ANALYTICALLY RIESZ OPERATORS AND WEYL AND BROWDER TYPE THEOREMS (original) (raw)

On polynomially Riesz operators

Filomat, 2014

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.

On the Riesz Idempotent for a Class of Operators

Mathematical Proceedings of the Royal Irish Academy, 2007

Let A be a bounded linear operator acting on infinite dimensional separable Hilbert space H. Browder's theorem and a-Browder's theorem are related to an important property which has a leading role on local spectral theory: the single valued extension property see the recent monograph by Laursen and Neumann . The study of operators satisfying Browder's theorem is of significant interest, and is currently being done by a number of mathematicians around the world. In order to generalize some recent results in the literature, we prove that a-Browder's theorem holds for a large class of operators containing the classes of normal, hyponormal, p-hyponormal and M -hyponormal operators. In , Stampfli proved that if A is hyponormal and λ ∈ σ(A) is isolated, then the Riesz idempotent E with respect to λ is self-adjoint and satisfies EH = ker(A − λ) = ker(A − λ) * . It is intersting to study whether Stampli's result holds for other classes of operators containing the class of hyponormal operators. In this paper we prove that Stampfli's result holds for algebraically class H(q) operators. *

On a Riesz basis of exponentials related to a family of analytic operators and application

Journal of Pseudo-Differential Operators and Applications, 2018

In this paper, we are interested by the perturbed operator T (ε) := T 0 + εT 1 + ε 2 T 2 + • • • + ε k T k + • • • where ε ∈ C, T 0 is a closed densely defined linear operator on a separable Hilbert space H with domain D(T 0) having isolated eigenvalues with multiplicity one whereas T 1 , T 2 ,. .. are linear operators on H having the same domain D ⊃ D(T 0) and satisfying a specific growing inequality. The basic idea here is to investigate under sufficient conditions the existence of Riesz bases of exponentials, where the exponents corresponding as a sequence of eigenvalues of T (ε), can be developed as entire series of ε. An application to a nonself-adjoint problem describing the radiation of a vibrating structure in a light fluid is presented.

Generalized Browder's and Weyl's theorems for Banach space operators

Journal of Mathematical Analysis and Applications, 2007

We find necessary and sufficient conditions for a Banach space operator T to satisfy the generalized Browder's theorem, and we obtain new necessary and sufficient conditions to guarantee that the spectral mapping theorem holds for the B-Weyl spectrum and for polynomials in T . We also prove that the spectral mapping theorem holds for the B-Browder spectrum and for analytic functions on an open neighborhood of σ(T ). As applications, we show that if T is algebraically M -hyponormal, or if T is algebraically paranormal, then the generalized Weyl's theorem holds for f (T ), where f ∈ H((T )), the space of functions analytic on an open neighborhood of σ(T ). We also show that if T is reduced by each of its eigenspaces, then the generalized Browder's theorem holds for f (T ), for each f ∈ H(σ(T )).

A Riesz theory in von Neumann algebras

Pacific Journal of Mathematics, 1991

An operator T is called a Riesz operator relative to a von Neumann algebra si if T-λl is Fredholm relative to si for each 1^0. Properties of Riesz operators are studied and a geometrical characterization of these operators are given. This characterization is used to show that a Riesz type of decomposition holds. Introduction. The main theme of this paper is to introduce Riesz operators relative to a von Neumann algebra and to obtain a Riesz type of decomposition for these operators. The theory of compact and Fredholm operators relative to a von Neumann algebra has been studied in detail by various authors (cf. [3], [4], [7], [8], [10], etc.). In the present paper Riesz operators are defined in a natural way via the Fredholm operators relative to a von Neumann algebra si, i.e. T will be called Riesz relative to sf if T-λl is Fredholm relative to si for every λφO. After some preliminaries in § 1 we develop the basic results on Riesz operators in §2. These results are similar to results known for the classical case and will be used in the sequel. Section 3 contains a geometrical characterization of the Riesz operators. This may be considered as the main result of this paper, since it allows one to use the techniques of [4] and [5] to obtain the required Riesz decomposition in §4. Whereas in the classical case the theory of Riesz operators has an intimate connection with spectral theory, it should be noted that in our representation we do not use spectral theory at all. Actually one cannot hope to obtain any results on the spectrum of a Riesz operator relative to a von Neumann algebra. In finite von Neumann algebras for instance all operators are Riesz. One can thus find Riesz operators with spectral properties very different from the classical case.