for every z ∈ K. Furthermore, the map z 7→ (zI − A)−1 , from K to Cvm is continuous. T Proof. According to the proof of Theorem 2.1, A ∈ v∈V Cv , and each weight v ∈ V satisfies GRS-condition. As σB(`2 ) (A) = σCv (A) for every v ∈ V , the map K → Cv , z 7→ (zI−A)−1 is well defined and continuous. Hence { (zI − A)−1 ; z ∈ K } is a compact set in Cv for every v ∈ V . For every z ∈ K and j ∈ Zd we put fz (j) := sup |bz (k, k − j)|, k∈Zd

where bz (j, k) are the matrix elements ofS (zI −A)−1 . Then T { f1z ; z ∈ K } 1 is a bounded set in the inductive limit n∈N `(vn ) = v∈V `(v) . We can apply [6, Theorem 2.3] to conclude that { fz ; z ∈ K } is a bounded subset of `1(vm ) for some m ∈ N. Consequently { (zI − A)−1 ; z ∈ K } is contained in Cvm . After enlarging m if it is necessary we also have that { (zI − A) ; z ∈ K } is contained in Cvm . Since Cvm is a Banach algebra, the result follows.  Apart from the classes Cv of convolution dominated matrices, there are other ways to measure the off-diagonal decay. We now concentrate on strict off-diagonal decay. Definition 2.9. Let v be a submultiplicative weight on Rd . (a) Av is the set of all matrices A = (a(j, k))j,k∈Zd such that kAkAv := sup |a(j, j − k)|v(k) < ∞. j,k∈Zd

(b) A0v consists of those matrices A ∈ Av such that ! lim

k→∞

sup |a(j, j − k)| v(k) = 0. j∈Zd

13

As in the proof of Theorem 2.1, for a matrix A = (a(j, k))j,k∈Zd we denote by dA , the sequence with entries dA (k) = supj∈Zd |a(j, j − k)|, 0 k ∈ Zd . Then A ∈ Av if and only if dA ∈ `∞ (v) , while A ∈ Av if and only if dA ∈ c0 (v). The submultiplicative weight v on Zd is called sub-convolutive if 1/v ∈ `1 and there is C > 0 such that v1 ∗ v1 ≤ Cv . By [13, Lemma 1. (b)] if v is sub-convolutive then `∞ v is a Banach algebra for convolution and Av is a Banach algebra of bounded operators on `2 (Zd ). The weight τs (k) := (1 + |k|)s is sub-convolutive in Zd when s > d. Moreover, if u is an arbitrary submultiplicative weight and w is sub-convolutive, then v := uw is also sub-convolutive. In fact, since u(j)−1 u(k − j)−1 ≤ u(k)−1 , we get X u(j)−1 u(k − j)−1 1 1 1 1 ( ∗ )(k) = ≤ u(k)−1 ( ∗ )(k). v v w(j)w(k − j) w w j In particular, v = uτs is sub-convolutive for s > d. We omit the proof of the following result, since the first part is contained in [3, Theorem 2], and the second part follows from [13, Corollary 4.2]. Theorem 2.10. (a) Assume that v is sub-convolutive and satisfies the GRS-condition. Then A0v is inverse closed in B(`2 (Zd )). (b) Let u be a weight with GRS-condition and s > d. Then, for v = uτs , Av is inverse closed in B(`2 (Zd )). We aim to obtain a version of Theorem 2.1 for the matrix algebras Av and for this we need to require that the sequence (vn )n∈N satisfies the so-called condition (D). Definition 2.11. The sequence of weights (vn )n∈N is said to satisfy condition (D) if there exists an increasing sequence (Im )m∈N of subsets Im of I = Zd such that v` (j) (a) for each m ∈ N there is nm ∈ N with inf > 0 ∀ ` > nm . j∈Im vnm (j) (b) for each n ∈ N and each J ⊂ I with J ∩ (I \ Im ) 6= ∅ for all vn0 (j) m ∈ N there exists n0 > n such that inf = 0. j∈J vn (j) The condition (D) was introduced in [5] by Bierstedt and Meise to address the question under which conditions the strong dual of certain Fr´echet spaces of sequences can be represented as a countable inductive limit of Banach spaces. As a concrete example we could consider an arbitrary weight µ on Z and a decreasing sequence (λn )n∈N of weights λn+1 (j) on Z such that lim = 0 for every n ∈ N. Then the sequence j→∞ λn (j) 14

(vn )n∈N of weights on Z2 , defined by vn (j, k) = λn (j)µ(k), satisfies condition (D): take Im = {(j, k) ∈ Z2 : |j| ≤ m}. Theorem 2.12. Let s > d and vn = un τs , n ∈ N, where (un )n∈N is a decreasing sequence of weights on Zd , and assume that (vn )n∈N satisfies condition (D) and (0.3). If A ∈ Avn for some n ∈ N and A is invertible on `2 (Zd ), then A−1 ∈ Avm for some m ∈ N. Proof. We shall argue in a similar way as in the proof of Theorem 2.1, after the weighted `1 norms have been replaced by weighted `∞ norms. Clearly (vn )n∈N satisfies (D) condition if and only if (un )n∈N does. Let   u(k) d W = u : Z → R+ ; sup n such that vm /vn ∈ `1 (Zd ),

(2.4)

then [

Avn =

n

[

Cvn .

n

Therefore, the main interest of the previous result lies in the case that condition (2.4) is not satisfied, since otherwise it is covered by Theorem 2.1. Corollary 2.14. Let vn = un w where w is a sub-convolutive weight and (un )n∈N is a decreasing sequence of submultiplicative weights on Zd such that (0.3) holds and vn+1 (j) = 0 ∀ n ∈ N. j→∞ vn (j) lim

If A ∈ Avn for some n ∈ N and A is invertible on `2 (Zd ), then A−1 ∈ Avm for some m ∈ N. 15

Proof. TheSsequence (vn )n∈N satisfies condition (D). On the other hand S 0 n Avn = n Avn and we can proceed as before but using part (a) of Theorem 2.10.  3. Algebras of pseudodifferential operators The aim of this section is to apply some of the results previously obtained to the study of certain algebras of pseudodifferential operators, defined in terms of weights lacking GRS-condition. If v is a submultiplicative weight on R2d , then Θv is submultiplicative 2d 2 d on R4d . By [14], Opw (M∞,1 (Θv) (R )) is inverse closed in B(L (R )), if and only if v satisfies the GRS-condition. The proof relays on the almost diagonalization of pseudo-differential operators with respect to Gabor frames. For a refined version of the spectral invariance see [15]. By combining this property with (1.5) and Proposition 1.4, it follows that ∞,1 (R2d )) is inverse closed in B(L2 (Rd )), if t ∈ R is fixed, then Opt (M(Θv) if and only if v satisfies the GRS-condition. We want to analyze what can be said about the inverse of Opt (a) in ∞,1 the limit case that a ∈ M(Θv) (R2d ) and v lacks the GRS-condition. In order to do so, it suffices to consider the case of Weyl quantization and 2d letting a ∈ M∞,1 (Θv) (R ). The next lemma is a reformulation of Lemma 3.1 in [10]. For X = (x, ξ) ∈ R2d , π(X) denotes the corresponding time-frequency shift π(X)f (y) = eihy,ξi f (y − x),

X = (x, ξ).

Lemma 3.1. Let v be submultiplicative, φ ∈ Mv1 (Rd ) and let Φ = Wφ,φ . 2d Given a ∈ M∞,1 (Θv) (R ), |(Opw (a)(π(X)φ), π(Y )φ)| = (2π)−d/2 |(VΦ a)((Y + X)/2, (Y − X)/2)| . The first part of the next result is [11, Theorem 13.1.1], while the second part is Theorem 3.2 in [10], except that we have relaxed the assumptions on the involved weight v. More precisely, here it is assumed that v is submultiplicative, while in [10] it is in addition assumed that v should satisfy the GRS condition. This permits the identification of 2d 2d elements a ∈ M∞,1 (Θv) (R ) with matrices, indexed by Z , in Cv˜ for some weight v˜ defined in terms of v. In order to simplify the notation, we let J = Zd × Zd and, whenever α, β > 0 are fixed, we put XJ = (αk, βn) for each j = (k, n) ∈ J. 1 Theorem 3.2. Let v be submultiplicative on R2d and let φ ∈ M(v) (Rd )\ {0}. Then the following is true: (a) There is δ > 0 such that, for every 0 < α, β ≤ δ, X Sf = hf, π(Xj )φi π(Xj )φ j∈J

16

1 1 is invertible on M(v) (Rd ) and γ 0 := S −1 φ ∈ M(v) (Rd ).

(b) The following conditions are equivalent: 2d (1) a ∈ M∞,1 (Θv) (R ); (2) a ∈ S 0 (R2d ) and there exists a function H ∈ L1(v) (R2d ) such that |(Opw (a)(π(X)φ), π(Y )φ)| ≤ H(X − Y ),

X, Y ∈ R2d ;

(3) a ∈ S 0 (R2d ) and |(Opw (a)(π(Xj )φ), π(Xk )φ)| ≤ h(k − j), j, k ∈ J, where X

h(j)v(Xj ) < ∞.

j∈J

Proof. The statement (a) is the content of [11, Theorem 13.1.1]. Concerning part (b), the equivalence between (1) and (2) follows by the same arguments as for the proof of Theorem 3.2 in [10], but using Lemma 3.1. For the equivalence between (2) and (3), the argument from [10] still works since we already have that the dual window γ 0 := S −1 φ 1 (Rd ) by part (a).  is in M(v) In [10], the condition GRS is used to prove that for any lattice Λ in R2d with the property that G(φ, Λ) is a Gabor frame we have that 1 the dual window γ 0 is also in M(v) . However, in the previous theorem we are not dealing with arbitrary Gabor frames but only with frames associated to a lattice Λ = αZd × βZd for α and β small enough so that the conclusion of part (a) holds. Remark 3.3. Let v be a submultiplicative weight on R2d and Λ = αZd × βZd a lattice chosen according to Theorem 3.2 (a). Then the equivalence between conditions (1)-(3) in Theorem 3.2 (b) holds after replacing v by any submultiplicative weight dominated by v. Condition (3) in Theorem 3.2 means that the matrix A with entries a(j, j0 ) = hOpw (a)(π(Xj0 )φ), π(Xj )φi , j, j0 ∈ J, is in the algebra Cv˜, where v˜(j) = v(Xj ). We are now able to formulate and prove the main result of the present section. Theorem 3.4. Let t ∈ R, and let (vn )n∈N be a decreasing sequence of submultiplicative weights on R2d such that inf lim vn (`X)1/` = 1, n `→∞

∀ X ∈ R2d .

∞,1 If a ∈ M(Θv (R2d ) for some n ∈ N and T = Opt (a) is invertible on n) L2 (Rd ), then there exists m ∈ N such that T −1 is equal to Opt (b) for ∞,1 some b ∈ M(Θv (R2d ). m)

17

Proof. By Proposition 1.4 we may assume that t = 1/2. Furthermore, by (1.5), the result follows if we prove that T −1 = Opw (b) for some ∞,1 2d 2d b ∈ M∞,1 (Θvm ) (R ) when a ∈ M(Θvn ) (R ). 1 According to Theorem 3.2 there are φ ∈ M(v (Rd ) and α, β > 0 such n) that G(φ, α, β) = {π(Xj )φ, j ∈ J} is a frame for L2 (Rd ), and γ 0 , the canonical dual window of φ, belongs 1 (Rd ). We now define the matrix A with entries to M(v n) a(j, j0 ) = hT π(Xj0 )φ, π(Xj )φi ,

j, j0 ∈ J.

According to Theorem 3.2, A ∈ Cv˜n where v˜n (j) = vn (Xj ). In particular A defines a bounded operator on `2 (J). Moreover, denoting by Cφ and Cγ 0 the analysis operators associated to the frame G(φ, α, β) and its canonical dual frame, it turns out that Cφ ◦ T = A ◦ Cγ 0 , Cφ (L2 (Rd )) = Cγ 0 (L2 (Rd )) and A maps the range of Cφ into itself with ran Cφ⊥ ⊂ ker A (see [10, Lemma 3.4]). We now consider b ∈ S 0 (R2d ) the unique tempered distribution such that Opw (b) = T −1 and define the matrix B with entries

 b(j, j0 ) = T −1 π(Xj0 )γ 0 , π(Xj )γ 0 , j, j0 ∈ J. We claim that B is a pseudoinverse for A in B(`2 (J)). In fact, we consider R := Cφ (L2 (Rd )) = Cγ 0 (L2 (Rd )) and observe that also Cγ 0 ◦ T −1 = B ◦ Cφ . Then (1) A ◦ B ◦ Cφ f = A ◦ Cγ 0 ◦ T −1 f = Cφ f , (2) B ◦ A ◦ Cγ 0 f = B ◦ Cφ ◦ T f = Cγ 0 , (3) R⊥ = ker A. Also ker B = R⊥ . Consequently B is a pseudoinverse of A and the claim is proved. Now, B is given by Z 1 1 B= (zI − A)−1 dz, 2πi Γ z for a suitable closed path Γ surrounding σB(`2 (J)) (A)\{0}. From Lemma 2.8, the previous integral defines an element in Cv˜m for some m > n. 2d After applying Theorem 3.2 once again we obtain b ∈ M∞,1 (Θvm ) (R ).  The next example is based in [12]. It shows that, given a symbol 2d 2 d a ∈ M∞,1 (Θvn ) (R ) such that Opt (a) is invertible in B(L (R )), the decay of the symbol of the inverse operator does not only depend on the weight vn but also on a. 2 Let ϕ denote the Gaussian ϕ(X) = e−|X| , with X = (x, ξ) ∈ R2 , and consider cZ (X) := e2iσ(Z,X) ,

Z = (z, ζ) ∈ R2 . 18

By straight-forward computations, it follows that (Vϕ cZ )(X, Y ) = e2iσ(Y +Z,X) ϕ(Y + Z) = e2iσ(Y,X) cZ (X)ϕ(Y + Z), and that the Weyl operator with symbol cZ is Opw (cZ ) = e2i(x−z)ζ SZ , where SZ denotes the translation (SZ f )(x) = f (x − 2z). In particular, (3) gives kcZ kM∞,1 ≤ v(Z)kϕkL1(v) ,

(3.1)

(Θv)

when ϕ is used as window function in the definition of the M∞,1 (Θv) norm. Example 3.5. Let vn (Y ) = expT((|y| + |η|)/n), n ∈ N. We claim that ∞,1 (R2 ) such that Opw (a) is for every m ∈ N there exist a ∈ n∈N M(Θv n) 2 invertible on L2 (R) but (Opw (a))−1 ∈ / Opw (M∞,1 (Θvm ) (R )). In fact, let m be fixed, Zk = (k, 0) for k ≥ 1, ε > 0, and let a = 1 − e−ε cZ1 . Obviously, \ ∞,1 a∈ M(Θv (R2 ) n) n∈N

in view of (3.1), and the Weyl operator Opw (a) = Id − e− SZ1 is invertible on L2 (R) with inverse w

Op (a)

−1

=

∞ X

−kε

e

SZk 1

k=0

=

∞ X

e−kε SZk .

k=0

Hence, the Weyl symbol of Opw (a)−1 is given by b=

∞ X

e−kε cZk .

k=0

By (3) we have 2iσ(Y,X)

(Vϕ b)(X, Y ) = e

∞ X

e−kε cZk (X)ϕ(Y + Zk ),

k=0

which gives, after selecting X = (0, π), sup |(Vϕ b)(X, Y )| = X∈R2

∞ X k=0

19

e−kε ϕ(Y + Zk ).

By applying the L1(vm ) norm on the latter equality we get kbkM∞,1

(Θvm )

=

∞ X

−kε

Z

e

ϕ(Y + Zk )vm (Y ) dY

k=0

≥ C1

∞ X k=0

e

−kε

Z e

|Y |/m

|Y +Zk |≤1

dY ≥ C2

∞ X

e−kε+k/m ,

k=0

for some positive constants C1 and C2 . By choosing ε < 1/m, it follows that the sum on the right-hand side diverges. Consequently, b∈ / M∞,1 2 (Θvm ) , which proves the assertion. Acknowledgement. The research of the first two authors was partially supported by MEC and FEDER Project MTM2010-15200 and GVA Prometeo II/2013/013. References [1] M. Akkar, C. Nacir: Structure m-convexe d’une alg`ebre limite inductive localement convexe d’alg`ebres de Banach, Rend. Sem. Mat. Univ. Padova 95 (1996), 107-126. [2] A. Arosio: Locally convex inductive limits of normed algebras, Rend. Sem. Mat. Uni. Padova 51 (1974), 332-359. [3] A.G. Baskakov: Asymptotic estimates for the entries of the matrices of inverse operators and harmonic analysis. Siberian Mathematical Journal 38 (1997), 10-22. [4] K.D. Bierstedt, J. Bonet: Stefan Heinrich’s density condition for Fr´echet spaces and the characterization of the distinguished K¨ othe echelon spaces, Math. Nachr. 135 (1988), 149-180. [5] K.D. Bierstedt, R. Meise: Distinguished echelon spaces and the projective description of weighted inductive limits of type Vd C(X), Aspects of Mathematics and its applications, North-Holland Math. Library 34, Amsterdam, 1986, pp. 169-226. [6] K.D. Bierstedt, R.G. Meise, W.H. Summers: K¨ othe sets and K¨ othe sequence spaces in: Functional analysis, holomorphy and approximation theory (Rio de Janeiro, 1980), North-Holland Math. Stud. 71, Amsterdam NewYork, 1982, pp. 27-91. [7] S. Demko, W.F. Moss, P.W. Smith: Decay rates for inverses of band matrices, Math. Comp. 43 (1984), 491-499. [8] S. Dierolf, J. Wengenroth: Inductive limits of topological algebras, Linear Topol. Spaces Complex Anal. 3 (1997), 45-49. [9] I. M. Gelfand, G. E. Shilov: Generalized functions, I-III, Academic Press, NewYork London, 1968. [10] K. Gr¨ ochenig: Time-frequency analysis of Sj¨ ostrand’s class, Rev. Mat. Iberoam. 22 (2006), 703-724. [11] K. Gr¨ ochenig: Foundations of Time-Frequency Analysis, Birkh¨auser, 2001. [12] K. Gr¨ ochenig: Wiener’s lemma: Theme and variations. An introduction to spectral invariance and its applications In: Four short courses on harmonic analysis. Wavelets, frames, time-frequency methods, and applications to signal 20

[13]

[14]

[15]

[16] [17] [18] [19] [20] [21]

and image analysis, Applied and Numerical Harmonic Analysis, Birkh¨auser, Basel, 2010, pp. 175-244. K. Gr¨ ochenig, M. Leinert: Symmetry and inverse-closedness of matrix algebras and functional calculus for infinite matrices, Trans. Amer. Math. Soc. 358 (2006), 2695-2711. K. Gr¨ ochenig, Z. Rzeszotnik: Banach algebras of pseudodifferential operators and their almost diagonalization, Ann. Inst. Fourier (Grenoble) 58 (2008), 2279-2314. K. Gr¨ ochenig, J. Toft: The range of localization operators and lifting theorems for modulation and Bargmann-Fock spaces, Trans. Amer. Math. Soc. 365 (2013), 4475-4496. L. H¨ ormander: The Analysis of Linear Partial Differential Operators, I-III, Springer-Verlag, Berlin Heidelberg NewYork Tokyo, 1983, 1985. S. Jaffard: Propri´et´es des matrices bien localis´ees pr`es de leur diagonale et quelques applications, Ann. Inst. Henri Poincar´e 7 (1990), 461-476. H. Komatsu: Ultradistributions I, J. Fac. Sci. Univ. Tokyo Sect. IA Math. 20 (1973), 25-205. S. Pilipovi´c: Tempered ultradistributions, Boll. Un. Mat. Ital. B. 2 (1988), 235251 T. Strohmer: Pseudodifferential operators and Banach algebras in mobile communications, Appl. Comput. Harmon. Anal. 20 (2006), 237-249. J. Toft: The Bargmann transform on modulation and Gelfand-Shilov spaces, with applications to Toeplitz and pseudo-differential operators, J. PseudoDiffer. Oper. Appl. 3 (2012), 145-227.

´ lisis Matema ´ tico, Universidad de Valencia, VaDepartamento de Ana lencia, Spain E-mail address: [email protected], [email protected] ¨ xjo ¨ , Sweden Department of Mathematics, Linnæus University, Va E-mail address: [email protected]

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