## 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