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The Fredholm Spectrum of the Sum and Product of Two Operators

Jack Shapiro and Morris Snow
Transactions of the American Mathematical Society
Vol. 191 (Apr., 1974), pp. 387-393
DOI: 10.2307/1997004
Stable URL: http://www.jstor.org/stable/1997004
Page Count: 7
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Since scans are not currently available to screen readers, please contact JSTOR User Support for access. We'll provide a PDF copy for your screen reader.
The Fredholm Spectrum of the Sum and Product of Two Operators
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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 Φ(X) denote the set of all Fredholm operators on X, and σΦ(A) the set of all complex numbers λ such that $(\lambda - A) \nonin \Phi(X)$. In this paper we establish conditions under which $\sigma_\Phi(A + B) \subseteq \sigma_\Phi(A) + \sigma_\Phi(B), \sigma_\Phi({\overline BA}) \subseteq \sigma_\Phi(A) \cdot \sigma_\Phi(B)$, and $\sigma_\Phi(AB) \subseteq \sigma_\Phi(A) \sigma_\Phi(B)$.

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