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, , depending on the parameter, , and having the following properties:

Let be the complement of , and let , where denotes the set of complex valued functions which are analytic on and at . We then use the operator, , to construct an operational calculus for *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, , to construct an analytic basis for .

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DOI:
https://doi.org/10.1090/S0002-9947-1973-0313853-0

Keywords:
Fredholm operator,
operational calculus,
analytic basis

Article copyright:
© Copyright 1973
American Mathematical Society