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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The Fredholm spectrum of the sum and product of two operators


Authors: Jack Shapiro and Morris Snow
Journal: Trans. Amer. Math. Soc. 191 (1974), 387-393
MSC: Primary 47A10
MathSciNet review: 0454682
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Abstract: Let $ C(X)$ denote the set of closed operators with dense domain on a Banach space X, and $ L(X)$ the set of all bounded linear operators on X. Let $ {\mathbf{\Phi }}(X)$ denote the set of all Fredholm operators on X, and $ {\sigma _{\mathbf{\Phi }}}(A)$ the set of all complex numbers $ {\mathbf{\lambda }}$ such that $ ({\mathbf{\lambda }} - A) \notin {\mathbf{\Phi }}(X)$. In this paper we establish conditions under which $ {\sigma _{\mathbf{\Phi }}}(A + B) \subseteq {\sigma _{\mathbf{\Phi }}}(A) + {\... ...} ) \subseteq {\sigma _{\mathbf{\Phi }}}(A) \cdot {\sigma _{\mathbf{\Phi }}}(B)$, and $ {\sigma _\Phi }(AB) \subseteq {\sigma _\Phi }(A){\sigma _\Phi }(B)$.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1974-0454682-8
PII: S 0002-9947(1974)0454682-8
Keywords: Fredholm operator, Fredholm spectrum, operational calculus
Article copyright: © Copyright 1974 American Mathematical Society