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# Properties which normal operators share with normal derivations and related operators

Anderson, Joel and Foias, Ciprian (1975) Properties which normal operators share with normal derivations and related operators. Pacific Journal of Mathematics, 61 (2). pp. 313-325. ISSN 0030-8730. https://resolver.caltech.edu/CaltechAUTHORS:ANDpmj75

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## Abstract

Let \$S\$ and \$T\$ be (bounded) scalar operators on a Banach space \$\scr X\$ and let \$C(T,S)\$ be the map on \$\scr B(\scr X)\$, the bounded linear operators on \$\scr X\$, defined by \$C(T,S)(X)=TX-XS\$ for \$X\$ in \$\scr B(\scr X)\$. This paper was motivated by the question: to what extent does \$C(T,S)\$ behave like a normal operator on Hilbert space? It will be shown that \$C(T,S)\$ does share many of the special properties enjoyed by normal operators. For example, it is shown that the range of \$C(T,S)\$ meets its null space at a positive angle and that \$C(T,S)\$ is Hermitian if \$T\$ and \$S\$ are Hermitian. However, if \$\scr X\$ is a Hilbert space then \$C(T,S)\$ is a spectral operator if and only if the spectrum of \$T\$ and the spectrum of \$S\$ are both finite.

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Additional Information:Euclid Identifier: euclid.pjm/1102868028 Zentralblatt Math Identifier : 0324.47018 Mathmatical Reviews number (MathSciNet): MR0412889
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