Asymptotic estimates for the entries of the matrices of inverse operators and harmonic analysis
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 \(M_0>0\), \(\gamma_0 \in (0,1)\). The author uses this result to generalize the well-known N. Wiener theorem on absolutely convergent Fourier series. Select some function \(f\) in the algebra \(L_1(\mathbb R)\) whose Fourier transform possesses the properties \(\hat f(0)=1\) and \(\text{supp}\hat f\subset [-1/2,1/2]\). Given \(\varphi\in C(\mathbb R)\), define the function \(d_{\varphi}(\lambda )=|(fe_{\lambda})*\varphi|\), where \(e_{\lambda}(t)=\exp i\lambda t\). Suppose that the function \(\varphi\in C(\mathbb R)\) satisfies the condition \(\inf\limits_{t\in \mathbb R}|\varphi(t)|>0\) and one of the following conditions: (1) \(\int\limits_{\mathbb R}d_{\varphi}(\lambda)|\lambda|^qd\lambda<\infty\) for some \(q>1\); (2) \(\lim\limits_{|\lambda|\to\infty} d_{\varphi}(\lambda)|\lambda|^q=0\) for some \(q>1\); (3) \(d_{\varphi}(\lambda)\leq C\exp(-\varepsilon|\lambda|)\) for some \(C,\varepsilon >0\). Then the function \(\psi(t)=1/\varphi(t)\) satisfies the respective conditions: (1') \(\int\limits_{\mathbb R}d_{\varphi}(\lambda)|\lambda |^qd\lambda < \infty\); (2') \(\lim\limits_{|\lambda |\to\infty}d_{\varphi}(\lambda)|\lambda |^q=0\); (3') \(d_{\varphi}(\lambda)\leq C_0\exp(-{\varepsilon}_0|\lambda |)\) for some \(C_0, {\varepsilon}_0>0\). The article generalizes some earlier results of the author, see [Mat. Zametki 52, No.2, 17-26 (1992; Zbl 0805.43002)]and [Funkts. Anal. Prilozh. 29, No.2, 61-64 (1995; Zbl 0849.47001)].
scientific article; zbMATH DE number 1046581
scientific article