Asymptotic estimates for the entries of the matrices of inverse operators and harmonic analysis
type: Item
instance of: scholarly article
title: Asymptotic estimates for the entries of the matrices of inverse operators and harmonic analysis
zbMATH Open document ID: 0870.43003
DOI: 10.1007/BF02674895
published in: Siberian Mathematical Journal
publication date: 1997-08-07T00:00:00Z
review text: Let \(G\) be a countable discrete Abelian group and let \(S\) be a subset of \(G\). Let \(\text{End }X\) be the Banach algebra of bounded linear operators acting on a complex Banach space \(X\). Fix a projection-valued function \(P\: S\to\text{End }X\) such that \(P(i)P(j)=0\;(i\neq j)\) and, for every \(x\in X\), the series \(\sum\limits_{i\in S}P(i)x\) converges unconditionally to \(x\). By use of this resolution of identity, each \(A\in\text{End }X\) is represented in the matrix form \((A_{ij})\) with \(A_{ij}=P(i)AP(j)\), \(i,j\in S\). We set \(d_A(g)=\sup\limits_{i,j\in S,\;i-j=g}|A_{ij}|\), \(g\in G\). Suppose that \(A\in \text{End }X\) is invertible with inverse \(B=A^{-1}\). In the paper under review, the author studies how the rate of decrease of \(\{d_A(g)\}\) is related to that of \(\{d_B(g)\}\). The main theorem (Theorem 2) is as follows. Let \(A\in\text{End }X\) be invertible and satisfy one of the following conditions: (1) \(\sum\limits_{k\in G}d_A(k)\alpha (k)<\infty\) (where \(\alpha\) is a weight function on \(G\) satisfying \(\alpha (g)\geq 1\), \(\alpha (g_1+g_2)\leq \alpha (g_1)\alpha (g_2)\), and \(\lim\limits_{n\to \infty} n^{-1}\log \alpha (gn)=0\) for all \(g,g_1,g_2\in G\)); (2) \(\lim\limits_{k\to\infty}d_A(k)/\beta (k)=0\) (where \(\sum_{g\in G}\beta (g)\leq\infty\), \(\sum_{s\in G}\beta(g-s)\beta(s)\leq L\beta(g)\), and \(\lim_{n\to\infty} n^{-1}\log1/\beta(ng)\) \(=0\) for some \(L>0\) and for all \(g\in G\)); (3) \(G={\mathbb Z}^n\) and there are constants \(M=M(A)\) and \(\gamma =\gamma (A)\in (0,1)\) such that \(d_A(k)\leq M{\gamma}^{|k|}\), \(k\in{\mathbb Z}^n\), \(k=(k_1,\dots ,k_n)\), \(|k|=\max\limits_{1\leq i\leq n}|k_i|\). Then the matrix of the inverse operator \(B=A^{-1}\) satisfies the respective conditions: (1') \(\sum\limits_{k\in G} d_B(k)\alpha (k)<\infty\); (2') \(\lim\limits_{k\to\infty}d_B(k)/\beta (k)=0\); (3') \(d_B(k)\leq M_0\gamma_0^{|k|}\) for all \(k\in {\mathbb Z}^n\) and some \(M_0>0\), \(\gamma_0 \in (0,1)\). …
Mathematics Subject Classification ID: 43A50
zbMATH DE Number: 1046581
zbMATH Keywords: Banach algebra
zbMATH Keywords: discrete Abelian group
zbMATH Keywords: Fourier transform
zbMATH Keywords: asymptotic estimates
zbMATH Keywords: Beurling spectrum
author: Anatoly G. Baskakov
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf02674895
OpenAlex ID: W2084914001
description: scientific article; zbMATH DE number 1046581