The Fredholm spectrum of the sum and product of two operators
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- by Jack Shapiro and Morris Snow PDF
- Trans. Amer. Math. Soc. 191 (1974), 387-393 Request permission
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 ${\sigma _{\mathbf {\Phi }}}(A + B) \subseteq {\sigma _{\mathbf {\Phi }}}(A) + {\sigma _{\mathbf {\Phi }}}(B),{\sigma _{\mathbf {\Phi }}}(\overline {BA} ) \subseteq {\sigma _{\mathbf {\Phi }}}(A) \cdot {\sigma _{\mathbf {\Phi }}}(B)$, and ${\sigma _\Phi }(AB) \subseteq {\sigma _\Phi }(A){\sigma _\Phi }(B)$.References
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Additional Information
- © Copyright 1974 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 191 (1974), 387-393
- MSC: Primary 47A10
- DOI: https://doi.org/10.1090/S0002-9947-1974-0454682-8
- MathSciNet review: 0454682