## Spectrally compact operators.(English)Zbl 1278.47041

Summary: We define the concept of a spectrally compact operator and study the basic properties of these operators. We show that the class of spectrally compact operators is strictly contained in the class of compact operators and in the class of spectrally bounded operators. It is also proved that the set of spectrally compact operators on a spectrally normed space $$E$$ is a right ideal of $$SB(E)$$ and, in certain cases, it is a two sided ideal. We also study the spectral adjoint of a spectrally compact operator.

### MSC:

 47B48 Linear operators on Banach algebras 46B99 Normed linear spaces and Banach spaces; Banach lattices 47L10 Algebras of operators on Banach spaces and other topological linear spaces
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