On polynomially Riesz operators

Abstract

A bounded linear operator A on a Banach space X is said to be ?polynomially Riesz?, if there exists a nonzero complex polynomial p such that the image p(A) is Riesz. In this paper we give some characterizations of these operators.

A characterization of unbounded generalized meromorphic operators

“…We recall that, an operator A ∈ L(X) is of Riesz-type if λI − A is a Fredholm operator and a(λI − A) = d(λI − A) < ∞, for each λ 0 and is meromorphic, if every non-zero isolated point of its spectrum is a pôle of the resolvent of A. In the bounded case, the generalized meromorphic operators are studied by S.Č.Živković Zlatanović et al in [17], they give some characterizing properties and they have established an equivalence between this class of operators and that of polynomially meromorphic operators (see Definition 3.8), which generalizes some of their results obtained in [16] on the structure of bounded linear generalized Riesz operators on Banach spaces. As an example of generalized meromorphic operators, polynomially compact more generally polynomially Riesz and generalized Riesz operators introduced in [11] (see Remark 3.4 ii).…”

Generalized Kato-Riesz decomposition

“…Every self-adjoint, as well as, unitary operator on Hilbert space have the spectrum contained in a line. The spectrum of a polynomially Riesz operator [14] or polynomially meromorphic operator [8] is most countable.…”

“…We shall say that an operator T ∈ B(X) is polynomially Riesz and write T ∈ Poly −1 R(X) if there exists a nonzero complex polynomial p(z) such that p(T ) ∈ R(X). Recall that if T ∈ Poly −1 R(X), then there exists a unique polynomial π T of minimal degree with leading coefficient 1 such that π T (T ) ∈ R(X) which we call the minimal polynomial of T (see [14]).…”

Polynomially Riesz perturbations II

“…An operator A ∈ B(X) is called polynomially Riesz if A ∈ Poly −1 (R(X)). We recall the following result [20,Theorem 11.1], [22,Theorem 2.3].…”