Spectral Properties for Matrix Algebras

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 $\ell ^2$ we analyze the decay of its inverse matrix entries in the case where the matrix algebra is not inverse closed in ${\mathcal B} (\ell ^2),$ the Banach algebra of bounded operators on $\ell ^2.$ To this end we consider a condition on sequences of weights which extends the notion of GRS-condition. 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.

1 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 $\ell ^2=\ell ^2(\mathbb Z^{d})$) is inverse closed in ${\mathcal B}(\ell ^2).$ Recall that a Banach algebra of matrices on $\ell ^2$ is called inverse closed in ${\mathcal B}(\ell ^2)$ if whenever an element in the algebra is invertible in ${\mathcal B}(\ell ^2)$, its inverse is also in the algebra. (For other definitions, see Sect. 2).

In the paper we especially consider invertibility questions for the Banach algebra $\mathcal C_v=\mathcal C_v(\mathbb Z^{d})$, the set of all matrices $(a(j,k))_{j,k\in \mathbb Z^{d}}$ which satisfy

$$\begin{aligned} \Vert A\Vert _{\mathcal C_v}\equiv \sum _{k\in \mathbb Z^{d}} \sup _{j\in \mathbb Z^{d}}|a(j,j-k)|v(k)<\infty , \quad A = (a(j,k))_{j,k\in \mathbb Z^{d}}. \end{aligned}$$

(1.1)

Here $v$ is a weight on $\mathbb R^{d}$ which is symmetric and submultiplicative.

We remark that there are strong links between $\mathcal C_v$ and $\mathrm{Op }_t(M^{\infty ,1}_{(v)})$, the set of all pseudo-differential operators $\mathrm{Op }_t(a)$ with the symbol $a$ in the modulation space (or the weighted Sjöstrand class) $M^{\infty ,1} _{(v)}=M^{\infty ,1} _{(v)}(\mathbb R^{2d})$.

There are several situations when matrices in $\mathcal C_v$ or pseudo-differential operators in $\mathrm{Op }_t(M^{\infty ,1}_{(v)})$ appear. For example, matrices in $\mathcal C_v$ appear when considering discrete slow time-varying systems, and when discretizing non-stationary filters in time-frequency analysis and signal processing (see [11] and the references therein). In the latter situation, the link between $\mathrm{Op }_t(M^{\infty ,1}_{(v)})$ and the matrices in $\mathcal C_v$ is essential. Another example can be found in [20], where Strohmer when modeling mobile wireless channels, considers pseudodifferential operators in the class $\mathrm{Op }_t(M^{\infty ,1}_{(v)})$.

The set $\mathcal C_v$ is contained in $\mathcal B(\ell ^2)$. Moreover, $\mathcal C_v$ is inverse closed in $\mathcal B(\ell ^2)$, if and only if $v$ satisfies the GRS-condition (Gelfand–Raikov–Shilov condition), i.e.,

$$\begin{aligned} \lim _{\ell \rightarrow \infty }v(\ell x)^{1/\ell }=1,\quad \text {when}\quad x\in \mathbb R^{d} \end{aligned}$$

(1.2)

(cf. [11, Corollary 5.31]). A similar fact is true when $\mathcal C_v$ and $\ell ^2$ are replaced by $\mathrm{Op }_t(M^{\infty ,1}_{(v)})$ and $L^2$, respectively.

We note that the set of weights on $\mathbb R^{d}$ which satisfy the GRS-condition is a monoid with respect to the pointwise multiplication, and contains important weights of the form

$$\begin{aligned} x\mapsto C(1+|x|)^{t}\quad \text {and}\quad x\mapsto e^{t|x|^s}, \end{aligned}$$

when $C\ge 1, t\ge 0$ and $0\le s<1$.

Exponential growth in some direction is the reason why GRS-condition can fail. In the limit case where $v(x) = e^{t|x|}$ some information about the entries of the inverse of a matrix $A\in {\mathcal C}_v$ was obtained by Jaffard [17] (see also Baskakov [3]), despite the fact that ${\mathcal C}_v$ is not inverse closed in ${\mathcal B}(\ell ^2)$. In fact, it was proved in [17] that the existence of $A^{-1}\in {\mathcal B}(\ell ^2)$ implies

$$\begin{aligned} \sum _{{k}} \sup _{j}|c(j,j-k)|e^{s |k|}<\infty ,\quad A^{-1}=(c(j,k))_{j,k\in \mathbb Z^{d}}, \end{aligned}$$

for some $0<s<t$.

In several situations, the lack of inverse closedness in ${\mathcal B}(\ell ^2)$, 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 Sect. 2 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 Sect. 3 we prove that $\mathcal C_v$ possess a weaker form of inverse closedness in ${\mathcal B}(\ell ^2)$ under this sequence version of GRS-condition.

More precisely, we consider decreasing sequences $(v_n)_{n\in \mathbb N}$ of (submultiplicative) weights $v_n$ on $\mathbb R^{d}$ satisfying

$$\begin{aligned} \inf _n \left( \lim _{\ell \rightarrow \infty } v_n(\ell x)^{1/\ell } \right) =1, \quad \text {when}\quad x\in \mathbb R^{d}. \end{aligned}$$

(1.3)

In Theorem 3.1 we then prove that

$$\begin{aligned} \bigcup _{n}{\mathcal C}_{v_{n}} \end{aligned}$$

(1.4)

is equal to an (uncountable) intersection of algebras ${\mathcal C}_v$, where each $v$ satisfies the GRS condition, by using the projective description technique due to Bierstedt, Meise and Summers (cf. [6]). Since any ${\mathcal C}_v$ is inverse closed in ${\mathcal B}(\ell ^2)$, it follows that the union (1.4) is inverse closed in ${\mathcal B}(\ell ^2).$ In particular, if $A\in \mathcal C_{v_n}$ for some $n$ is invertible on $\ell ^2$, then its inverse $A^{-1}$ belongs to $\mathcal C_{v_m}$ for some $m$.

If $v_n=v$ is independent of $n$, then (1.3) agrees with the GRS-condition. Hence Theorem 3.1 in this case means that $\mathcal C_v$ is inverse closed in ${\mathcal B}(\ell ^2)$ when $v$ satisfies the GRS-condition.

If instead $v_n(k)=e^{t|x|/n}$, for some $t>0$, then Theorem 3.1 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 3.5.

For a submultiplicative weight $\omega$ on $\mathbb R^{2d}$, it is convenient to let $\Theta$ be the extension operator to weights on $\mathbb R^{4d}$, i.e., we put $(\Theta \omega )(X,Y) = \omega (Y)$ for $X,Y\in \mathbb R^{2d}$.

According to Gröchenig and Rzeszotnik in [14], $\mathrm{Op }_t(M^{\infty ,1}_{(\Theta v)} (\mathbb R^{2d}))$ is inverse closed in ${\mathcal B}(L^2(\mathbb R^{d}))$ if and only if $v$ satisfies GRS-condition. In Sect. 4 we apply our results to analyze the behavior of inverses of pseudo-differential operators in $\mathrm{Op }_t(M^{\infty ,1}_{(\Theta v)}(\mathbb R^{2d}))$, where the GRS condition might be violated for the weight $v$.

2 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.

2.1 Weights

First we consider weight functions. Let $\mathbb X$ denote either $\mathbb X =\mathbb R^{d}$ or $\mathbb X=\mathbb Z^{d}.$ A weight on $\mathbb X$ is a positive and locally bounded function on $\mathbb X$. The weight $v$ on $\mathbb X$ is called submultiplicative, if it is even and satisfies

$$\begin{aligned} v(x+y)\le v(x)v(y),\quad \text{ for }\quad x,y \in \mathbb X. \end{aligned}$$

For two positive and locally bounded functions $f$ and $g$ on $\mathbb X, f\lesssim g$ means that there exists $C > 0$ such that $f(x) \le C g(x)$ for all $x\in \mathbb X.$ It is well-known that

$$\begin{aligned} v(x)\lesssim e^{t|x|} \end{aligned}$$

for some $t>0$, when $v$ is submultiplicative. In the sequel, $v$ and $v_j$ always denote submultiplicative weights on $\mathbb R^{d}$ or on $\mathbb Z^{d}$.

Let $\omega$ be a weight on $\mathbb R^{d}$. Then $\omega$ is called $v$-moderate if

$$\begin{aligned} \omega (x+y)\lesssim \omega (x)v(y),\quad \text{ for } x,y \in \mathbb R^{d}. \end{aligned}$$

(2.1)

A weight $\omega$ which is $v$-moderate for some $v$ is also called moderate. If $\omega$ is $v$-moderate, then

$$\begin{aligned} v(x)^{-1}\lesssim \omega (x)\lesssim v(x). \end{aligned}$$

If in addition $v$ satisfies the GRS-condition [cf. (1.2)], then $v$ is called a GRS-weight.

More generally, the sequence $( v_n) _{n\in \mathbb N}$ of submultiplicative weights on $\mathbb R^{d}$ is called a GRS-sequence of weights, if (1.3) is fulfilled. By letting $v_n=v$ 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 $\omega$ is a weight on $\mathbb R^{d}$, which is $v$-moderate for some $v$, then $\ell ^p_{(\omega )}=\ell ^p_{(\omega )}(\mathbb Z^{d})$ denotes the set of all sequences $a$ on $\mathbb Z^{d}$ such that $j\mapsto a_j\omega (j)$ belongs to $\ell ^p(\mathbb Z^{d})$.

2.2 Matrices

Next we consider matrices indexed on $\mathbb Z^{d}$. Let $v$ be submultiplicative on $\mathbb R^{d}$. We recall that ${\mathcal C}_v ={\mathcal C}_v(\mathbb Z^{d})$ consists of all matrices $A = \left( a_{i,j}\right) _{(i,j)\in \mathbb Z^{d} \times \mathbb Z^{d}}$ such that (1.1) holds. In particular, the off-diagonal decay of the entries of the matrix $A$ is controlled by the weight $v$.

We note that ${\mathcal C}_v$ is a Banach $*$-algebra with respect to matrix multiplication and taking the adjoint matrix as involution. As ${\mathcal C}_v$ is a subalgebra of the Banach algebra ${\mathcal B}(\ell ^2)$ it makes sense to consider when ${\mathcal C}_v$ is inverse closed in ${\mathcal B}(\ell ^2).$ This is the case if and only if $v$ satisfies the GRS-condition.

Remark 2.1

Any weight on $\mathbb R^{d}$ is also a weight on $\mathbb Z^{d}$. On the other hand, let $\omega$ be a weight on $\mathbb R^{d}$ which is $v$-moderate for some $v$, and let $\omega _0$ be the weight on $\mathbb R^{d}$, given by

$$\begin{aligned} \omega _0(x)=\omega (k),\quad \text {when}\quad k\in \mathbb Z^{d},\ x\in k+Q. \end{aligned}$$

(2.2)

Here $Q$ is the unit cube such that $2Q=[-1,1)^d$. The condition (2.1) now implies that $\omega _0$ is equivalent to $\omega$, in the sense

$$\begin{aligned} \omega _0\lesssim \omega \lesssim \omega _0. \end{aligned}$$

Here we note that the classes $\ell ^p_{(\omega )}, \mathcal C_v, M^{\infty ,1}_{(v)}$ and other similar weighted spaces, do not change if the weights $\omega$ and $v$ are replaced by other equivalent ones. Hence, when investigating such weighted spaces it suffices to consider weights $\omega _0$ on $\mathbb R^{d}$, given by (2.2), where $\omega$ is a weight on $\mathbb Z^{d}$.

2.3 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 $0<h,s\in \mathbb R$ be fixed, and let $\mathcal S_{s,h}(\mathbb R^{d})$ be the set of all $f\in C^\infty (\mathbb R^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{\mathcal S_{s,h}}\equiv \sup \frac{|x^\beta \partial ^\alpha f(x)|}{h^{|\alpha | + |\beta |}\alpha !^s\, \beta !^s} \end{aligned}$$

is finite. Here the supremum should be taken over all $\alpha ,\beta \in \mathbb N_0^d$ and $x\in \mathbb R^{d}$. We have that $\mathcal S_{s,h}(\mathbb R^{d})$ is a Banach space which increases with $h$ and $s$ and is contained in $\fancyscript{S}(\mathbb R^{d})$, the set of all Schwartz functions on $\mathbb R^{d}$.

The Gelfand–Shilov space $\mathcal S_{s}(\mathbb R^{d})$ is the inductive limit of $\mathcal S_{s,h}(\mathbb R^{d})$. In particular,

$$\begin{aligned} \mathcal S_{s}(\mathbb R^{d}) = \bigcup _{h>0}\mathcal S_{s,h}(\mathbb R^{d}), \end{aligned}$$

(2.3)

and the Gelfand–Shilov distribution space $\mathcal S_{s}'(\mathbb R^{d})$ is the projective limit of $\mathcal S_{s,h}'(\mathbb R^{d})$. This implies that

$$\begin{aligned} \mathcal S_s'(\mathbb R^{d}) = \bigcap _{h>0}\mathcal S_{s,h}'(\mathbb R^{d}). \end{aligned}$$

(2.3')

We remark that in [9, 18, 19] it is proved that $\mathcal S_s'(\mathbb R^{d})$ is the dual of $\mathcal S_s(\mathbb R^{d})$ (also in topological sense).

Evidently, $\mathcal S_s(\mathbb R^{d})$ increases with $s$, and $\mathcal S_s'(\mathbb R^{d})$ decreases with $s$. If $s<1/2$, then $\mathcal S_s(\mathbb R^{d})$ is trivial. Otherwise $\mathcal S_s(\mathbb R^{d})$ is embedded and dense in $\fancyscript{S}(\mathbb R^{d})$, and $\fancyscript{S}'(\mathbb R^{d})$ is contained in $\mathcal S_s'(\mathbb R^{d})$.

We let the Fourier transform on $\fancyscript{S}'(\mathbb R^{d})$ be the linear and continuous map which takes the form

$$\begin{aligned} (\fancyscript{F}f)(\xi )= \widehat{f}(\xi ) \equiv (2\pi )^{-d/2}\int \limits _{\mathbb R^{d}} f(x)e^{-i\langle x,\xi \rangle }\, dx \end{aligned}$$

when $f\in L^1(\mathbb R^{d})$. For every $s\ge 1/2$, the Fourier transform is continuous and bijective on $\mathcal S_s(\mathbb R^{d})$, and extends uniquely to a continuous and bijective map on $\mathcal S_s'(\mathbb R^{d})$.

2.4 Modulation Spaces

Let $1/2\le s_0<s$, and let $\phi \in \mathcal S_{s_0}(\mathbb R^{d})\setminus \{0\}$ be fixed. Then the short-time Fourier transform $V_\phi f$ of $f\in \mathcal S_s'(\mathbb R^{d})$ with respect to the window function $\phi$ is the element in $\mathcal S_s'(\mathbb R^{2d}) \cap C^\infty (\mathbb R^{2d})$, defined by the formula

$$\begin{aligned} (V_\phi f)(x,\xi ):= (2\pi )^{-d/2}\left( f,\phi (\, \cdot \, -x)e^{i\langle \, \cdot \, ,\xi \rangle }\right) , \end{aligned}$$

where $(\, \cdot \, ,\, \cdot \, )$ is the (unique) extension of the $L^2$-form on $\mathcal S_{s_0}(\mathbb R^{d})$. If in addition $f$ is locally integrable, then $V_\phi f$ is given by

$$\begin{aligned} (V_\phi f)(x,\xi ) = (2\pi )^{-d/2}\int \limits _{\mathbb R^{d}} f(y)\overline{\phi (y-x)} e^{-i\langle y,\xi \rangle }\, dy. \end{aligned}$$

Let $\phi \in \mathcal S_{1/2}(\mathbb R^{d}), p,q\in [1,\infty ], v$ be submultiplicative on $\mathbb R^{2d}$, and let $\omega$ be a $v$-moderate weight on $\mathbb R^{2d}$. Then the modulation space $M^{p,q}_{(\omega )}(\mathbb R^{d})$ is the Banach space which consists of all $f\in \mathcal S_{1/2} '(\mathbb R^{d})$ such that

$$\begin{aligned} \Vert f\Vert _{M^{p,q}_{(\omega )}}:= \left( \;\;\int \limits _{\mathbb R^{d}} \left( \,\,\,\int \limits _{\mathbb R^{d}} \left| V_\phi f (x,\xi )\omega (x,\xi ) \right| ^p\, dx \right) ^{q/p}\, d\xi \right) ^{1/q} \end{aligned}$$

(2.4)

is finite (with obvious modifications when $p=\infty$ or $q=\infty$). We remark that the definition of $M^{p,q}_{(\omega )}(\mathbb R^{d})$ is independent of the choice of the window $\phi \in \mathcal S_{1/2} (\mathbb R^{d})\setminus \left\{ 0\right\}$, and different $\phi$ gives rise to equivalent norms. See e.g., [12, Proposition 11.4.2] or [21, Proposition 1.11]. Furthermore, if $s<1$ and $\omega$ is a moderate weight, then $M^{p,q}_{(\omega )}$ contains $\mathcal S_s$ and is contained in $\mathcal S_s'$ (see e.g., [21]).

For distributions defined on the phase space $\mathbb R^{2d}$, we may define modulation spaces in terms of the symplectic short-time Fourier transform, which is given by

$$\begin{aligned} ({\mathcal V}_\Phi a)(X,Y):= \pi ^{-d} \left( a,\Phi (\, \cdot \, -X)e^{2i \sigma (\, \cdot \, ,Y)}\right) , \end{aligned}$$

when $\Phi \in \mathcal S_{s_0}(\mathbb R^{2d})\setminus \left\{ 0\right\}$ is fixed, $a\in \mathcal S_s'(\mathbb R^{2d})$ and $X,Y\in \mathbb R^{2d}$. Here $\sigma$ is the symplectic form, given by $\sigma (X,Y) = \langle y,\xi \rangle - \langle x,\eta \rangle$ when $X = (x,\xi )\in \mathbb R^{2d}$ and $Y = (y,\eta )\in \mathbb R^{2d}$. Again we note that if in addition $a$ is locally integrable, then

$$\begin{aligned} ({\mathcal V}_\Phi a)(X,Y) = \pi ^{-d} \int \limits _{\mathbb R^{2d}}a(Z) \overline{\Phi (Z-X)}e^{2 i \sigma (Y,Z)}\, dZ. \end{aligned}$$

If $\omega$ is a $v$-moderate weight on $\mathbb R^{4d}$ for some submultiplicative weight $v$ on $\mathbb R^{4d}$, then the modulation space $\mathcal M^{p,q} _{(\omega )}(\mathbb R^{2d})$ 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

$$\begin{aligned} ({\mathcal V}_\Phi a)(X,Y) = 2^{d}(V_\Phi a)(\jmath (X,Y)), \end{aligned}$$

when $\jmath (X,Y) = (x,\xi ,-2\eta ,2y), X=(x,\xi )$ and $Y=(y,\eta )$, which implies that

$$\begin{aligned} \mathcal M^{p,q}_{(\omega \circ \jmath )}(\mathbb R^{2d}) = M^{p,q}_{(\omega )} (\mathbb R^{2d}), \end{aligned}$$

(2.5)

with equivalent norms.

2.5 Pseudo-Differential Operators

Next we recall some properties in pseudo-differential calculus. Let $s\ge 1/2, a\in \mathcal S_s (\mathbb R^{2d})$, and $t\in \mathbb R$ be fixed. Then the pseudo-differential operator $\mathrm{Op }_t(a)$ is a linear and continuous operator on $\mathcal S_s (\mathbb R^{d})$, given by

$$\begin{aligned} (\mathrm{Op }_t(a)f)(x) = (2\pi ) ^{-d}\int \limits _{{\mathbb R^{d}}}\int \limits _{{\mathbb R^{d}}} a((1-t)x+ty,\xi )f(y)e^{i\langle x-y,\xi \rangle }\, dyd\xi . \end{aligned}$$

(2.6)

For $t=0$, then $\mathrm{Op }_0(a)$ agrees with Kohn–Nirenberg or normal representation $a(x,D)$. If instead $t=1/2$, then $\mathrm{Op }_t(a)$ is the Weyl quantization, and is denoted by $\mathrm{Op }^w(a)$.

For general $a\in \mathcal S_s'(\mathbb R^{2d})$, the pseudo-differential operator $\mathrm{Op }_t(a)$ is defined as the continuous operator from $\mathcal S_s(\mathbb R^{d})$ to $\mathcal S_s'(\mathbb R^{d})$ with distribution kernel given by

$$\begin{aligned} K_{a,t}(x,y)=(2\pi )^{-d/2}(\fancyscript{F}_2^{-1}a)((1-t)x+ty,x-y), \end{aligned}$$

(2.7)

that is,

$$\begin{aligned} \langle \mathrm{Op }_t(a)f,g\rangle = \langle K_{a,t}, g\otimes f\rangle , \quad f,g \in \mathcal S_s (\mathbb R^{d}). \end{aligned}$$

Here $\langle \, \cdot \, , \, \cdot \, \rangle$ is the dual form between $\mathcal S_s$ and $\mathcal S_s'$, and $\fancyscript{F}_2F$ is the partial Fourier transform of $F(x,y)\in \mathcal S_s'(\mathbb R^{2d})$ with respect to the $y$ variable. This definition of $\mathrm{Op }_t (a)$ makes sense, since both of the mappings

$$\begin{aligned} \fancyscript{F}_2\quad \text {and}\quad F(x,y)\mapsto F((1-t)x+ty,y-x) \end{aligned}$$

(2.8)

are homeomorphisms on $\mathcal S_s'(\mathbb R^{2d})$. In particular, the map $a\mapsto K_{a,t}$ is a homeomorphism on $\mathcal S_s'(\mathbb R^{2d})$.

For future references we recall the link between the Weyl quantization and the (cross)-Wigner distribution $W_{\phi ,\psi }$ of $\phi ,\psi \in \mathcal S_s'(\mathbb R^{d})$, which belongs to $\mathcal S_s'(\mathbb R^{2d})$, and is defined by the formula

$$\begin{aligned} W_{\phi ,\psi }(x,\xi ) = \fancyscript{F}\big (\phi (x+\, \cdot \, /2)\overline{\psi (x-\, \cdot \, /2)}\big )(\xi ). \end{aligned}$$

We note that $W_{\phi ,\psi }$ is given by

$$\begin{aligned} W_{\phi ,\psi }(x,\xi ) = (2\pi )^{-d/2}\int \limits _{\mathbb R^{d}} \phi (x+y/2)\overline{\psi (x-y/2)}e^{-i\langle y,\xi \rangle }\, dy, \end{aligned}$$

when $\phi ,\psi \in L^2(\mathbb R^{d})$. By straight-forward computations we have

$$\begin{aligned} (\mathrm{Op }^w(a)\psi ,\phi ) = (a,W_{\phi ,\psi }), \end{aligned}$$

for admissible $a, \phi$ and $\psi$.

The set of pseudo-differential operators $\mathrm{Op }_t(a)$ with the symbols $a$ in the spaces

$$\begin{aligned} \mathcal S_s(\mathbb R^{2d}),\quad \fancyscript{S}(\mathbb R^{2d}),\quad \fancyscript{S}'(\mathbb R^{2d}) \quad \text {or}\quad \mathcal S_s'(\mathbb R^{2d}), \end{aligned}$$

(2.9)

is independent of the choice of $t\in \mathbb R$. In fact, by [16, Section 18.5] and its analysis, we have

$$\begin{aligned} \mathrm{Op }_{t_1}(a_1) = \mathrm{Op }_{t_2}(a_2) \quad \Longleftrightarrow \quad a_2=e^{i(t_2-t_1)\langle D_\xi ,D_x\rangle }a_1, \end{aligned}$$

(2.10)

where the map $e^{i(t_2-t_1)\langle D_\xi ,D_x\rangle }$ is a continuous bijection on each one of the spaces in (2.9).

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 $\fancyscript{P}_E(\mathbb R^{d})$ is the set of all moderate weights on $\mathbb R^{d}$.

Theorem 2.2

Let $t\in \mathbb R$ and $p,q\in [1,\infty ]$. Also let $\omega \in \fancyscript{P}_E(\mathbb R^{4d})$ and $\omega _1,\omega _2\in \fancyscript{P}_E(\mathbb R^{2d})$ be such that

$$\begin{aligned} \frac{\omega _2(x-ty,\xi +(1-t)\eta )}{\omega _1 (x+(1-t)y,\xi -t\eta )} \lesssim \omega (x,\xi ,\eta ,y). \end{aligned}$$

If $a\in M^{\infty ,1}_{(\omega )}(\mathbb R^{2d})$, then $\mathrm{Op }_t(a)$ from $\mathcal S_{1/2}(\mathbb R^{d})$ to $\mathcal S_{1/2}'(\mathbb R^{d})$ extends uniquely to a continuous map from $M^{p,q}_{(\omega _1)}(\mathbb R^{d})$ to $M^{p,q}_{(\omega _2)}(\mathbb R^{d})$.

We note that the previous result agrees with [12, Theorem 14.5.2] when the weights are trivial ($\omega =\omega _j=1$) and $t=0$.

Remark 2.3

Theorem 2.2 attains the following convenient form in the case of the Weyl quantization. Let $p,q\in [1,\infty ]$. Also let $\omega \in \fancyscript{P}_E(\mathbb R^{4d})$ and $\omega _1,\omega _2\in \fancyscript{P}_E(\mathbb R^{2d})$ be such that

$$\begin{aligned} \frac{\omega _2(X-Y)}{\omega _1 (X+Y)} \lesssim \omega (X,Y). \end{aligned}$$

If $a\in \mathcal M^{\infty ,1} _{(\omega )}(\mathbb R^{2d})$, then $\mathrm{Op }^w(a)$ from $\mathcal S_{1/2}(\mathbb R^{d})$ to $\mathcal S_{1/2}'(\mathbb R^{d})$ extends uniquely to a continuous map from $M^{p,q}_{(\omega _1)}(\mathbb R^{d})$ to $M^{p,q}_{(\omega _2)}(\mathbb R^{d})$.

Finally we also need the following result which is an immediate consequence of [21, Proposition 6.14] and (2.10). Here recall that $\Theta$ is the extension operator $(\Theta \omega ) (X,Y) =\omega (Y)$.

Proposition 2.4

Let $\omega \in \fancyscript{P}_E(\mathbb R^{2d})$. Then the operator class

$$\begin{aligned} \mathrm{Op }_t(M^{\infty ,1}_{(\Theta \omega )}) \end{aligned}$$

is independent of $t\in \mathbb R$.

3 Off-Diagonal Decay Matrices

In this section we investigate off-diagonal decay of inverses of matrices in ${\mathcal C}_v$ in absence of GRS-condition. We consider unions of the form (1.4) where $( v_n)_{n\in \mathbb {N}}$ is a GRS sequence of weights on $\mathbb R^{d}$ (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 ${\mathcal C}_v$, in such a way that each $v$ satisfies the GRS-condition. Since any ${\mathcal C}_v$ is inverse closed in ${\mathcal B}(\ell ^2)$, it follows that the union is also inverse closed in ${\mathcal B}(\ell ^2).$

Theorem 3.1

Let $(v_n)_{n\in {\mathbb N}}$ be a decreasing sequence of submultiplicative weights on $\mathbb R^{d}$ such that (1.3) holds. Then, there is a family $V$ of submultiplicative weights satisfying GRS-condition such that

$$\begin{aligned} \bigcup _{n}{\mathcal C}_{v_n}=\bigcap _{v \in V} {\mathcal C}_{v}. \end{aligned}$$

(3.1)

In particular, if $A\in {\mathcal C}_{v_n}$ for some $n\in {\mathbb N}$ and $A$ is invertible on $\ell ^2(\mathbb Z^{d})$, then $A^{-1}\in {\mathcal C}_{v_m}$ for some $m\in {\mathbb N}$.

Proof

By Remark 2.1 we may assume that $(v_n)_{n\in \mathbb {N}}$ is a family of weights on $\mathbb Z^{d}$ (instead of $\mathbb R^{d}$). Let $W$ be the family of positive even functions on $\mathbb Z^{d}$ which are dominated by $v_n$, for every $n$. That is,

$$\begin{aligned} {W} = \left\{ w: \mathbb Z^{d}\rightarrow \mathbb R_+ \,;\, w\ \text{ even },\ \sup _{{k}}\frac{w(k)}{v_n(k)} < \infty \quad \forall \ n\in {\mathbb N}\right\} , \end{aligned}$$

and consider the sets

$$\begin{aligned} \bigcap _{w\in W} \ell ^1_{(w)}(\mathbb Z^{d}) \quad \text {and}\quad \bigcup _{n\in {\mathbb N}} \ell ^1_{(v_n)}(\mathbb Z^{d}). \end{aligned}$$

(3.2)

We equip the intersection in (3.2) by the locally convex topology, induced by the system of semi-norms $\Vert \, \cdot \, \Vert _{\ell ^1_{(w)}}, w\in W$. The union in (3.2) is equipped with the inductive limit topology, which is possible due to the fact that it consists of a union of an increasing sequence of Banach algebras (under convolution).

According to [6, Theorem 2.3], the sets and their topologies in (3.2) agree, i.e.,

$$\begin{aligned} \bigcap _{w\in W} \ell ^1_{(w)}(\mathbb Z^{d}) = \bigcup _{n\in {\mathbb N}} \ell ^1_{(v_n)}(\mathbb Z^{d}). \end{aligned}$$

Since the right-hand side is the inductive limit of a sequence of commutative Banach algebras it admits a fundamental system of seminorms $(q_i)_{i\in I}$ which are submultiplicative [2, Prop. 12]. Consequently, for every $w_1\in W$ there are $i\in I$ and $w_2\in W$ such that

$$\begin{aligned} \Vert \, \cdot \, \Vert _{\ell ^1_{(w_1)}} \le q_i \le \Vert \, \cdot \, \Vert _{\ell ^1_{(w_2)}}. \end{aligned}$$

From

$$\begin{aligned} q_i(e_{j+k}) = q_i(e_{j}*e_{k}) \le q_i(e_{k} ) q_i(e_{j}), \end{aligned}$$

we get that $w(k):= q_i(e_{k})$ defines a submultiplicative function with $w_1 \le w \le w_2$. Here $e_{j}$ is the function on $\mathbb Z^{d}$, given by $e_j(k)=\delta _{j,k}$. In particular, $w_1$ is dominated by the weight $w_0\in W$ given by $w_0(k) = \max (w(k),w(-k))$. It follows that

$$\begin{aligned} \bigcup _{n\in {\mathbb N}}\ell ^1_{(v_n)} = \bigcap _{v\in V} \ell ^1_{(v)}, \end{aligned}$$

(3.3)

where $V$ is the (possibly uncountable) set consisting of those $w\in W$ which are submultiplicative and symmetric.

For a matrix $A = (a(j,k))_{j,k\in \mathbb Z^{d}}$ we denote by $d_A$ the sequence with components $d_A(k):= \sup _{j\in \mathbb Z^{d}}\left| a(j,j-k)\right| , k\in \mathbb Z^{d}$. Obviously, if $v$ is submultiplicative, then $A\in {\mathcal C}_v$ if and only if $d_A \in \ell ^1_{(v)}$. Consequently, (3.3) gives (3.1).

Moreover, each weight $v\in V$ satisfies GRS condition. In fact, there is a positive sequence $\alpha _n, n\in {\mathbb N}$, with $v \le \inf _n\alpha _nv_n$ from where it follows

$$\begin{aligned} \lim _{\ell \rightarrow \infty } v(\ell k)^{1/\ell } \le \lim _{\ell \rightarrow \infty } v_n(\ell k)^{1/\ell } \end{aligned}$$

for every $n\in {\mathbb N}$. Consequently, $\lim _{\ell \rightarrow \infty }v(\ell k)^{1/\ell } = 1$.

The last statement of the theorem follows from the fact that $\mathcal C_v$ is inverse closed in ${\mathcal B}(\ell ^2)$ when $v$ is submultiplicative and satisfies the GRS-condition. (See e.g., [3, 11]). $\square$

Remark 3.2

Wiener’s Lemma can be rephrased by saying that the inverse of a convolution operator on $\ell ^2$ with symbol in $\ell ^1$ is again a convolution operator with symbol in $\ell ^1$ [11, Theorem 5.18]. Let us assume that the hypothesis of Theorem 3.1 are satisfied. Let $a\in \ell ^1_{(v_n)}(\mathbb Z^{d})$ be such that the convolution operator $b\mapsto a*b$ is invertible on $\ell ^2.$ Then the corresponding matrix $A$ is in ${\mathcal C}_{v_n}$ and, according to Theorem 3.1, $A^{-1}\in {\mathcal C}_{v_m}$ for some $m.$ Since $A^{-1}$ is also the matrix of a convolution operator we easily conclude that $a$ has an inverse in $\ell ^1_{(v_m)}(\mathbb Z^{d}).$

Corollary 3.3

Let $v$ be a submultiplicative weight on $\mathbb R^{d}$. If $A\in {\mathcal C}_{v}$ and $A$ is invertible on $\ell ^2(\mathbb Z^{d})$ then $A^{-1}\in {\mathcal C}_{v^t}$ for some $t>0$.

Proof

Let $v_n:=v^{1/n}$ when $n\in \mathbb {N}$. As

$$\begin{aligned} 1\le v_n(\ell x)^{1/\ell }\le v(x)^{1/n}, \end{aligned}$$

the previous theorem can be applied. $\square$

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 3.4

Let $v$ be submultiplicative on $\mathbb R^{d_1}\times \mathbb R^{d_2}$ of the form $v(x,y)=u(x)\cdot e^{t|y|},$ where $u$ is a weight on $\mathbb R^{d_1}$ satisfying GRS-condition and $t>0$. If $A\in {\mathcal C}_{v}$ is invertible on $\ell ^2(\mathbb Z^{d})$ then $A^{-1}\in {\mathcal C}_{w}$ where $w(x,y)=u(x)\cdot e^{s|y|}$ for some $s>0$.

Proof

It suffices to apply Theorem 3.1 to the sequence of weights $v_n(x,y)=u(x)\cdot e^{t|y|/n}$. $\square$

The previous corollary can be extended and reformulated as follows.

Corollary 3.5

Let $v_j$ on $\mathbb R^{d_j}, j=1,2$, be submultiplicative weights such that $v_1$ satisfies the GRS-condition. If $A\in {\mathcal C}_{v_1\otimes v_2}$ is invertible on $\ell ^2(\mathbb Z^{d})$ then $A^{-1}\in {\mathcal C}_{v_1\otimes v_2^t}$, for some $t>0$.

In [7] (see also [13, Remark 1]) it is shown that for any exponential weight $v_t(k)=e^{t|k|}$ there exists a banded matrix $A$ which is invertible as a bounded operator on $\ell ^2({\mathbb Z})$ and yet $A^{-1}\notin {\mathcal C}_{v_t}$. As $A$ is banded, we have $A\in \bigcap _{s>0} {\mathcal C}_{v_s}$. 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 $\sigma _X(x),$ is the set of those complex numbers $\lambda$ for which the element $\lambda e-x$ 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 $(q_i)_{i\in I}.$ 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 $(v_n)_{n\in {\mathbb N}}$ a decreasing sequence of submultiplicative weights on ${\mathbb R}^d$ the spaces

$$\begin{aligned} { k}_1:= \bigcup _{n\in {\mathbb N}} \ell ^1_{(v_n)}({\mathbb Z}^d) \end{aligned}$$

and

$$\begin{aligned} \bigcup _{n}{\mathcal C}_{v_n} \end{aligned}$$

equipped with the corresponding inductive limit topologies are locally m-convex algebras.

Given a locally m-convex algebra $X,$ its spectrum, denoted by $\text{ Spec }(X),$ is the set of all non-trivial, multiplicative, continuous and linear functionals $\varphi :X \rightarrow {\mathbb C}$. If $x\in X$ is invertible then $\varphi (x)\ne 0$ for each $\varphi \in \text{ Spec }(X).$ By [2, Theorem 1] the converse holds if $X$ is a countable inductive limit of commutative Banach algebras. Consequently, in this case

$$\begin{aligned} \{\varphi (x):\varphi \in \text{ Spec }(X) \} = \sigma _X(x). \end{aligned}$$

In particular, given $a\in k_1$,

$$\begin{aligned} \sigma _{{ k}_1}(a) = \left\{ \varphi (a): \varphi \in \mathrm{Spec}({ k}_1)\right\} . \end{aligned}$$

On the other hand, it is clear that $\mathrm{Spec}({ k}_1)=\bigcap _{n=1}^{\infty }\mathrm{Spec}(\ell ^1_{(v_n)}).$

For $m = 1, \ldots , d, J_m\in {\mathbb Z}^d$ 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 $\{e_j: j\in {\mathbb Z}^d\}$ in $\ell ^1_{(v_n)}$ can be obtained from the $e_{J_m}, m=1,\ldots , d$, and their inverses, by convolution. Therefore, every $\varphi \in \text{ Spec }\left( \ell ^1_{(v_n)}\right)$ is completely determined by $\xi = \left( \varphi (e_{J_1}),\ldots , \varphi (e_{J_d})\right) \in {\mathbb C}^d.$

We recall that, given two algebras $X \subset Y$ with a common unit, $X$ is inverse closed in $Y$ if and only if $\sigma _X(x) = \sigma _Y(x)$ for all $x\in X.$ In particular, under the assumptions of Theorem 3.1, every $A\in \bigcup _{n}{\mathcal C}_{v_n}$ has the same spectrum in the algebras ${\mathcal B}\left( \ell ^2\right)$ and $\bigcup _{n}{\mathcal C}_{v_n}.$

Proposition 3.6

Let $(v_n)_{n\in {\mathbb N}}$ be a decreasing sequence of weights on $\mathbb R^{d}$ that do not satisfy GRS-condition but (1.3). Then there is $A\in {\mathcal C}_{v_1}$ such that $\sigma _{{\mathcal B} (\ell ^2)}(A)\ne \sigma _{{\mathcal C}_{v_n}}(A)$ for each $n \in {\mathbb N}$.

Proof

We may assume that $v_n$ are weights on $\mathbb Z^{d}$. Let $a\in \ell ^1_{(v_1)}(\mathbb Z^{d})$ be a sequence with all the coordinates positive and let $A$ be the matrix of the convolution operator $b\mapsto a*b$ on $\ell ^2(\mathbb Z^{d})$. The spectrum of $A$ in the locally m-convex algebra $\bigcup _n{\mathcal C}_{v_n}$ coincides with the spectrum of $a$ in the commutative locally m-convex algebra ${ k}_1:= \bigcup _n\ell ^1_{(v_n)}$ which, according to [2, Theorem 1], is given by

$$\begin{aligned} \sigma _{{ k}_1}(a) = \left\{ \varphi (a): \varphi \in \bigcap _{n=1}^{\infty }\mathrm{Spec}(\ell ^1_{(v_n)})\right\} . \end{aligned}$$

For $\varphi \in \text{ Spec }(\ell ^1_{(v_n)})$ we have $\varphi (a) = \sum \nolimits _{j\in \mathbb Z^{d}}a_j \xi ^j$ where $\xi \in {\mathbb C}^d$ is given by $\xi _m = \varphi (e_{J_m}), m=1,\dots ,d$. It easily follows that

$$\begin{aligned} r_{n,m}^{-1} \le |\xi _m| \le r_{n,m} \quad \text{ where }\quad r_{n,m} = \lim _{\ell \rightarrow \infty }v_n(\ell J_m)^{{1}/{\ell }}. \end{aligned}$$

From the condition (1.3) and Theorem 3.1 we conclude that

$$\begin{aligned} \sigma _{{\mathcal B}(\ell ^2)}(A) = \sigma _{k_1}(a) = \left\{ \sum _{j\in \mathbb Z^{d}} a_j \xi ^j: \xi \in \mathbb {T}^d \right\} . \end{aligned}$$

On the other hand, for a fixed $n\in {\mathbb N},$ there is $k\in \mathbb Z^{d}$ such that $\lim \limits _{\ell \rightarrow \infty }v_n(\ell k)^{1/ {\ell }} > 1$. From

$$\begin{aligned} v_n(\ell k) \le \prod _{m=1}^d v_n(\ell k_mJ_m) \end{aligned}$$

we get $r_{n,m} = \lim _{\ell \rightarrow \infty } v_n(\ell J_m)^{1/{\ell }} > 1$ for some $m = 1,\ldots , d$. For simplicity we assume $m = 1$ and denote $U_n = \left\{ z\in {\mathbb C}:\ r_{n,1}^{-1} < |z| < r_{n,1}\right\} .$ Then for every $z\in U_n$ we define the element of $\text{ Spec }(\ell ^1_{(v_n)})$ given by

$$\begin{aligned} \varphi _z(b) = \sum _{j\in \mathbb Z^{d}}b_{j} z^{j_1},\ b\in \ell ^1_{(v_n)}, \end{aligned}$$

that is, $\varphi _z$ is the unique element in the spectrum of $\ell ^1_{(v_n)}$ satisfying $\varphi _z(e_{J_1}) = z$ and $\varphi _z(e_{J_m}) = 1$ for $m\ne 1.$ Consequently,

$$\begin{aligned} h(U_n) \subset \sigma _{\ell ^1_{(v_n)}}(a) = \sigma _{{\mathcal C}_{v_n}}(A) \end{aligned}$$

being $h$ the holomorphic function

$$\begin{aligned} h(z) = \varphi _z(a) = \sum _{j\in \mathbb Z^{d}}a_j z^{j_1} = \sum _{k\in \mathbb Z}\left( \sum _{s\in \mathbb Z^{d-1}}a_{k,s}\right) z^{k}, \ z\in U_n. \end{aligned}$$

Therefore $\sigma _{{\mathcal C}_{v_n}}(A)$ has non empty interior, from where it follows $\sigma _{{\mathcal B}(\ell ^2)}(A)\ne \sigma _{{\mathcal C}_{v_n}}(A)$ for any $n\in {\mathbb N}$. $\square$

According to Theorem 3.1, for every $\lambda \notin \sigma _{{\mathcal B}(\ell ^2)}(A)$ there is $m\in {\mathbb N}$ such that $\lambda \notin \sigma _{{\mathcal C}_{v_m}}(A)$. The previous proposition shows that, in general, $m$ depends on $\lambda$.

Remark 3.7

Arguing as in the proof of Proposition 3.6, if condition (1.3) is violated then we may assume that

$$\begin{aligned} \lim _{\ell \rightarrow \infty }v_n(\ell J_1)^{1/{\ell }} > \rho > 1 \end{aligned}$$

for all $n\in {\mathbb N}$. Then, proceeding as in [13, Remark 1], there exists a banded matrix $A$ which is invertible as a bounded operator on $\ell ^2(\mathbb Z^{d})$ and yet $A^{-1}\notin \bigcup _{n\in {\mathbb N}}{\mathcal C}_{v_n}$. Hence the condition (1.3) in Theorem 3.1 is also necessary.

The next result will be needed to prove the main result of Sect. 4.

Lemma 3.8

Let $(v_n)_{n\in {\mathbb N}}$ be a decreasing sequence of submultiplicative weights on $\mathbb R^{d}$ such that (1.3) holds. Let $A\in {\mathcal C}_{v_n}$ for some $n\in \mathbb N$, and let $K\subset {\mathbb C}$ be compact with $K \cap \sigma _{{\mathcal B}(\ell ^2)}(A) = \emptyset$. Then there exists $m\in {\mathbb N}$ such that $(zI - A)^{-1}\in {\mathcal C}_{v_m}$ for every $z\in K$. Furthermore, the map

$$\begin{aligned} z\mapsto (zI - A)^{-1}, \end{aligned}$$

from $K$ to ${\mathcal C}_{v_m}$ is continuous.

Proof

According to the proof of Theorem 3.1, $A\in \bigcap _{v \in V}{\mathcal C}_{{v}}$, and each weight ${v}\in V$ satisfies GRS-condition. As $\sigma _{{\mathcal B}(\ell ^2)}(A) = \sigma _{{\mathcal C}_{v}}(A)$ for every $v\in V$, the map $K\rightarrow {\mathcal C} _{v},\ z\mapsto (zI - A)^{-1}$ is well defined and continuous.

Hence $\{ \, (zI - A)^{-1}\, ;\, \ z\in K\, \}$ is a compact set in ${\mathcal C}_{v}$ for every $v\in V$. For every $z\in K$ and $j\in \mathbb Z^{d}$ we put

$$\begin{aligned} f_z(j):= \sup _{k\in \mathbb Z^{d}} |b_{z}(k,k-j)|, \end{aligned}$$

where $b_{z}(j,k)$ are the matrix elements of $(zI - A)^{-1}$. Then $\{ \, f_z\, ;\, z\in K\, \}$ is a bounded set in the inductive limit $\bigcup _{n\in {\mathbb N}}\ell ^1_{(v_n)} = \bigcap _{v\in V}\ell ^1_{(v)}$. We can apply [6, Theorem 2.3] to conclude that $\{ \, f_z\, ;\, z\in K\, \}$ is a bounded subset of $\ell ^1_{(v_m)}$ for some $m\in {\mathbb N}$. Consequently $\{ \, (zI - A) ^{-1}\, ;\, \ z\in K\, \}$ is contained in ${\mathcal C}_{v_m}$. After enlarging $m$ if it is necessary we also have that $\{ \, ( zI - A)\, ;\, \ z\in K\, \}$ is contained in ${\mathcal C}_{v_m}$. Since ${\mathcal C}_{v_m}$ is a Banach algebra, the result follows.

Apart from the classes ${\mathcal C}_v$ of convolution dominated matrices, there are other ways to measure the off-diagonal decay. We now concentrate on strict off-diagonal decay.

Definition 3.9

Let $v$ be a submultiplicative weight on $\mathbb R^{d}$.

• (a) ${\mathcal A}_v$ is the set of all matrices $A = (a(j,k) )_ {j,k\in \mathbb Z^{d}}$ such that

  $$\begin{aligned} \Vert A\Vert _{{\mathcal A}_v}:= \sup _{j,k\in \mathbb Z^{d}} |a(j,j-k)|v(k)< \infty . \end{aligned}$$

• (b) ${\mathcal A}^0_v$ consists of those matrices $A\in {\mathcal A}_v$ such that

  $$\begin{aligned} \lim _{k\rightarrow \infty }\left( \sup _{j\in \mathbb Z^{d}}\left| a(j,j-k) \right| \right) v(k) = 0. \end{aligned}$$

As in the proof of Theorem 3.1, for a matrix $A = ( a(j,k) ) _{j,k\in \mathbb Z^{d}}$ we denote by $d_A$, the sequence with entries $d_A(k) = \sup _{j\in \mathbb Z^{d}} \left| a(j,j-k)\right| , k\in \mathbb Z^{d}$. Then $A\in {\mathcal A}_v$ if and only if $d_A \in \ell ^\infty _{(v)}$, while $A\in {\mathcal A}^0_v$ if and only if $d_A \in c_0 (v)$.

The submultiplicative weight $v$ on $\mathbb Z^{d}$ is called sub-convolutive if $1/v \in \ell ^1$ and there is $C > 0$ such that $\frac{1}{v}*\frac{1}{v} \le \frac{C}{v}$. By [13, Lemma 1. (b)] if $v$ is sub-convolutive then $\ell ^\infty _v$ is a Banach algebra for convolution and ${\mathcal A}_v$ is a Banach algebra of bounded operators on $\ell ^2(\mathbb Z^{d})$.

The weight $\tau _s(k):=(1+|k|)^s$ is sub-convolutive in $\mathbb Z^{d}$ when $s>d$. Moreover, if $u$ is an arbitrary submultiplicative weight and $w$ is sub-convolutive, then $v:=u w$ is also sub-convolutive. In fact, since $u(j)^{-1}u(k-j) ^{-1}\le u(k)^{-1}$, we get

$$\begin{aligned} \left( \frac{1}{v} *\frac{1}{v}\right) (k) = \sum _ {j} \frac{u(j)^{-1} u(k-j)^{-1}}{w(j)w(k-j)} \le u(k)^{-1} \left( \frac{1}{w} *\frac{1}{w}\right) (k). \end{aligned}$$

In particular, $v = u \tau _s$ 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 3.10

• (a) Assume that $v$ is sub-convolutive and satisfies the GRS-condition. Then ${\mathcal A}^0_v$ is inverse closed in ${\mathcal B}(\ell ^2(\mathbb Z^{d}))$.

• (b) Let $u$ be a weight with GRS-condition and $s > d$. Then, for $v = u \tau _s, {\mathcal A}_v$ is inverse closed in ${\mathcal B}(\ell ^2(\mathbb Z^{d}))$.

We aim to obtain a version of Theorem 3.1 for the matrix algebras ${\mathcal A}_v$ and for this we need to require that the sequence $\left( v_n\right) _{n\in {\mathbb N}}$ satisfies the so-called condition (D).

Definition 3.11

The sequence of weights $\left( v_n\right) _{n\in {\mathbb N}}$ is said to satisfy condition (D) if there exists an increasing sequence $(I_m)_{m\in {\mathbb N}}$ of subsets $I_m$ of $I = \mathbb Z^{d}$ such that

• (a) for each $m\in {\mathbb N}$ there is $n_m\in {\mathbb N}$ with $\inf _{j\in I_m}\frac{v_{\ell }(j)}{v_{n_m}(j)} > 0\, \forall \ \ell > n_m$.

• (b) for each $n\in {\mathbb N}$ and each $J\subset I$ with $J\cap (I\setminus I_{m})\ne \emptyset$ for all $m\in {\mathbb N}$ there exists $n^\prime > n$ such that $\inf _{j\in J}\frac{v_{n^\prime }(j)}{v_n(j)}=0.$

The condition (D) was introduced in [5] by Bierstedt and Meise to address the question under which conditions the strong dual of certain Fréchet spaces of sequences can be represented as a countable inductive limit of Banach spaces. As a concrete example we could consider an arbitrary weight $\mu$ on $\mathbb Z$ and a decreasing sequence $\left( \lambda _n\right) _{n\in {\mathbb N}}$ of weights on $\mathbb Z$ such that $\lim _{j\rightarrow \infty }\frac{\lambda _{n+1}(j)}{\lambda _n(j)} =0$ for every $n\in {\mathbb N}$. Then the sequence $\left( v_n\right) _{n\in {\mathbb N}}$ of weights on $\mathbb Z^{2}$, defined by $v_n(j,k) = \lambda _n(j) \mu (k)$, satisfies condition (D): take $I_m = \left\{ (j,k)\in \mathbb Z^{2}:\ |j|\le m \right\}$.

Theorem 3.12

Let $s > d$ and $v_n = u_n\tau _s, n\in {\mathbb N},$ where $(u_n) _{n\in {\mathbb N}}$ is a decreasing sequence of weights on $\mathbb Z^{d}$, and assume that $(v_n)_{n\in {\mathbb N}}$ satisfies condition (D) and (1.3). If $A\in {\mathcal A}_{v_n}$ for some $n\in {\mathbb N}$ and $A$ is invertible on $\ell ^2(\mathbb Z^{d})$, then $A^{-1}\in {\mathcal A}_{v_m}$ for some $m\in {\mathbb N}$.

Proof

We shall argue in a similar way as in the proof of Theorem 3.1, after the weighted $\ell ^1$ norms have been replaced by weighted $\ell ^\infty$ norms. Clearly $(v_n)_{n\in {\mathbb N}}$ satisfies (D) condition if and only if $(u_n)_{n\in {\mathbb N}}$ does.

Let

$$\begin{aligned} W = \left\{ \, u \,: \, \mathbb Z^{d} \rightarrow \mathbb R_+\,;\, \sup _{k} \frac{u(k)}{u_n(k)} < \infty \quad \forall \ n\in {\mathbb N} \, \right\} . \end{aligned}$$

Then it follows from [4] that

$$\begin{aligned} \bigcap _{u\in W} \ell ^\infty _{(u)}(\mathbb Z^{d}) = \bigcup _{n\in {\mathbb N}} \ell ^\infty _{(u_n)}(\mathbb Z^{d}), \end{aligned}$$

topologically if and only if condition (D) holds. Since

$$\begin{aligned} \ell ^\infty _{(u\tau _s)}(\mathbb Z^{d}) = \{ \, \tau _s^{-1}a\, ;\, a\in \ell ^\infty _{(u)}(\mathbb Z^{d})\, \} , \end{aligned}$$

and proceeding as in the proof of Theorem 3.1 we conclude that

$$\begin{aligned} \bigcap _{u\in U} \ell ^\infty _{(\tau _su)}(\mathbb Z^{d}) = \bigcup _{n\in {\mathbb N}}\ell ^\infty _{(v_n)}(\mathbb Z^{d}), \end{aligned}$$

where $U$ consists of those $u\in W$ which are a submultiplicative. Since each weight in $U$ satisfies the GRS-condition, an application of (b) of Theorem 3.10 gives the result. $\square$

Remark 3.13

If the sequence $(v_n)_{n\in {\mathbb N}}$ satisfies

$$\begin{aligned} \forall \ n\ \exists \ m>n\quad \text{ such } \text{ that }\quad v_m/v_n \in \ell ^1(\mathbb Z^{d}), \end{aligned}$$

(3.4)

then

$$\begin{aligned} \bigcup _{n} {\mathcal A}_{v_n}= \bigcup _{n} {\mathcal C}_{v_n}. \end{aligned}$$

Therefore, the main interest of the previous result lies in the case that condition (3.4) is not satisfied, since otherwise it is covered by Theorem 3.1.

Corollary 3.14

Let $v_n = u_n w$ where $w$ is a sub-convolutive weight and $\left( u_n \right) _{n\in {\mathbb N}}$ is a decreasing sequence of submultiplicative weights on $\mathbb Z^{d}$ such that (1.3) holds and

$$\begin{aligned} \lim _{j \rightarrow \infty }\frac{v_{n+1}(j)}{v_n(j)} = 0 \quad \forall \ n\in {\mathbb N}. \end{aligned}$$

If $A\in {\mathcal A}_{v_n}$ for some $n\in {\mathbb N}$ and $A$ is invertible on $\ell ^2(\mathbb Z^{d})$, then $A^{-1}\in {\mathcal A}_{v_m}$ for some $m\in {\mathbb N}$.

Proof

The sequence $\left( v_n\right) _{n\in {\mathbb N}}$ satisfies condition (D). On the other hand $\bigcup _{n} {\mathcal A}_{v_n} = \bigcup _{n} {\mathcal A}^0_{v_n}$ and we can proceed as before but using part (a) of Theorem 3.10. $\square$

4 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 $\mathbb R^{2d}$, then $\Theta v$ is submultiplicative on $\mathbb R^{4d}$. By [14], $\mathrm{Op }^w({\mathcal M}^{\infty ,1}_{(\Theta v)}(\mathbb R^{2d}))$ is inverse closed in ${\mathcal B}(L^2(\mathbb R^{d}))$, 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 (2.5) and Proposition 2.4, it follows that if $t\in \mathbb R$ is fixed, then $\mathrm{Op }_t(M^{\infty ,1}_{(\Theta v)}(\mathbb R^{2d}))$ is inverse closed in ${\mathcal B}(L^2(\mathbb R^{d}))$, if and only if $v$ satisfies the GRS-condition.

We want to analyze what can be said about the inverse of $\mathrm{Op }_t(a)$ in the limit case that $a \in M^{\infty ,1}_{(\Theta v)}(\mathbb R^{2d})$ and $v$ lacks the GRS-condition. In order to do so, it suffices to consider the case of Weyl quantization and letting $a\in {\mathcal M} ^{\infty ,1}_{(\Theta v)}(\mathbb R^{2d})$.

The next lemma is a reformulation of Lemma 3.1 in [10]. For $X=(x,\xi )\in \mathbb R^{2d}, \pi (X)$ denotes the corresponding time-frequency shift

$$\begin{aligned} \pi (X)f(y) = e^{i \langle y,\xi \rangle }f(y-x),\quad X=(x,\xi ). \end{aligned}$$

Lemma 4.1

Let $v$ be submultiplicative, $\phi \in M^1_{v}({\mathbb R}^d)$ and let $\Phi = W_{\phi , \phi }.$ Given $a\in {\mathcal M}^{\infty ,1}_{(\Theta v)} (\mathbb R^{2d}),$

$$\begin{aligned} \left| \left( \mathrm{Op }^w(a)(\pi (X)\phi ),\ \pi (Y)\phi \right) \right| = (2\pi )^{-{d}/{2}} \left| ({\mathcal V}_{\Phi }a)((Y+X)/2, (Y-X)/2)\right| . \end{aligned}$$

The first part of the next result is [12, 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 elements $a\in {\mathcal M}^{\infty ,1}_{(\Theta v)} ({\mathbb R}^{2d})$ with matrices, indexed by ${\mathbb Z}^{2d}$, in ${\mathcal C}_{\tilde{v}}$ for some weight $\tilde{v}$ defined in terms of $v$.

In order to simplify the notation, we let ${\mathbf J} = \mathbb Z^{d} \times \mathbb Z^{d}$ and, whenever $\alpha , \beta > 0$ are fixed, we put $X_\mathbf J = (\alpha k, \beta n)$ for each $\mathbf{j} = (k,n)\in \mathbf J$.

Theorem 4.2

Let $v$ be submultiplicative on $\mathbb R^{2d}$ and let $\phi \in M^1_{(v)}(\mathbb R^{d}) \setminus \{0\}$. Then the following is true:

• (a) There is $\delta > 0$ such that, for every $0 < \alpha , \beta \le \delta ,$

  $$\begin{aligned} S f = \sum _{\mathbf{j}\in \mathbf J}\left<f,\ \pi (X_{\mathbf{j}})\phi \right> \pi (X_{\mathbf{j}})\phi \end{aligned}$$

  is invertible on $M^1_{(v)}({\mathbb R}^d)$ and $\gamma ^0: = S^{-1} \phi \in M^1_{(v)}({\mathbb R}^d)$.

• (b) The following conditions are equivalent:

  • (1) $a\in {\mathcal M}^{\infty ,1}_{(\Theta v)}({\mathbb R}^{2d})$;

  • (2) $a\in \fancyscript{S}'(\mathbb R^{2d})$ and there exists a function $H\in L^1_{(v)}(\mathbb R^{2d})$ such that

    $$\begin{aligned} \left| \left( \mathrm{Op }^w(a)(\pi (X)\phi ),\ \pi (Y)\phi \right) \right| \le H(X-Y), \quad X,Y\in \mathbb R^{2d}; \end{aligned}$$

  • (3) $a\in \fancyscript{S}'(\mathbb R^{2d})$ and

    $$\begin{aligned} \left| \left( \mathrm{Op }^w(a)(\pi (X_\mathbf{j})\phi ),\ \pi (X_\mathbf{k})\phi \right) \right| \le h(\mathbf{k} - \mathbf{j}),\ \ \mathbf{j}, \mathbf{k}\in \mathbf J, \end{aligned}$$

    where

    $$\begin{aligned} \sum _{\mathbf{j}\in \mathbf J}h(\mathbf{j})v(X_\mathbf{j}) < \infty . \end{aligned}$$

Proof

The statement (a) is the content of [12, 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 4.1. For the equivalence between (2) and (3), the argument from [10] still works since we already have that the dual window $\gamma ^0: = S^{-1} \phi$ is in $M^1_{(v)}({\mathbb R}^d)$ by part (a). $\square$

In [10], the condition GRS is used to prove that for any lattice $\Lambda$ in ${\mathbb R}^{2d}$ with the property that ${\mathcal G}(\phi , \Lambda )$ is a Gabor frame we have that the dual window $\gamma ^0$ is also in $M^1_{(v)}$. However, in the previous theorem we are not dealing with arbitrary Gabor frames but only with frames associated to a lattice $\Lambda = \alpha {\mathbb Z}^d \times \beta {\mathbb Z}^d$ for $\alpha$ and $\beta$ small enough so that the conclusion of part (a) holds.

Remark 4.3

Let $v$ be a submultiplicative weight on $\mathbb R^{2d}$ and $\Lambda = \alpha {\mathbb Z}^d \times \beta {\mathbb Z}^d$ a lattice chosen according to Theorem 4.2 (a). Then the equivalence between conditions (1)–(3) in Theorem 4.2 (b) holds after replacing $v$ by any submultiplicative weight dominated by $v$.

Condition (3) in Theorem 4.2 means that the matrix $A$ with entries

$$\begin{aligned} a(\mathbf{j},\mathbf{j^{\prime }}) = \left<\mathrm{Op }^w(a)(\pi (X_{\mathbf{j^\prime }})\phi ),\ \pi (X_\mathbf{j})\phi \right>,\ \ \mathbf{j,j^\prime } \in \mathbf J, \end{aligned}$$

is in the algebra ${\mathcal C}_{\tilde{v}},$ where $\tilde{v}(\mathbf{j}) = v(X_{\mathbf{j}})$.

We are now able to formulate and prove the main result of the present section.

Theorem 4.4

Let $t\in \mathbb R$, and let $(v_n)_{n\in {\mathbb N}}$ be a decreasing sequence of submultiplicative weights on $\mathbb R^{2d}$ such that

$$\begin{aligned} \inf _n\lim _{\ell \rightarrow \infty }v_n(\ell X)^{{1}/{\ell }} = 1,\quad \forall \ X\in \mathbb R^{2d}. \end{aligned}$$

If $a\in M^{\infty ,1}_{(\Theta v_n)}(\mathbb R^{2d})$ for some $n\in {\mathbb N}$ and $T = \mathrm{Op }_t(a)$ is invertible on $L^2(\mathbb R^{d})$, then there exists $m\in {\mathbb N}$ such that $T^{-1}$ is equal to $\mathrm{Op }_t(b)$ for some $b\in M^{\infty ,1}_{(\Theta v_m)}(\mathbb R^{2d})$.

Proof

By Proposition 2.4 we may assume that $t=1/2$. Furthermore, by (2.5), the result follows if we prove that $T^{-1}=\mathrm{Op }^w(b)$ for some $b\in {\mathcal M}^{\infty ,1}_{(\Theta v_m)}(\mathbb R^{2d})$ when $a\in {\mathcal M}^{\infty ,1}_{(\Theta v_n)}(\mathbb R^{2d})$.

According to Theorem 4.2 there are $\phi \in M^1_{(v_n)} (\mathbb R^{d})$ and $\alpha , \beta > 0$ such that

$$\begin{aligned} {\mathcal G}(\phi , \alpha , \beta ) = \{\pi (X_{\mathbf{j}})\phi ,\ \mathbf{j}\in \mathbf J\} \end{aligned}$$

is a frame for $L^2({\mathbb R}^d),$ and $\gamma ^0$, the canonical dual window of $\phi$, belongs to $M^1_{(v_{n})}(\mathbb R^{d})$. We now define the matrix $A$ with entries

$$\begin{aligned} a(\mathbf{j},\mathbf{j^{\prime }}) = \left<T \pi (X_{\mathbf{j^\prime }})\phi ,\ \pi (X_\mathbf{j})\phi \right>,\quad \mathbf{j,j^\prime } \in \mathbf J. \end{aligned}$$

According to Theorem 4.2, $A\in \mathcal C_{\tilde{v}_n}$ where $\tilde{v}_n(\mathbf{j}) = v_n(X_{\mathbf{j}}).$ In particular $A$ defines a bounded operator on $\ell ^2(\mathbf J).$ Moreover, denoting by $C_\phi$ and $C_{\gamma ^0}$ the analysis operators associated to the frame ${\mathcal G}(\phi ,\alpha ,\beta )$ and its canonical dual frame, it turns out that

$$\begin{aligned} C_\phi \circ T = A \circ C_{\gamma ^0},\ C_\phi (L^2({\mathbb R}^d)) = C_{\gamma ^0}(L^2({\mathbb R}^d)) \end{aligned}$$

and $A$ maps the range of $C_\phi$ into itself with $\text{ ran }\ C_\phi ^{\bot }\subset \text{ ker }\ A$ (see [10, Lemma 3.4]). We now consider $b\in \mathcal S^{\prime }(\mathbb R^{2d})$ the unique tempered distribution such that $\mathrm{Op }^w(b) = T^{-1}$ and define the matrix $B$ with entries

$$\begin{aligned} b(\mathbf{j},\mathbf{j^{\prime }}) = \left<T^{-1} \pi (X_{\mathbf{j^\prime }})\gamma ^0,\ \pi (X_ \mathbf{j})\gamma ^0\right>,\ \ \mathbf{j,j^\prime } \in \mathbf J. \end{aligned}$$

We claim that $B$ is a pseudoinverse for $A$ in ${\mathcal B}(\ell ^2(J)).$ In fact, we consider $R:= C_\phi (L^2(\mathbb R^{d})) = C_{\gamma ^0}(L^2(\mathbb R^{d}))$ and observe that also $C_{\gamma ^0} \circ T^{-1} = B \circ C_\phi$. Then

• (1) $A\circ B \circ C_\phi f = A\circ C_{\gamma ^0} \circ T^{-1} f = C_\phi f$,

• (2) $B \circ A \circ C_{\gamma ^0} f = B\circ C_\phi \circ T f = C_{\gamma ^0}$,

• (3) $R^{\bot } = \text{ ker }\ A$.

Also $\text{ ker }\ B = R^{\bot }$. Consequently $B$ is a pseudoinverse of $A$ and the claim is proved. Now, $B$ is given by

$$\begin{aligned} B = \frac{1}{2\pi i}\int \limits _{\Gamma }\frac{1}{z}(z I - A)^{-1}\ dz, \end{aligned}$$

for a suitable closed path $\Gamma$ surrounding $\sigma _{{\mathcal B}(\ell ^2(J))}(A)\setminus \{0\}.$ From Lemma 3.8, the previous integral defines an element in ${\mathcal C}_{{\tilde{v}_m}}$ for some $m>n.$ After applying Theorem 4.2 once again we obtain $b\in {\mathcal M}^{\infty ,1}_{(\Theta v_m)}(\mathbb R^{2d})$. $\square$

The next example is based in [11]. It shows that, given a symbol $a\in {\mathcal M}^{\infty ,1}_{(\Theta v_n)}(\mathbb R^{2d})$ such that $\mathrm{Op }_t(a)$ is invertible in ${\mathcal B}(L^2(\mathbb R^{d}))$, the decay of the symbol of the inverse operator does not only depend on the weight $v_n$ but also on $a$.

Let $\varphi$ denote the Gaussian $\varphi (X) = e^{-|X|^2}$, with $X=(x,\xi ) \in \mathbb R^{2}$, and consider

$$\begin{aligned} c_Z(X):= e^{2i\sigma (Z,X)},\quad Z=(z,\zeta )\in \mathbb R^{2}. \end{aligned}$$

By straight-forward computations, it follows that

$$\begin{aligned} ({\mathcal V}_{\varphi }c_Z)(X,Y)&=e^{2i\sigma (Y+Z,X)}\varphi (Y+Z)\\&= e^{2i\sigma (Y,X)}c_Z(X)\varphi (Y+Z), \end{aligned}$$

and that the Weyl operator with symbol $c_Z$ is

$$\begin{aligned} \mathrm{Op }^w(c_Z) = e^{2i(x-z)\zeta }S_Z, \end{aligned}$$

where $S_Z$ denotes the translation

$$\begin{aligned} (S_Zf)(x) = f(x-2z). \end{aligned}$$

In particular, (3) gives

$$\begin{aligned} \Vert c _Z\Vert _{\mathcal M^{\infty ,1}_{(\Theta v)}}\le v(Z)\Vert \varphi \Vert _{L^1_{(v)}}, \end{aligned}$$

(4.1)

when $\varphi$ is used as window function in the definition of the $\mathcal M^{\infty ,1} _{(\Theta v)}$ norm.

Example 4.5

Let $v_n(Y) = \exp \left( (|y|+|\eta |)/n\right) , n\in {\mathbb N}$. We claim that for every $m\in {\mathbb N}$ there exist $a\in \bigcap _{n\in {\mathbb N}}{\mathcal M}^{\infty ,1}_{(\Theta v_n)}(\mathbb R^{2})$ such that $\mathrm{Op }^w(a)$ is invertible on $L^2({\mathbb R})$ but $(\mathrm{Op }^w (a))^{-1} \notin \mathrm{Op }^w({\mathcal M}^{\infty ,1}_{(\Theta v_m)}(\mathbb R^{2}))$.

In fact, let $m$ be fixed, $Z_k=(k,0)$ for $k\ge 1, \varepsilon > 0$, and let $a = 1 - e^{-\varepsilon }c_{Z_1}$. Obviously,

$$\begin{aligned} a \in \bigcap _{n\in {\mathbb N}}{\mathcal M}^{\infty ,1}_{(\Theta v_n)}(\mathbb R^{2}) \end{aligned}$$

in view of (4.1), and the Weyl operator $\mathrm{Op }^w(a) = \text{ Id } - e^{-\epsilon }S_{Z_1}$ is invertible on $L^2({\mathbb R})$ with inverse

$$\begin{aligned} \mathrm{Op }^w(a)^{-1} = \sum _{k=0}^{\infty }e^{-k\varepsilon }S_{Z_1}^k = \sum _{k=0}^{\infty }e^{-k\varepsilon }S_{Z_k}. \end{aligned}$$

Hence, the Weyl symbol of $\mathrm{Op }^w(a)^{-1}$ is given by

$$\begin{aligned} b = \sum _{k=0}^{\infty } e^{-k\varepsilon }c_{Z_k}. \end{aligned}$$

By (3) we have

$$\begin{aligned} ({\mathcal V}_\varphi b)(X,Y) = e^{2i\sigma (Y,X)} \sum _{k=0}^{\infty } e^{-k\varepsilon } c_{Z_k}(X)\varphi (Y+Z_k), \end{aligned}$$

which gives, after selecting $X=(0,\pi ),$

$$\begin{aligned} \sup _{X\in \mathbb R^{2}}|({\mathcal V}_\varphi b)(X,Y)| = \sum _{k=0}^{\infty } e^{-k\varepsilon }\varphi (Y+Z_k). \end{aligned}$$

By applying the $L^1_{(v_m)}$ norm on the latter equality we get

$$\begin{aligned} \Vert b\Vert _{{\mathcal M}^{\infty ,1}_{(\Theta v_m)}}&= \sum _{k=0}^{\infty } e^{-k\varepsilon }\int \varphi (Y+Z_k)v_m(Y)\, dY\\&\ge C_1 \sum _{k=0}^{\infty } e^{-k\varepsilon }\int \limits _{|Y+Z_k|\le 1}e^{|Y|/m}\, dY \ge C_2 \sum _{k=0}^{\infty } e^{-k\varepsilon +k/m}, \end{aligned}$$

for some positive constants $C_1$ and $C_2$. By choosing $\varepsilon <1/m$, it follows that the sum on the right-hand side diverges. Consequently, $b\notin {\mathcal M}^{\infty ,1}_{(\Theta v_m)}$, which proves the assertion. $\square$