Abstract harmonic analysis and asymptotic estimates of elements of inverse matrices
Let \(X\) be an infinite dimensional complex Banach space and let \(P_ 1, P_ 2,\dots\) be a sequence of disjoint projections in the Banach algebra \(\text{End }X\) of all bounded linear operators in \(X\) such that \(\sum_{k \geq 1}P_ k x\) is unconditionally convergent for any \(x \in X\). By use of these projections each \(A \in \text{End }X\) is represented in matrix form as (\(A_{jk}\)) with \(A_{ij} = P_ i AP_ j\). We set \(d_ k(A) = \sup_{i - j = k}| A_{ij}|\) for \(k = 0,\pm1,\pm2,\dots\). Suppose that \(A \in \text{End }X\) is invertible with inverse \(B = A^{-1}\). In the paper under review the author discusses how the rate of decrease of \(\{d_ k(A)\}\) is related to that of \(\{d_ k (B)\}\). His main theorem (Theorem 1) reads as follows: Let \(A \in \text{End }X\) be invertible with inverse \(B\) and satisfy one of the following: (1) \(\sum_{k = 1}^ \infty d_ k(A) \alpha(k) < \infty\) where \(\alpha\) is a weight function on \(Z\) satisfying \(\lim_{| k| \to \infty} \log \alpha(k)/| k| = 0\); (2) \(\lim_{| k| \to \infty} d_ k(A)| k|^{\gamma} = 0\) with \(\gamma > 1\); (3) \(d_ k(A) \leq C \exp(-\varepsilon | k|)\) with \(\varepsilon > 0\). Then, \(B\) satisfies the corresponding condition: \((1')\) \(\sum_{k = 1}^ \infty d_ k(B)\alpha(k) < \infty\), \((2')\) \(\lim_{| k| \to \infty} d_ k(B) | k|^ \gamma = 0\); or \((3')\) \(d_ k(B) \leq C_ 0 \exp (-\varepsilon_ 0 | k|)\), where \(C_ 0\) and \(\varepsilon_ 0\) depend on \(C\) and \(\varepsilon\) in the condition (3).
scientific article; zbMATH DE number 148630
scientific article