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Adjoint-abelian and iso-abelian operators on Banach spaces. (English) Zbl 1026.47001

Let \(X\) be a Banach space. An operator \(T\) on \(X\) is said to be an adjoint-abelian operator if \(T^*\varphi=\varphi T\), where \(\varphi:X\to X^*\) is a duality mapping, and \(T^*\) is the adjoint of \(T\). An invertible operator \(T\) is said to be iso-abelian if there is a duality mapping \(\varphi\) such that \(\varphi T=T^{*-1} \varphi\).
In this paper, certain properties of these operators are studied. For example: If \(X\) is smooth, strictly convex and reflexive, and \(T\) is adjoint-abelian, then \(T\) is adjoint-abelian, too. Or: If \(X\) is strictly convex and reflexive and \(T\) is iso-abelian, then the residual spectrum of \(T\) coincides with its approximate point spectrum.

MSC:

47A05 General (adjoints, conjugates, products, inverses, domains, ranges, etc.)
46C15 Characterizations of Hilbert spaces
47A10 Spectrum, resolvent
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