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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

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

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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A generalized operational calculus developed from Fredholm operator theory
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by Jack Shapiro and Martin Schechter PDF
Trans. Amer. Math. Soc. 175 (1973), 439-467 Request permission

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: \[ \begin {array}{*{20}{c}} {(\lambda - A){{R’}_\lambda }(A) = I + {F_1},} \\ {{{R’}_\lambda }(A)(\lambda - A) = I + {F_2}} \\ \end {array} \] 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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Additional Information
  • © Copyright 1973 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 175 (1973), 439-467
  • MSC: Primary 47A60; Secondary 47B30
  • DOI: https://doi.org/10.1090/S0002-9947-1973-0313853-0
  • MathSciNet review: 0313853