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This paper is concerned with the convergence rates of two processes \(\left\{A_{x}\right\}\) and \(\left\{B_{x}\right\}\), under the assumption that \(\left\|A_{x}\right\|=O(1)\) and there is a closed operator \(A\) such that \(B_{x} A \subset A B_{x}=I-A_{x},\left\|A A_{x}\right\|=O(e(\alpha))\), and \(B_{x}^{*} x^{*}=\varphi(\alpha) x^{*}\) for \(x^{*} \in R(A)^{\perp}\), where \(e(\alpha) \rightarrow 0\) and \(|\varphi(x)| \rightarrow \infty\). It was previously proved that \(\left\{A_{x}\right\}\) converges strongly on \(N(A) \oplus \overline{R(A)}\) to \(P\), the projection onto \(N(A)\) along \(\overline{R(A)}\), and \(\left\{B_{x}\right\}\) converges strongly on \(A(D(A) \cap \overline{R(A)})\) to \(A_{1}^{-1}\), the inverse operator of \(A_{1}=A \mid \widehat{R(A)}\). In this paper, the two processes are shown to be saturated with order \(O(e(\alpha))\), and their saturation classes are characterized. The result provides a unified approach to convergence rates for many particular mean ergodic theorems and for various methods of solving the equation \(A x=y\). We discuss in particular applications to integrated semigroups, cosine operator functions, and tensor product semigr

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Author

S.Y. Shaw

Publisher

Elsevier Science; Elsevier ; Elsevier Inc.; Elsevier BV (ISSN 0021-9045)

Published

1993

Language

EN

Field

Mathematics (Physical Sciences)