# Comptes Rendus
Mathématique

Qiquan Fang and Chang Eon Shin

Norm-Controlled Inversion of Banach algebras of infinite matrices

Volume 358, issue 4 (2020), p. 407-414

Published online: 28 July 2020

https://doi.org/10.5802/crmath.54

<figure><img src="image_1.png" /></figure>

This article is licensed under the
CREATIVE COMMONS ATTRIBUTION 4.0 INTERNATIONAL LICENSE.
http://creativecommons.org/licenses/by/4.0/

<figure><img src="image_2.png" /></figure>

Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l'édition scientifique ouverte
www.centre-mersenne.org

e-ISSN : 1778-3569
# Norm-Controlled Inversion of Banach algebras of infinite matrices

Qiquan Fang<sup>a</sup> and Chang Eon Shin<sup>b</sup>

<sup>a</sup> Department of Mathematics, Zhejiang University of Science and Technology, Hangzhou, Zhejiang, 310023, China

<sup>b</sup> Chang Eon Shin: Department of Mathematics, Sogang University, Seoul, 04107, Korea. Email: shinc@sogang.ac.kr.

E-mail: qiquanfang@163.com.

**Abstract.** In this paper we provide a polynomial norm-controlled inversion of Baskakov-Gohberg-Sjöstrand Banach algebra in a Banach algebra $\mathcal{B}(\ell^q)$, $1 \le q \le \infty$, which is not a symmetric $*$-Banach algebra.

**2020 Mathematics Subject Classification.** 47G10, 45P05, 47B38, 31B10, 46E30.

**Funding.** The project is partially supported by NSF of China (Grant Nos. 11701513, 11771399, 11571306) and the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (NRF-2019R1F1A1051712).

Manuscript received 13th March 2020, revised 10th April 2020 and 20th April 2020, accepted 20th April 2020.

## 1. Introduction

N. Wiener in [19] proved that if a periodic function with absolutely convergent Fourier series never vanishes, then it also has an absolutely convergent Fourier series.

A Banach subalgebra $\mathcal{A}$ of a Banach algebra $\mathcal{B}$ having a common identity is called inverse-closed in $\mathcal{B}$ if $A \in \mathcal{A}$ with $A^{-1} \in \mathcal{B}$ implies $A^{-1} \in \mathcal{A}$. For a Banach subalgebra $\mathcal{A}$ which is inverse-closed in $\mathcal{B}$, we say that $\mathcal{A}$ admits a norm-controlled inversion in $\mathcal{B}$ if there exists a function $h: \mathbb{R}_+ \times \mathbb{R}_+ \to \mathbb{R}_+$ such that

$$ \|A^{-1}\|_{\mathcal{A}} \le h(\|A^{-1}\|_{\mathcal{B}}, \|A\|_{\mathcal{A}}) $$

for all $A \in \mathcal{A}$ with an inverse $A^{-1}$ in $\mathcal{B}$, where $\|\cdot\|_{\mathcal{A}}$ and $\|\cdot\|_{\mathcal{B}}$ are norms on $\mathcal{A}$ and $\mathcal{B}$, respectively.

N. Nikolski in [9] showed that the algebra of absolutely convergent Fourier series does not admit norm-controlled inversion in the algebra of continuous periodic functions.

Let a discrete set $\Lambda \subset \mathbb{R}^d$ be relatively-separated, that is,

$$ R(\Lambda) = \sup_{x \in \mathbb{R}^d} \sum_{\lambda \in \Lambda} \chi_{\lambda+[0,1]^d}(x) < \infty. \quad (1) $$
The set $\Lambda$ may not form a group. Our prime models are paraboloids $\{(x, y, z) : z = ax^2 + by^2, x, y \in \mathbb{Z}\}$, and elliptical hyperboloids $\{(x, y, z) : z^2 = ax^2 + by^2, x, y \in \mathbb{Z}\}$, where $a, b > 0$, and the set $\{k + \delta_k : k \in \mathbb{Z}^d, \delta_k \in [0, 1)^d\}$.

For $1 \le p \le \infty$ and $r \ge 0$, define the Baskakov–Gohberg–Sjöstrand (BGS) class $\mathscr{C}_{p,r}(\Lambda)$ by

$$ \mathscr{C}_{p,r}(\Lambda) = \{A = (a(\lambda, \lambda'))_{\lambda, \lambda' \in \Lambda} : \|A\|_{\mathscr{C}_{p,r}(\Lambda)} < \infty\} $$

where for $1 \le p < \infty$,

$$ \|A\|_{\mathscr{C}_{p,r}(\Lambda)} = \left( \sum_{k \in \mathbb{Z}^d} \sup_{\lambda, \lambda' \in \Lambda} |a(\lambda, \lambda')|^p (1 + |\lambda - \lambda'|)^{pr} \chi_{k+[0,1]^d}(\lambda - \lambda') \right)^{1/p}, \quad (2) $$

and for $p = \infty$,

$$ \|A\|_{\mathscr{C}_{\infty,r}(\Lambda)} = \sup_{\lambda, \lambda' \in \Lambda} |a(\lambda, \lambda')|(1 + |\lambda - \lambda'|)^r, \quad (3) $$

where for $x = (x_1, \dots, x_d) \in \mathbb{R}^d$, $|x| = \max(|x_1|, \dots, |x_d|)$. The above classes of infinite matrices form Banach algebras. In particular, when $p = \infty$, $\mathscr{C}_{\infty,r}(\Lambda)$ is called a Jaffard algebra and written as $J_r(\Lambda)$ with the norm $\| \cdot \|_{J_r(\Lambda)}$.

Let $\mathcal{A}$ and $\mathcal{B}$ be Banach $*$-algebras with common identity and involution. If $\mathcal{B}$ is a symmetric algebra (see [4]) and $\mathcal{A}$ is a differential subalgebra of $\mathcal{B}$, then $\mathcal{A}$ admits norm-controlled inversion in $\mathcal{B}$ (see [6, 7, 12]). Several algebras of infinite matrices with certain off-diagonal decay including Gröchenig–Schur algebra, Baskakov–Gohberg–Sjöstrand algebra and Jaffard algebra are shown to be differential $*$-subalgebra of $\mathcal{B}(\ell^2(\mathbb{Z}^d))$ (see [3, 5–8, 10–13, 15–18]), where for $1 \le q \le \infty$, $\mathcal{B}(\ell^q(V))$ denotes the space of all bounded linear operators on $\ell^q(V)$ with the norm $\| \cdot \|_{\mathcal{B}(\ell^q)}$ and $\ell^q(V)$ is the set of all $q$-summable sequences on $V$ with the norm $\| \cdot \|_q$.

Using the commutator trick and the partition of the identity, J. Sjöstrand in [14] showed Wiener's lemma for $\mathscr{C}_{1,0}(\mathbb{Z}^d)$. The polynomial norm-controlled inversion is studied in [6] for a differential subalgebra of a symmetric Banach algebra and in [7] for matrices in Besov algebras, Bessel algebras, Dales–Davie algebras, Baskakov–Gohberg–Sjöstrand algebras and Jaffard algebras. A. G. Baskakov in [1, 2] depending on Bochner–Phillips theorem proved that Jaffard algebras and Baskakov–Gohberg–Sjöstrand algebras with $p = 1$ admit norm-controlled inversion in $B(\ell^2)$. E. Samei and V. Shepelska in [11] showed that the convolution algebras as a subalgebra of a $C^*$-algebra admits an inversion controlled by a subexponential function. In [13], it is shown that a Beurling algebra admits a polynomial norm-controlled inversion in a symmetric Banach algebra $\mathcal{B}(\ell^2(V))$, where $V, E$ are the sets of vertices and edges in the graph $\mathcal{G} = (V, E)$, respectively, which has a complicated structure to prove the norm-controlled inversion.

In some applications in the field of mathematics and engineering, widespread-used algebras $\mathcal{B}$ of infinite matrices are Banach algebras $\mathcal{B}(\ell^p)$ for $p \in [1, \infty]$, while they are symmetric only when $p = 2$. The results in [1, 2, 6, 7, 11, 13] deal with the norm-controlled inversions in symmetric algebras, on the other hand, we provide the norm-controlled inversion in a nonsymmetric algebra. In this paper, for $1 \le p, q \le \infty$, $r > d(1 - 1/p)$ and a relatively-separated subset $\Lambda$ of $\mathbb{R}^d$, we give the simple proof for the norm-controlled inversion of the Baskakov–Gohberg–Sjöstrand subalgebra $\mathscr{C}_{p,r}(\Lambda)$ of a nonsymmetric Banach algebra $\mathcal{B}(\ell^q(\Lambda))$. We expect that the method in this paper can be applied to algebras of infinite matrices having off-diagonal decay with different weights from polynomial functions. The proof of the main theorem is based on commutator trick and the partition of the identity in [14].

For $a = (a_1, \dots, a_d) \in \mathbb{R}$, we write $[a] = ([a_1], \dots, [a_d])$, where for $b \in \mathbb{R}$, $[b]$ denotes the largest preceding integer of $b$.
## 2. Norm-Controlled Inversion

To state our result on norm-controlled inversion for localized infinite matrices, we recall some concepts. For a relatively-separated subset $\Lambda$ of $\mathbb{R}^d$ satisfying (1), we define Schur norm of an infinite matrix $A = (a(\lambda, \lambda'))_{\lambda, \lambda' \in \Lambda}$ by

$$ \|A\|_{\mathcal{S}(\Lambda)} = \max \left( \sup_{\lambda' \in \Lambda} \sum_{\lambda \in \Lambda} |a(\lambda, \lambda')|, \sup_{\lambda' \in \Lambda} \sum_{\lambda \in \Lambda} |a(\lambda, \lambda')| \right). \quad (4) $$

For any $1 \le q \le \infty$, one can show that the Schur class $\mathcal{S}(\Lambda)$ is a subalgebra of the Banach algebra $\mathcal{B}(\ell^q(\Lambda))$ and

$$ \|A\|_{\mathcal{B}(\ell^q(\Lambda))} \le \|A\|_{\mathcal{S}(\Lambda)}. \quad (5) $$

Let $A = (a(\lambda, \lambda'))_{\lambda, \lambda' \in \Lambda}$ be an infinite matrix in a BGS algebra, we define its approximation matrices $A_N, N \ge 1$, with finite bandwidth by

$$ A_N := (a(\lambda, \lambda')\chi_{[0,1]}(|\lambda - \lambda'|/N))_{\lambda, \lambda' \in \Lambda}. \quad (6) $$

We have the following properties of the algebra $\mathcal{C}_{p,r}(\Lambda)$ for $1 \le p \le \infty$ and $r > 0$.

**Proposition 1.** Let $1 \le p, q \le \infty$ and $r > d(1 - 1/p)$, and let $\Lambda$ be a relatively-separated subset of $\mathbb{R}^d$ satisfying (1). Then the following statements hold.

(1) The BGS algebra $\mathcal{C}_{1,0}(\Lambda)$ is a subalgebra of Schur algebra $\mathcal{S}(\Lambda)$, and

$$ \|A\|_{\mathcal{S}(\Lambda)} \le 2R(\Lambda) \|A\|_{\mathcal{C}_{1,0}(\Lambda)} \quad \text{for all } A \in \mathcal{C}_{1,0}(\Lambda). \quad (7) $$

(2) The BGS algebra $\mathcal{C}_{1,0}(\Lambda)$ is a subalgebra of the Banach algebra $\mathcal{B}(\ell^q(\Lambda))$, and

$$ \|Ac\|_{\ell^q(\Lambda)} \le 2R(\Lambda) \|A\|_{\mathcal{C}_{1,0}(\Lambda)} \|c\|_{\ell^q(\Lambda)} \quad \text{for all } A \in \mathcal{C}_{1,0}(\Lambda) \text{ and } c \in \ell^q(\Lambda). \quad (8) $$

(3) The BGS algebra $\mathcal{C}_{p,r}(\Lambda)$ is a subalgebra of the algebra $\mathcal{C}_{1,0}(\Lambda)$, and

$$ \|A\|_{\mathcal{C}_{1,0}(\Lambda)} \le \left( \frac{3^d r}{r - d(1 - 1/p)} \right)^{1-1/p} \|A\|_{\mathcal{C}_{p,r}} \quad \text{for all } A \in \mathcal{C}_{p,r}(\Lambda). \quad (9) $$

(4) The BGS algebra $\mathcal{C}_{p,r}(\Lambda)$ is a Banach algebra, and there exists a positive constant $C_1$ such that

$$ \|AB\|_{\mathcal{C}_{p,r}(\Lambda)} \le C_1 \|A\|_{\mathcal{C}_{p,r}(\Lambda)} \|B\|_{\mathcal{C}_{p,r}(\Lambda)} \quad \text{for all } A, B \in \mathcal{C}_{p,r}(\Lambda). \quad (10) $$

(5) A matrix $A$ in $\mathcal{C}_{p,r}(\Lambda)$ is well approximated by its truncated matrix $A_N, N \ge 1$, in the norm $\| \cdot \|_{\mathcal{C}_{1,0}(\Lambda)}$, and

$$ \|A - A_N\|_{\mathcal{C}_{1,0}(\Lambda)} \le \|A\|_{\mathcal{C}_{p,r}(\Lambda)} \times \begin{cases} \left( \frac{d(1-1/p)}{r-d(1-1/p)} \right)^{1-1/p} N^{-r+d(1-1/p)} & \text{if } p \ne 1, \\ N^{-r} & \text{if } p = 1. \end{cases} \quad (11) $$

### Proof.

(i) and (ii). Observing that for $\lambda \in \Lambda$,

$$ \sup_{\lambda \in \Lambda} \sum_{\tilde{\lambda} \in \Lambda} |a(\lambda, \tilde{\lambda})| \le R(\Lambda) \sup_{k \in \mathbb{Z}^d} \sum_{\lambda, \tilde{\lambda} \in \Lambda} |a(\lambda, \tilde{\lambda})| \chi_{k+[0,1]^d}(\lambda - \tilde{\lambda}) \quad (12) $$

and

$$ \sup_{\tilde{\lambda} \in \Lambda} \sum_{\lambda \in \Lambda} |a(\lambda, \tilde{\lambda})| \le 2R(\Lambda) \sup_{k \in \mathbb{Z}^d} \sum_{\lambda, \tilde{\lambda} \in \Lambda} |a(\lambda, \tilde{\lambda})| \chi_{k+[0,1]^d}(\lambda - \tilde{\lambda}), \quad (13) $$

which imply (7) in (i). Combining (5) and (7), one can get (8) in (ii).

(iii) and (v). By direct computation, we obtain (iii) and (v).
(iv). Let $1 \le p < \infty$ and $r > d(1-1/p)$, and take the matrices $A = (a(\lambda, \lambda'))_{\lambda, \lambda' \in \Lambda} \in \mathscr{C}_{p,r}(\Lambda)$ and $B = (b(\lambda, \lambda'))_{\lambda, \lambda' \in \Lambda} \in \mathscr{C}_{p,r}(\Lambda)$. Then by the fact that $|a+b|^r \le 2^r (|a|^r + |b|^r)$ for $a, b \in \mathbb{R}$, we have

$$
\begin{align}
\|AB\|_{\mathscr{C}_{p,r}(\Lambda)} &\le 2^r \left( \sum_{k \in \mathbb{Z}^d} \sup_{\lambda, \lambda' \in \Lambda} \left( \sum_{\lambda'' \in \Lambda} |a(\lambda, \lambda'')|(1+|\lambda-\lambda''|)^r |b(\lambda'', \lambda')| \right)^p \chi_{k+[0,1)d}(\lambda-\lambda') \right)^{1/p} \nonumber \\
&\quad + 2^r \left( \sum_{k \in \mathbb{Z}^d} \sup_{\lambda, \lambda' \in \Lambda} \left( \sum_{\lambda'' \in \Lambda} |a(\lambda, \lambda'')||b(\lambda'', \lambda')|(1+|\lambda''-\lambda'|)^r \right)^p \chi_{k+[0,1)d}(\lambda-\lambda') \right)^{1/p} \nonumber \\
&=: J_1 + J_2. \tag{14}
\end{align}
$$

Observing from (12) that

$$
\begin{align*}
J_1/2^r &\le \left( \sum_{k \in \mathbb{Z}^d} \sup_{\lambda, \lambda' \in \Lambda} (R(\Lambda) \sum_{\ell \in \mathbb{Z}^d} |a(\lambda, \tilde{\lambda})|(1+|\lambda-\tilde{\lambda}|)^r \chi_{k-\ell+(-1,1)d}(\lambda-\tilde{\lambda}) \right. \\
&\qquad \left. \times |b(\tilde{\lambda}, \lambda')| \chi_{\ell+[0,1)d}(\tilde{\lambda}-\lambda') \right)^{1/p} \\
&\le R(\Lambda) \|A\|_{\mathscr{C}_{p,r}(\Lambda)} \|B\|_{\mathscr{C}_{1,0}(\Lambda)},
\end{align*}
$$

and similarly $J_2/2^r \le R(\Lambda)\|A\|_{\mathscr{C}_{1,0}(\Lambda)}\|B\|_{\mathscr{C}_{p,r}(\Lambda)}$, these together with (14) and (9) in (iii) imply (10) with $C_1 = 2^{r+1}R(\Lambda) \left( \frac{3^d r}{r-d(1-1/p)} \right)^{1-1/p}$ for $1 \le p < \infty$ and $r > d(1-1/p)$.

For $p = \infty$, we have

$$
\begin{align}
\|AB\|_{\mathscr{C}_{\infty,r}(\Lambda)} &\le 2^r \left( \sup_{\lambda, \lambda' \in \Lambda} \sum_{\lambda'' \in \Lambda} |a(\lambda, \lambda'')|(1+|\lambda-\lambda''|)^r |b(\lambda'', \lambda')| \right) \nonumber \\
&\quad + 2^r \left( \sup_{\lambda, \lambda' \in \Lambda} \sum_{\lambda'' \in \Lambda} |a(\lambda, \lambda'')||b(\lambda'', \lambda')|(1+|\lambda''-\lambda'|)^r \right) \nonumber \\
&\le 2^r \left( \|B\|_{\mathscr{S}(\Lambda)} \|A\|_{\mathscr{C}_{\infty,r}(\Lambda)} + \|A\|_{\mathscr{S}(\Lambda)} \|B\|_{\mathscr{C}_{\infty,r}(\Lambda)} \right). \tag{15}
\end{align}
$$

The desired result (10) follows from (7) and (15) for $p = \infty$. $\square$

Let $h(t) := \min(\max(2 - |t|, 0), 1)$ be the trapezoidal-shaped function. The function $h$ is Lipschitz continuous.

For $1 \le q \le \infty$, a positive integer $N$ and $A \in \mathscr{B}(\ell^q(\Lambda))$, define localization operators $\Psi_i^N$, $\chi_i^N$ and commutators $[\Psi_i^N, A]$, $i \in \mathbb{Z}^d$, by

$$
\Psi_i^N c := (h(\lambda/N - i)c(\lambda))_{\lambda \in \Lambda} \qquad (16)
$$

$$
\chi_i^N c := (\chi_{[0,1]}(|i - \lambda|/N) c(\lambda))_{\lambda \in \Lambda} \quad (17)
$$

and

$$
[\Psi_i^N, A]c = \Psi_i^N Ac - A\Psi_i^N c \quad \text{for } c := (c(\lambda))_{\lambda \in \Lambda} \in \ell^q(\Lambda),
$$

where for a set $I$, $\chi_I(\cdot)$ denotes the characteristic function on $I$.

In the next theorem, we show the norm-controlled inversion of a Banach algebra $\|A\|_{\mathscr{C}_{p,r}(\Lambda)}$ in $\mathscr{B}(\ell^q)(\Lambda)$ which is not a symmetric*-Banach algebra.

**Theorem 2.** Let $1 \le p, q \le \infty$, $r > d(1-1/p)$, $\Lambda$ be a relatively-separated subset of $\mathbb{R}^d$ satisfying (1), and let $A \in \mathscr{C}_{p,r}(\Lambda)$ be invertible in $\mathscr{B}(\ell^q(\Lambda))$. Then there exists an absolute constant $C$, independent of $A$, such that

$$
\|A^{-1}\|_{\mathscr{C}_{p,r}(\Lambda)} \le C \|A^{-1}\|_{\mathscr{B}(\ell^q)} (\|A^{-1}\|_{\mathscr{B}(\ell^q)} \|A\|_{\mathscr{C}_{p,r}(\Lambda)})^{(d/p+r)/\min(1,r-d(1-1/p))} \\
\times \begin{cases} 1 & \text{if } r \ne d(1-1/p)+1, \\ (\ln(\|A^{-1}\|_{\mathscr{B}(\ell^q)} \|A\|_{\mathscr{C}_{p,r}(\Lambda)}))^{(d/p+r)(1-1/p)} & \text{if } r = d(1-1/p)+1. \end{cases} \tag{18}
$$
**Proof.** We follow the arguments in [18]. Let $1 \le q < \infty$ and $1 < p < \infty$. When $q = \infty$, $p = 1$ or $p = \infty$, we can follow the same proof. Write $p' = p/(p-1)$. Define the linear operator $\Phi_N$ on $\ell^q(\Lambda)$ by

$$ \Phi_N c := (H(\lambda/N)c(\lambda))_{\lambda \in \Lambda} \quad \text{for } c := (c(\lambda))_{\lambda \in \Lambda} \in \ell^q(\Lambda), $$

where $H(t) = (\sum_{i \in \mathbb{Z}^d} h(t-i))^{-1}$, $t \in \mathbb{R}^d$. By the invertibility of $A$, we have that

$$ \begin{aligned} \|A^{-1}\|_{\mathcal{B}(\ell^q)}^{-1} \|\Psi_i^N c\|_q &\le \|\Psi_i^N A c\|_q + \|[\Psi_i^N, A]c\|_q \\ &\le \|\Psi_i^N A c\|_q + \|\chi_{iN}^{3N}[\Psi_i^N, A]c\|_q + \|(I - \chi_{iN}^{3N})A\chi_{iN}^{2N}\Psi_i^N c\|_q \\ &\le \|\Psi_i^N A c\|_q + \sum_{j \in \mathbb{Z}^d} \|\chi_{iN}^{3N}[\Psi_i^N, A]\Phi_N \Psi_j^N c\|_q + \|A - A_N\|_{\mathcal{C}_{1,0}(\Lambda)} \|\Psi_i^N c\|_q. \end{aligned} \quad (19) $$

Choose $N$ so large that

$$ N \ge \left( 2 \left( \frac{d}{rp' - d} \right)^{1/p'} \|A^{-1}\|_{\mathcal{B}(\ell^q)} \|A\|_{\mathcal{C}_{p,r}(\Lambda)} \right)^{1/(r-d/p')} \quad (20) $$

It follows from (11), (19) and (20) that

$$ \|A^{-1}\|_{\mathcal{B}(\ell^q)}^{-1} \|\Psi_i^N c\|_q \le 2 \|\Psi_i^N A c\|_q + 2 \sum_{j \in \mathbb{Z}^d} \|\chi_{iN}^{3N}[\Psi_i^N, A] \Phi_N \Psi_j^N c\|_q. \quad (21) $$

For $i, j \in \mathbb{Z}^d$ with $|i-j| \le 10$, we obtain from Lipschitz property of $h$ that

$$ \begin{aligned} \|\chi_{iN}^{3N}[\Psi_i^N, A]\Phi_N \Psi_j^N c\|_q &\le \left( \sum_{\lambda \in \Lambda} \left( \sum_{\lambda' \in \Lambda} \chi_{iN}^{3N}(\lambda) |a(\lambda, \lambda')| \|h(\lambda/N - i) - h(\lambda'/N - i)|h(\lambda'/N - j)|c(\lambda')| \right) \right)^{1/q} \\ &\le \left( \sum_{\lambda' \in \Lambda} \left( \sum_{|\lambda - \lambda'| \le 15N} \min(|\lambda - \lambda'|/N, 1) |a(\lambda, \lambda')| \|h(\lambda'/N - j)|c(\lambda')| \right) \right)^{1/q} \\ &\le R(\Lambda) \left( \sum_{|k| \le 15N} \min((|k|+1)/N, 1) A_k \right) \|\Psi_j^N c\|_q, \end{aligned} \quad (22) $$

and for $i, j \in \mathbb{Z}^d$ with $|i-j| > 10$, we have that

$$ \begin{aligned} \|\chi_{iN}^{3N}[\Psi_i^N, A]\Phi_N \Psi_j^N c\|_q &\le \left( \sum_{\lambda \in \Lambda} \left( \sum_{\lambda' \in \Lambda} \chi_{iN}^{3N}(\lambda) |a(\lambda, \lambda')| \|h(\lambda'/N - j)|c(\lambda')| \right) \right)^{1/q} \\ &\le \left( \sum_{\lambda \in \Lambda} \left( \sum_{(|i-j|-5)N \le |\lambda-\lambda'| \le (|i-j|+5)N} |a(\lambda, \lambda')| \|h(\lambda'/N - j)|c(\lambda')| \right) \right)^{1/q} \\ &\le 2R(\Lambda) \left( \sum_{(|i-j|-5)N \le |k| \le (|i-j|+5)N} A_k \right) \|\Psi_j^N c\|_q, \end{aligned} \quad (23) $$

where

$$ A_k = \sup_{\lambda, \lambda' \in \Lambda} |a(\lambda, \lambda')| \chi_{k+[0,1]^d}(\lambda - \lambda'). $$

We define a function

$$ \tilde{V}_{A,N}(i) = 2R(\Lambda) \times \begin{cases} \sum_{|k| \le 15N} \min((|k|+1)/N, 1) A_k & \text{if } |i| \le 10, \\ \sum_{(|i|-5)N \le |k| \le (|i|+5)N} A_k & \text{if } |i| > 10. \end{cases} \quad (24) $$

We have from (21), (22), (23) and (24) that

$$ \|\Psi_i^N c\|_q \le 2 \|A^{-1}\|_{\mathcal{B}(\ell^q)} \|\Psi_i^N A c\|_q + \sum_{j \in \mathbb{Z}^d} V_{A,N}(i,j) \|\Psi_j^N c\|_q, \quad (25) $$
where $V_{A,N} := (V_{A,N}(i, j))_{i,j \in \mathbb{Z}^d}$ and $V_{A,N}(i, j) = 2 \|A^{-1}\|_{\mathcal{B}(\ell^q)} \tilde{V}_{A,N}(i-j)$. We write

$$V_{A,N}^{\ell} = ((V_{A,N})^{\ell}(i, j))_{i,j \in \mathbb{Z}^d}. \qquad (26)$$

We apply (25) repeatedly to have that

$$\|\Psi_i^N c\|_q \le 2\|A^{-1}\|_{\mathcal{B}(\ell^q)} \|\Psi_i^N A c\|_q + 2\|A^{-1}\|_{\mathcal{B}(\ell^q)} \sum_{\ell=1}^{n-1} \sum_{j \in \mathbb{Z}^d} V_{A,N}^{\ell}(i,j) \|\Psi_j^N A c\|_q \\ + \sum_{j \in \mathbb{Z}^d} V_{A,N}^n(i,j) \|\Psi_j^N c\|_q. \quad (27)$$

Note that

$$\|A\|_{\mathcal{C}_{p,r}(\Lambda)} \le 2^r \left( \sum_{k \in \mathbb{Z}^d} A_k^p (1+|k|)^{pr} \right)^{1/p} \quad (28)$$

and

$$\left( \sum_{k \in \mathbb{Z}^d} A_k^p (1 + |k|)^{pr} \right)^{1/p} \le 2^r \|A\|_{\mathcal{C}_{p,r}(\Lambda)}. \quad (29)$$

Observing from (29) that

$$\left( \sum_{|i|>10} (\tilde{V}_{A,N}(i))^p (1+|i|)^{pr} \right)^{1/p} \\ \le 2R(\Lambda)(10N)^{d(p-1)/p} 4^r N^{-r} \left( \sum_{|i|>10} \sum_{(|i|-5)N \le |k| \le ((|i|+5)N)} A_k^p (1+|k|)^{pr} \right)^{1/p} \\ \le 10^d 2^{3r+1} R(\Lambda) N^{-r+d/p'} \|A\|_{\mathcal{C}_{p,r}(\Lambda)} \quad (30)$$

and

$$\left( \sum_{|i| \le 10} (\tilde{V}_{A,N}(i))^p (1+|i|)^{pr} \right)^{1/p} \\ \le 2R(\Lambda) \left( \sum_{|i| \le 10} (1+|i|)^{pr} \left( \sum_{|k| \le 15N} \min\left(\frac{|k|+1}{N}, 1\right) A_k \right)^p \right)^{1/p} \\ \le 2R(\Lambda) 11^{r+d/p} \left( N^{-1} \sum_{|k| \le N-1} (|k|+1) A_k + \sum_{N \le |k| \le 15N} A_k \right) \\ \le R(\Lambda) 2^{r+2} 11^{r+d/p} \|A\|_{\mathcal{C}_{p,r}(\Lambda)} \\ \times \left\{ \begin{array}{ll} \left( \frac{d}{rp'-d} \right)^{1/p'} + \left( \frac{2d+rp'-d}{|rp'-p'-d|} \right)^{1/p'} & \text{if } r \ne d/p' + 1, \\ (2d)^{1/p'} N^{-1} (\ln(N+1))^{1/p'} & \text{if } r = d/p' + 1, \end{array} \right. \quad (31)$$

we have that

$$\|V_{A,N}\|_{\mathcal{C}_{p,r}(\mathbb{Z}^d)} \le D_{d,p,r} \|A^{-1}\|_{\mathcal{B}(\ell^q)} \|A\|_{\mathcal{C}_{p,r}(\Lambda)} \times \begin{cases} N^{-\min(1,r-d/p')} & \text{if } r \ne d/p' + 1, \\ N^{-1}(\ln(N+1))^{1/p'} & \text{if } r = d/p' + 1, \end{cases} \quad (32)$$

where

$$D_{d,p,r} = 2^{3r+3} 11^{d+r} R(\Lambda) \times \begin{cases} \left( \frac{d}{rp'-d} \right)^{1/p'} + \left( \frac{2d+rp'-d}{|rp'-p'-d|} \right)^{1/p'} & \text{if } r \ne d/p' + 1, \\ (2d)^{1/p'} & \text{if } r = d/p' + 1. \end{cases}$$

Choose the smallest integer $N_0$ satisfying

$$1 \ge 2C_1 D_{d,p,r} \|A^{-1}\|_{\mathcal{B}(\ell^q)} \|A\|_{\mathcal{C}_{p,r}(\Lambda)} \times \begin{cases} N_0^{-\min(1,r-d/p')} & \text{if } r \ne d/p' + 1, \\ N_0^{-1}(\ln N_0)^{1/p'} & \text{if } r = d/p' + 1, \end{cases} \quad (33)$$
where $C_1$ is the constant in (10). Then $N_0$ satisfies (20), so (21) and (27) hold. From (33) there exist absolute constants $C_2, C_3$ such that for $r \neq d/p' + 1$,

$$N_0 \leq C_2 (\|A^{-1}\|_{\mathcal{B}(\ell q)} \|A\|_{\mathcal{C}_{p,r}(\Lambda)})^{1/\min(1,r-d/p')} \quad (34)$$

and for $r = d/p' + 1$

$$N_0 \leq C_3 \|A^{-1}\|_{\mathcal{B}(\ell q)} \|A\|_{\mathcal{C}_{p,r}(\Lambda)} (\ln(\|A^{-1}\|_{\mathcal{B}(\ell q)} \|A\|_{\mathcal{C}_{p,r}(\Lambda)}))^{1/p'}. \quad (35)$$

It follows from (10), (32) and (33) that for $n \in \mathbb{N}$,

$$\|V_{A,N_0}^n \|_{\mathcal{C}_{p,r}(\mathbb{Z}^d)} \leq 2^{-n}. \quad (36)$$

This combining with Proposition 1 (ii) and (iii) implies that

$$\lim_{n \to \infty} \left\| \left( \sum_{j \in \mathbb{Z}^d} V_{A,N_0}^n (i,j) \|\Psi_j^{N_0} c\|_q \right)_{i \in \mathbb{Z}^d} \right\|_q = 0,$$

so taking $n \to \infty$ in (27), we have that

$$\|\Psi_i^{N_0} c\|_q \leq \|A^{-1}\|_{\mathcal{B}(\ell q)} \sum_{j \in \mathbb{Z}^d} W_{A,N_0}(i,j) \|\Psi_j^{N_0} A c\|_q, \quad (37)$$

where $W_{A,N_0}(i,j) = 2I + 2 \sum_{\ell=1}^{\infty} V_{A,N_0}^{\ell}(i,j)$. It follows from (36) that

$$\|W_{A,N_0}\|_{\mathcal{C}_{p,r}(\mathbb{Z}^d)} \leq 4. \quad (38)$$

We define an infinite matrix $H_{A,N_0} := (H_{A,N_0}(\lambda, \lambda'))_{\lambda, \lambda' \in \Lambda}$, where

$$H_{A,N_0}(\lambda, \lambda') = \sum_{|mN_0 - \lambda| \le N_0} \sum_{|nN_0 - \lambda'| \le 2N_0} W_{A,N_0}(m,n). \quad (39)$$

Since for $k \in \mathbb{Z}^d$,

$$\sup_{\lambda, \lambda' \in \Lambda} \chi_{k+[0,1]^d}(\lambda - \lambda') \sum_{|mN_0 - \lambda| \le N_0} \sum_{|nN_0 - \lambda'| \le 2N_0} W_{A,N_0}(m,n) \\ \le \sum_{\varepsilon = \{-4, -3, \dots, 3, 4\}^d} \sum_{m-n = [k/N_0] + \varepsilon} W_{A,N_0}(m,n),$$

we have from (38) that

$$\|H_{A,N_0}\|_{\mathcal{C}_{p,r}(\Lambda)} \le 10^{d/p+r+d+1} N_0^{d/p+r}. \quad (40)$$

Let $A^{-1} := (a^{-1}(\lambda, \lambda'))_{\lambda, \lambda' \in \Lambda}$, $a_{\lambda'}^{-1} = (a^{-1}(\lambda, \lambda'))_{\lambda \in \Lambda}$ and $a_{\lambda'}^{-1}(\lambda) = a^{-1}(\lambda, \lambda')$. Replace c by $a_{\lambda'}^{-1}$ in (37) to get that for $\lambda \in \Lambda$ and $m \in \mathbb{Z}^d$ with $|mN_0 - \lambda| \le N_0$

$$\begin{aligned} |a^{-1}(\lambda, \lambda')| &\le \|\Psi_m^{N_0} a_{\lambda'}^{-1}\|_q \\ &\le \|A^{-1}\|_{\mathcal{B}(\ell q)} \sum_{|mN_0 - \lambda| \le N_0} \sum_{|nN_0 - \lambda'| \le 2N_0} W_{A,N_0}(m,n) \|\Psi_n^{N_0} A a_{\lambda'}^{-1}\|_q \\ &\le \|A^{-1}\|_{\mathcal{B}(\ell q)} H_{A,N_0}(\lambda, \lambda'). \end{aligned} \quad (41)$$

It follows from (40) and (41) that

$$\|A^{-1}\|_{\mathcal{C}_{p,r}(\Lambda)} \le \|A^{-1}\|_{\mathcal{B}(\ell q)} \|H_{A,N_0}\|_{\mathcal{C}_{p,r}(\Lambda)} \le 10^{d/p+r+d+1} N_0^{d/p+r} \|A^{-1}\|_{\mathcal{B}(\ell q)}. \quad (42)$$

From (34), (35) and (42), (18) holds.

<figure><img src="image_3.png" /></figure>

Acknowledgements

The authors would like to thank the editor and an anonymous reviewer for their comments which improved the quality of this paper.
References

[1] A. G. Baskakov, "Wiener's theorem and asymptotic estimates for elements of inverse matrices", Funkts. Anal. Prilozh. **24** (1990), no. 3, p. 64-65, translation in *Funct. Anal. Appl. 24* (1990), no. 3, p. 222-224.

[2] ——, "Asymptotic estimates for elements of matrices of inverse operators, and harmonic analysis", Sib. Mat. Zh. **38** (1997), no. 1, p. 14-28, translation in *Sib. Math. J.* **38** (1997), no. 1, p. 10-22.

[3] L. H. Brandenburg, "On identifying the maximal ideals in Banach Algebras", *J. Math. Anal. Appl.* **50** (1975), p. 489-510.

[4] K. Gröchenig, "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 and Image Analysis*, Applied and Numerical Harmonic Analysis, Birkhäuser, 2010.

[5] K. Gröchenig, A. Klotz, "Noncommutative approximation: inverse-closed subalgebras and off-diagonal decay of matrices", Constr. Approx. **32** (2010), no. 3, p. 429-466.

[6] ——, "Norm-controlled inversion in smooth Banach algebras. I", *J. Lond. Math. Soc.* **88** (2013), no. 1, p. 49-64.

[7] ——, "Norm-controlled inversion in smooth Banach algebras. II", *Math. Nachr.* **287** (2014), no. 8-9, p. 917-937.

[8] S. Jaffard, "Propriétés des matrices "bien localisées" près de leur diagonale et quelques applications", *Ann. Inst. Henri Poincaré, Anal. Non Linéaire* **7** (1990), no. 5, p. 461-476.

[9] N. Nikolski, "In search of the invisible spectrum", *Ann. Inst. Fourier* **49** (1999), no. 6, p. 1925-1966.

[10] K. S. Rim, C. E. Shin, Q. Sun, "Stability of localized integral operators on weighted $L^p$ spaces", Numer. Funct. Anal. Optim. **33** (2012), no. 7-9, p. 1166-1193.

[11] E. Samei, V. Shepelska, "Norm-controlled inversion in weighted convolution algebras", *J. Fourier Anal. Appl.* **25** (2019), no. 6, p. 3018-3044.

[12] C. E. Shin, Q. Sun, "Differential subalgebras and norm-controlled inversion", submitted, https://arxiv.org/abs/1911.08679, 2019.

[13] ——, "Polynomial control on stability", *J. Funct. Anal.* **276** (2019), no. 1, p. 148-182.

[14] J. Sjöstrand, "Wiener type algebra of pseudodifferential operators", Sémin. Équ. Dériv. Partielles **1994-1995** (1995), article ID 4 (19 pages).

[15] Q. Sun, "Wiener's lemma for infinite matrices with polynomial off-diagonal decay", C. R. Math. Acad. Sci. Paris **340** (2005), p. 567-570.

[16] ——, "Wiener's lemma for infinite matrices", *Trans. Am. Math. Soc.* **359** (2007), no. 7, p. 3099-3123.

[17] ——, "Wiener's lemma for localized integral operators", *Appl. Comput. Harmon. Anal.* **25** (2008), no. 2, p. 148-167.

[18] ——, "Wiener's lemma for infinite matrices. II.", *Constr. Approx.* **34** (2011), no. 2, p. 209-235.

[19] N. Wiener, "Tauberian theorem", *Ann. Math.* **33** (1932), p. 1-100.