A Note on Weyl's Theorem for Operator Matrices (original) (raw)
Related papers
Generalized Weyl's theorem for some classes of operators
Kyungpook Mathematical Journal, 2006
Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set σBw(A) of all λ ∈ C such that A−λI is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently in [6] Berkani showed that if A is a hyponormal operator, then A satisfies generalized Weyl's theorem σ Bw (A) = σ(A) \ E(A), and the B-Weyl spectrum σ Bw (A) of A satisfies the spectral mapping theorem. In [51], H. Weyl proved that weyl's theorem holds for hermitian operators. Weyl's theorem has been extended from hermitian operators to hyponormal and Toeplitz operators [12], and to several classes of operators including semi-normal operators ([9], [10]). Recently W. Y. Lee [35] showed that Weyl's theorem holds for algebraically hyponormal operators. R. Curto and Y. M. Han [14] have extended Lee's results to algebraically paranormal operators. In [19] the authors showed that Weyl's theorem holds for algebraically p-hyponormal operators. As Berkani has shown in , if the generalized Weyl's theorem holds for A, then so does Weyl's theorem. In this paper all the above results are generalized by proving that generalized Weyl's theorem holds for the case where A is an algebraically (p, k)-quasihyponormal or an algebarically paranormal operator which includes all the above mentioned operators.
Operators obeying aaa-Weyl's theorem
Publicationes Mathematicae Debrecen
This article treatises several problems relevant to a-Weyl's theorem for bounded operators on Banach spaces. There are presented sufficient conditions for an operator T , such that a-Weyl's theorem holds for T. If a-Weyl's theorem holds for an a-isoloid operator T , and F is a finite rank operator commuting with T , then a-Weyl's theorem holds for T + F. The algebraic view point for a-Weyl's theorem is considered in the sense of the spectral mapping theorem for a special part of the spectrum. If T * is a quasihyponormal operator on a Hilbert space, f is a regular function in a neighbourhood of the spectrum of T and f is not constant on the connected components of its domain, we prove that a-Weyl's theorem holds for f (T). The article also contains some related results.
Proceedings of the American Mathematical Society
The Kato spectrum of an operator is deployed to give necessary and sufficient conditions for Browder’s theorem to hold.
Weyl’s Theorems for Some Classes of Operators
Integral Equations and Operator Theory, 2005
We introduce the class of operators on Banach spaces having property (H) and study Weyl's theorems, and related results for operators which satisfy this property. We show that a-Weyl's theorem holds for every decomposable operator having property (H). We also show that a-Weyl's theorem holds for every multiplier T of a commutative semi-simple regular Tauberian Banach algebra. In particular every convolution operator Tµ of a group algebra L 1 (G), G a locally compact abelian group, satisfies a-Weyl's theorem. Similar results are given for multipliers of other important commutative Banach algebras. . Primary 47A10, 47A11; Secondary 47A53, 47A55.
Weyl’s theorem for upper triangular operator matrices
Linear Algebra and its Applications, 2005
Let σ ab (T) = {λ ∈ C : T − λI is not an upper semi-Fredholm operator with finite ascent} be the Browder essential approximate point spectrum of T ∈ B(H) and let σ d (T) = {λ ∈ C : T − λI is not surjective} be the surjective spectrum of T. In this paper it is shown that if M C = A C 0 B is a 2 × 2 upper triangular operator matrix acting on the Hilbert space H ⊕ K, then the passage from σ ab (A) ∪ σ ab (B) to σ ab (M C) is accomplished by removing certain open subsets of σ d (A) ∩ σ ab (B) from the former, that is, there is equality σ ab (A) ∪ σ ab (B) = σ ab (M C) ∪ G, where G is the union of certain of the holes in σ ab (M C) which happen to be subsets of σ d (A) ∩ σ ab (B). Weyl's theorem and Browder's theorem are liable to fail for 2 × 2 operator matrices. In this paper, it also explores how Weyl's theorem, Browder's theorem, a-Weyl's theorem and a-Browder's theorem survive for 2 × 2 upper triangular operator matrices on the Hilbert space.
Weyl type theorems for bounded linear operators
Acta Scientiarum Mathematicarum
For a bounded linear operator T acting on a Banach space let σ SBF − + (T) be the set of all λ ∈ C such that T − λI is upper semi-B-Fredholm and ind (T − λI) ≤ 0, and let E a (T) be the set of all isolated eigenvalues of T in the approximate point spectrum σ a (T) of T. We say that T satisfies generalized a-Weyl's theorem if σ SBF − + (T) = σ a (T)\E a (T). Among other things, we show in this paper that if T satisfies generalized a-Weyl's theorem, then it also satisfies generalized Weyl's theorem σ BW (T) = σ(T) \ E(T), where σ BW (T) is the B-Weyl spectrum of T and E(T) is the set of all eigenvalues of T which are isolated in the spectrum of T. 2000 AMS subject classification: 47A53, 47A55
Generalized Weyl'S Theorem for Posinormal Operators
Mathematical Proceedings of the Royal Irish Academy, 2007
Let A be a bounded linear operator acting on infinite dimensional separable Hilbert space H. In this paper, we prove that the generalized Weyl's theorem holds for f (A) if A is conditionally totally posinormal or totally posinormal, where f is a function analytic in an open neighborhood of σ(A). Other related results are also given. *
Weyl's type theorems and perturbations
Divulgaciones Matematicas
Weyl's theorem for a bounded linear operator T on complex Banach spaces, as well as its variants, a-Weyl's theorem and property (w), in general is not transmitted to the perturbation T + K, even when K is a "good" operator, as a commuting finite rank operator or compact operator. Weyl's theorems do not survive also if K is a commuting quasi-nilpotent operator. In this paper we discuss some sufficient conditions for which Weyl's theorem, a-Weyl's theorem as well as property (w) is transmitted under such kinds of perturbations.