a-Weyl's Theorem Research Articles

Topological Uniform Descent and Judgement of A-Weyl's Theorem

In this paper, a-Browder's theorem and a-Weyl's theorem for bounded linear operators are studied by means of the property of the topological uniform descent. The sufficient and necessary conditions for a bounded linear operator defined on a Hilbert space holding a-Browder's theorem and a-Weyl's theorem are established. As a consequence of the main result, the new judgements of a-Browder's theorem and a-Weyl's theorem for operator function are discussed.

Criteria for the Property (UWE) and the a-Weyl Theorem

Criteria for the Property (UWE) and the a-Weyl Theorem

New approach to a-Weyl's theorem through localized SVEP and Riesz-type perturbations

ABSTRACT In this paper, we study the properties and which we had introduced in [Ben Ouidren K, Zariouh H. New approach to a-Weyl's theorem and some preservation results. Rend Circ Mat Palermo. doi: 10.1007/s12215-020-00525-2], for an operator having the SVEP on the complementary of distinguished parts of its spectrum. We prove in particular, that a bounded linear operator T acting on a Banach space has the SVEP on the complementary of if and only if T possesses property We also study the stability of these properties under several commuting perturbations, and we prove that if T possesses property then possesses property for every Riesz operator R commuting with T and We also give an example which shows that the property is generally unstable under this type of perturbations, and we prove that if T possesses property then T + R possesses property ⇔ Analogous results are proved for the property and some applications to the class of a-isoloid-type operators are given.

A-Weyl's theorem and hypercyclic property for bounded linear operators

A-Weyl's theorem and hypercyclic property for bounded linear operators

Property $(aBw)$ and perturbations

A bounded linear operator $T$ acting on a Banach space $X$ satisfies property $(aBw)$, a strong version of a-Weyl's theorem, if the complement in the approximate point spectrum $\sigma_{a}(T)$ of the upper semi-B-Weyl spectrum $\sigma_{USBW}(T)$ is the set of all isolated points of approximate point spectrum which are eigenvalues of finite multiplicity. In this paper we investigate the property $(aBw)$ in connection with Weyl type theorems. In particular, we show that $T$ satisfies property $(aBw)$ if and only if $T$ satisfies a-Weyl's theorem and $\sigma_{USBW}(T)=\sigma_{USW}(T)$, where $\sigma_{USW}(T)$ is the upper semi-Weyl spectrum of $T$. The preservation of property $(aBw)$ is also studied under commuting nilpotent, quasi-nilpotent, power finite rank or Riesz perturbations. The theoretical results are illustrated by some concrete examples.

On an elementary operator withM-hyponormal operator entries

A Hilbert space operator is M-hyponormal if there exists a positive real number M such that for all . Let be M-hyponormal and let denote either the generalized derivation or the elementary operator . We prove that if are M-hyponormal, then satisfies the generalized Weyl's theorem and satisfies the generalized a-Weyl's theorem for every f that is analytic on a neighborhood of .

Weyl's theorem for algebraically quasi-$*$-$A$ operators

In the paper, we prove the following assertions: (1) If $T$ is an algebraically quasi-$*$-$A$ operator, then $T$ is polaroid. (2) If $T$ or $T^{*}$ is an algebraically quasi-$*$-$A$ operator, then Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$. (3) If $T^{*}$ is an algebraically quasi-$*$-$A$ operator, then a-Weyl's theorem holds for $f(T)$ for every $f \in H(\sigma(T))$.

WEYL TYPE THEOREMS FOR FUNCTIONS OF OPERATORS

AbstractA-Weyl's theorem and property (ω), as two variations of Weyl's theorem, were introduced by Rakočević. In this paper, we study a-Weyl's theorem and property (ω) for functions of bounded linear operators. A necessary and sufficient condition is given for an operator T to satisfy that f(T) obeys a-Weyl's theorem (property (ω)) for all f ∈ Hol(σ(T)). Also we investigate the small-compact perturbations of operators satisfying a-Weyl's theorem (property (ω)) in the setting of separable Hilbert spaces.

Weyl's type theorems and hypercyclic operators

For a bounded operator T acting on an infinite dimensional separable Hilbert space H, we prove the following assertions: (i) If T or T* ∈▪, then generalized a-Browder's theorem holds for f(T) for every f ∈ Hol(σ(T)). (ii) If T or T* ∈▪ has topological uniform descent at all λ ∈ iso(σ(T)), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (iii) If T ∈▪ has topological uniform descent at all λ ∈E(T), then T satisfies generalized Weyl's theorem. (iv) Let T ∈▪. If T satisfies the growth condition Gd (d ≥ 1), then generalized Weyl's theorem holds for f(T) for every f ∈ Hol(σ(T)). (v) If T ∈▪, then, f(σSBF+−(T)) = σSBF+−(f(T)) for all f ∈ Hol(σ(T)). (vi) Let T be a-isoloid such that T* ∈▪. If T − λI has finite ascent at every λ ∈Ea(T) and if F is of finite rank on H such that TF = FT, then T + F obeys generalized a-Weyl's theorem.

WEYL'S TYPE THEOREMS FOR ALGEBRAICALLY (p, k)-QUASIHYPONORMAL OPERATORS

For a bounded linear operator T we prove the following assertions: (a) If T is algebraically (p, k)-quasihyponormal, then T is a-isoloid, polaroid, reguloid and a-polaroid. (b) If <TEX>$T^*$</TEX> is algebraically (p, k)-quasihyponormal, then a-Weyl's theorem holds for f(T) for every <TEX>$f{\in}Hol({\sigma}T))$</TEX>, where <TEX>$Hol({\sigma}(T))$</TEX> is the space of all functions that analytic in an open neighborhoods of <TEX>${\sigma}(T)$</TEX> of T. (c) If <TEX>$T^*$</TEX> is algebraically (p, k)-quasihyponormal, then generalized a-Weyl's theorem holds for f(T) for every <TEX>$f{\in}Hol({\sigma}T))$</TEX>. (d) If T is a (p, k)-quasihyponormal operator, then the spectral mapping theorem holds for semi-B-essential approximate point spectrum <TEX>$\sigma_{SBF_+^-}(T)$</TEX>, and for left Drazin spectrum <TEX>${\sigma}_{lD}(T)$</TEX> for every <TEX>$f{\in}Hol({\sigma}T))$</TEX>.

Browder and Weyl spectra of upper triangular operator matrices

Let MC = (A/0 C/B) ( B(X ( X ) be an upper triangulat Banach space operator. The relationship between the spectra of MC and M0, and their various distinguished parts, has been studied by a large number of authors in the recent past. This paper brings forth the important role played by SVEP, the single-valued extension property, in the study of some of these relations. Operators MC and M0 satisfying Browder's, or a-Browder's, theorem are characterized, and we prove necessary and sufficient conditions for implications of the type 'M0 satisfies a-Browder's (or a-Weyl's) theorem ( MC satisfies a-Browder's (resp., a-Weyl's) theorem' to hold. 2010 Mathematics Subject Classifications. Primary 47B47, 47A10, 47A11. .

WEYL TYPE THEOREMS FOR POSINORMAL OPERATORS

Let be a bounded linear operator acting on infinite dimensional separable Hubert space H. The study of operators satisfying Weyl's theorem, Browder's theorem, the SVEP and Bishop's property is of significant interest and is currently being done by a number of mathematicians around the world. It is known that Weyl's theorem holds for M-hyponormal operators, but does not hold for dominant operators. Hence it is an interesting problem to seek a condition that implies Weyl's theorem for dominant operators. Ho Jeon et al. proved that if A is dominant and satisfies σ((A - λI)|M) = {0} ⇒ (A -λI)\M for every M ∈ Lat(A), then Weyl's theorem holds for A. Recently Cao showed that the generalized a-Weyl's theorem holds for ƒ(A), where ƒ is an analytic function defined in an open neighbourhood of σ(A) in the case where A* is p-hyponormal or M-hyponormal. Also Aiena showed that a-Weyl's theorem holds for some classes of operators. In this paper we prove that if A* is conditionally totally posinormal (with certain condition) or totally posinormal, then the generalized a-Weyl's theorem holds for A and for ƒ(A), where ƒ is an analytic function defined in an open neighbourhood of σ(A).

WEYL TYPE THEOREMS FOR POSINORMAL OPERATORS

Let A be a bounded linear operator acting on infinite dimensional separable Hilbert space H. The study of operators sat- isfying Weyl's theorem, Browder's theorm, the SVEP and Bishop's property is of significant interest, and is currently being done by a number of mathematicians around the world. It is known that Weyl's theorem holds for M-hyponormal operators, but does not hold for dominant operators. Hence it is an interesting problem to seek a condition which implies Weyl's theorem for dominant oper- ators. In Ho Jeon etat () proved that if A is dominant and satisfies ((A I )|M) = {0} ) (A I )|M for every M 2 Lat(A), then Weyl's theorem holds for A. Recently X.Cao () showed that gen- eralized a-Weyl's theorem holds for f(A), where f is an analytic function defined on an open neighborhood of (A) in the case where A is p-hyponormal or M-hyponormal. Also Aiena () showed that a-Weyl's theorem holds for some classes of operators. In this pa- per we prove that if A is conditionally totally posinormal (with certain condition) or totally posinormal, then generalized a-Weyl's theorem holds for A and for f(A), where f is an analytic function defined on an open neighborhood of (A).

Hereditarily polaroid operators, SVEP and Weyl's theorem

A Banach space operator T ∈ B ( X ) is hereditarily polaroid, T ∈ HP , if every part of T is polaroid. HP operators have SVEP. It is proved that if T ∈ B ( X ) has SVEP and R ∈ B ( X ) is a Riesz operator which commutes with T, then T + R satisfies generalized a-Browder's theorem. If, in particular, R is a quasi-nilpotent operator Q, then both T + Q and T ∗ + Q ∗ satisfy generalized a-Browder's theorem; furthermore, if Q is injective, then also T + Q satisfies Weyl's theorem. If A ∈ B ( X ) is an algebraic operator which commutes with the polynomially HP operator T, then T + N is polaroid and has SVEP, f ( T + N ) satisfies generalized Weyl's theorem for every function f which is analytic on a neighbourhood of σ ( T + N ) , and f ( T + N ) ∗ satisfies generalized a-Weyl's theorem for every function f which is analytic on, and constant on no component of, a neighbourhood of σ ( T + N ) .

Weyl type theorems and hypercyclic operators II

In this note, the relation between hypercyclic operator matrices (or supercyclic operator matrices) and the operator matrices which satisfy Weyl type theorems is discussed. Also, using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for A for which a-Browder's theorem or a-Weyl's theorem holds.

ON WEYL'S THEOREM FOR QUASI-CLASS A OPERATORS

Let T be a bounded linear operator on a complex in- flnite dimensional Hilbert space H. We say that T is a quasi-class A operator if T ⁄ jT 2 jT ‚ T ⁄ jTj 2 T. In this paper we prove that if T is a quasi-class A operator and f is a function analytic on a neigh- borhood of the spectrum of T, then f(T) satisfles Weyl's theorem and f(T ⁄ ) satisfles a-Weyl's theorem.

Variations on Weyl's theorem

In this note we study the property ( w ) , a variant of Weyl's theorem introduced by Rakočević, by means of the localized single-valued extension property (SVEP). We establish for a bounded linear operator defined on a Banach space several sufficient and necessary conditions for which property ( w ) holds. We also relate this property with Weyl's theorem and with another variant of it, a-Weyl's theorem. We show that Weyl's theorem, a-Weyl's theorem and property ( w ) for T (respectively T * ) coincide whenever T * (respectively T) satisfies SVEP. As a consequence of these results, we obtain that several classes of commonly considered operators have property ( w ) .

A-Weyl's theorem and perturbations

We study the stability of a-Weyl's theorem under perturbations by operators in some known classes. We establish in particular that if $T$ is a finite a-isoloid operator, then a-Weyl's theorem is transmitted from $T$ to $T+R$ for every Riesz operator $R$ c

Weyl type theorem and hypercyclic operators

Using a variant of the essential approximate point spectrum, we give the necessary and sufficient conditions for T for which the a-Browder's theorem or the a-Weyl's theorem holds. Also, the relation between hypercyclic operators (or supercyclic operators) and the operators which satisfy Weyl type theorem is discussed.

Weyl’s Theorems for Some Classes of Operators

We introduce the class of operators on Banach spaces having prop- erty (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 decom- posable operator having property (H). We also show that a-Weyl's theorem holds for every multiplier T of a commutative semi-simple regular Taube- rian 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 theo- rem. Similar results are given for multipliers of other important commutative Banach algebras.