A generalized operational calculus developed from Fredholm operator theory
Authors:
Jack Shapiro and Martin Schechter
Journal:
Trans. Amer. Math. Soc. 175 (1973), 439467
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: where and are bounded finite rank operators. is defined and analytic in for all except for at most a countable set containing no accumulation point in . 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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 Martin Schechter, Basic theory of Fredholm operators, Ann. Scuola Norm. Sup. Pisa (3) 21 (1967), 261280. MR 36 #6977. MR 0223930 (36:6977)
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 , Principles of functional analysis, Academic Press, New York, 1971. MR 0445263 (56:3607)
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 , Spectral theory of closed distributive operators, Acta Math. 84 (1951), 189224. MR 12, 717. MR 0040582 (12:717c)
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029947197303138530
PII:
S 00029947(1973)03138530
Keywords:
Fredholm operator,
operational calculus,
analytic basis
Article copyright:
© Copyright 1973
American Mathematical Society
