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RESEARCH PRODUCT

Polaroid-Type Operators

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In this chapter we introduce the classes of polaroid-type operators, i.e., those operators T ∈ L(X) for which the isolated points of the spectrum σ(T) are poles of the resolvent, or the isolated points of the approximate point spectrum σap(T) are left poles of the resolvent. We also consider the class of all hereditarily polaroid operators, i.e., those operators T ∈ L(X) for which all the restrictions to closed invariant subspaces are polaroid. The class of polaroid operators, as well as the class of hereditarily polaroid operators, is very large. We shall see that every generalized scalar operator is hereditarily polaroid, and this implies that many classes of operators acting on Hilbert spaces, obtained by relaxing the condition of normality, are hereditarily polaroid. Multipliers of commutative semi-simple Banach algebras, and in particular every convolution operators Tμ, defined in the group algebras L1(G), where G is a locally compact abelian group, are also hereditarily polaroid.