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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



A generalized operational calculus developed from Fredholm operator theory

Authors: Jack Shapiro and Martin Schechter
Journal: Trans. Amer. Math. Soc. 175 (1973), 439-467
MSC: Primary 47A60; Secondary 47B30
MathSciNet review: 0313853
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Abstract: Let A be a closed operator on the Banach space X. We construct an operator, $ {R'_\lambda }(A)$, depending on the parameter, $ \lambda $, and having the following properties:

\begin{displaymath}\begin{array}{*{20}{c}} {(\lambda - A){{R'}_\lambda }(A) = I ... ...\ {{{R'}_\lambda }(A)(\lambda - A) = I + {F_2}} \\ \end{array} \end{displaymath}

where $ {F_1}$ and $ {F_2}$ are bounded finite rank operators. $ {R'_\lambda }(A)$ is defined and analytic in $ \lambda $ for all $ \lambda \in {\Phi _A}$ except for at most a countable set containing no accumulation point in $ {\Phi _A}$.

Let $ {\sigma _\Phi }(A)$ be the complement of $ {\Phi _A}$, and let $ f \in {\mathcal{A}'_\infty }(A)$, where $ {\mathcal{A}'_\infty }(A)$ denotes the set of complex valued functions which are analytic on $ {\sigma _\Phi }(A)$ and at $ (\infty )$. We then use the operator, $ {R'_\lambda }(A)$, to construct an operational calculus for A. $ f(A)$ is defined up to addition by a compact operator. We prove for our operational calculus analogues of the theorems for the classical operational calculus. We then extend a theorem of Kato by using the operator, $ {R'_\lambda }(A)$, to construct an analytic basis for $ N(A - \lambda )$.

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Keywords: Fredholm operator, operational calculus, analytic basis
Article copyright: © Copyright 1973 American Mathematical Society