a-Browder's Theorem Research Articles

Weyl's Theorem for Algebraically (\(\wp, \wr, \varrho\))-Paranormal and Algebraically (\(\wp, \wr, \varrho\))-*-Paranormal Operators

Present L be an algebraically (\(\wp, \wr, \varrho\))-Paranormal and algebraically (\(\wp, \wr, \varrho\))-*-Paranormal operators on \(L^2\) space. We examine Weyl's theorem, a-Browder's theorem and spectral mapping theorem holds for weyl's spectrum of L and essential approximate point spectrum of L.

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.

Weyl Type Theorems for Unbounded Hyponormal Operators

If T is an unbounded hyponormal operator on an infinite dimensional complex Hilbert space H with , then it is shown that T satisfies Weyl's theorem, generalized Weyl's theorem, Browder's theorem and generalized Browder's theorem. The equivalence of generalized Weyl's theorem with generalized Browder's theorem, property (gw) with property (gb) and property (w) with property (b) have also been established. It is also shown that a-Browder's theorem holds for T as well as its adjoint .

Riesz Idempotent and Browder's Theorem for Absolute-(p, r)-Paranormal Operators

An operator T is said to be paranormal if ‖T2x‖ ≥ ‖Tx‖2 holds for every unit vector x. Several extensions of paranormal operators are considered until now, for example absolute-k-paranormal and p-paranormal introduced in [10], [14], respectively. Yamazaki and Yanagida [38] introduced the class of absolute-(p, r)-paranormal operators as a further generalization of the classes of both absolute-k-paranormal and p-paranormal operators. An operator T ∈B(H) is called absolute-(p, r)-paranormal operator if ‖|T|p|T*|rx‖r ≥ |‖T*|rx‖p+r for every unit vector x ∈H and for positive real numbers p > 0 and r > 0. The famous result of Browder, that self adjoint operators satisfy Browder's theorem, isextended to several classes of operators. In this paper we show that for any absolute-(p, r)-paranormal operator T, T satisfies Browder's theorem and a-Browder's theorem. It is also shown that if E is the Riesz idempotent for a nonzero isolated point μ of the spectrum of a absolute-(p, r)-paranormal operator T, then E is self-adjoint if and only if the null space of T − μ, N(T − μ) ⊆N(T* −▪).

Generalized weyl's theorem for algebraically quasi-paranormal operators

Let T or T* be an algebraically quasi-paranormal operator acting on a Hilbert space. We prove: (i) generalized Weyl's theorem holds for f(T) for every f ∈ H(σ (T)); (ii) generalized a-Browder's theorem holds for f(S) for every S ≺ T and f ∈ H(σ(S)); (iii) the spectral mapping theorem holds for the B-Weyl spectrum of T. Moreover, we show that if T is an algebraically quasi-paranormal operator, then T + F satisfies generalized Weyl's theorem for every algebraic operator F which commutes with T.

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.

Local spectral properties for the Helton class of an operator

In this paper we study some local spectral properties of the Helton class of an operator. In particular, we show that the Helton class of an operator preserves the property (Q) under some condition. Also, we prove that if R has SVEP, S ? Heltonk (R), and T ? S then a-Browder's theorem holds for f (T) for each f ? H (?(T )).

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 ) .

On generalized a-Browder's theorem

We characterize the bounded linear operators $T$ satisfying generalized $a$-Browder's theorem, or generalized $a$-Weyl's theorem, by means of localized SVEP, as well as by means of the quasi-nilpotent part $H_0(\lambda I-T)$ as $\lambda$ belongs to certai

ON THE RIESZ IDEMPOTENT FOR A CLASS OF OPERATORS

A result from Stampili says that, for hyponormal operators on Hubert space, the Riesz idempotent at an isolated point of a spectrum is self-adjoint, with its range given by null space of both the operator and its adjoint. In this note, we extend this result to operators that are algebraically class H(q). The arguments involve local spectral theory and extensions of a-Browder's theorem.

ON THE RIESZ IDEMPOTENT FOR A CLASS OF OPERATORS

A result from Stampili says that, for hyponormal operators on Hubert space, the Riesz idempotent at an isolated point of a spectrum is self-adjoint, with its range given by null space of both the operator and its adjoint. In this note, we extend this result to operators that are algebraically class H(q). The arguments involve local spectral theory and extensions of a-Browder's theorem.

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.

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.

Some characterizations of operators satisfying a-Browder's theorem

We characterize the bounded linear operators T defined on Banach spaces satisfying a-Browder's theorem, or a-Weyl's theorem, by means of the discontinuity of some maps defined on certain subsets of C . Several other characterizations are given in terms of localized SVEP, as well as by means of the quasi-nilpotent part, the hyper-kernel or the analytic core of λ I − T .

Weyl type Theorems and the Approximate Point Spectrum

It is shown that, if an operator T on a complex Banach space or its adjoint T ⁄ has the single-valued extension property, then the generalized a-Browder's theorem holds for f(T) for every complex-valued analytic function f on a neigh- borhood of the spectrum of T. We also study the generalized a-Weyl's theorem in connection with the single-valued exten- sion property. Finally, we examine the stability of the gen- eralized a-Weyl's theorem under commutative perturbations by finite rank operators.

Essential approximate point spectra and Weyl's theorem for operator matrices

When A ∈ B ( H ) and B ∈ B ( K ) are given, we denote by M C the operator acting on the infinite dimensional separable Hilbert space H ⊕ K of the form M C = ( A C 0 B ) . In this paper, it is shown that a 2 × 2 operator matrix M C is upper semi-Fredholm and ind ( M C ) ⩽ 0 for some C ∈ B ( K , H ) if and only if A is upper semi-Fredholm and { n ( B ) < ∞ and n ( A ) + n ( B ) ⩽ d ( A ) + d ( B ) or n ( B ) = d ( A ) = ∞ , if R ( B ) is closed , d ( A ) = ∞ , if R ( B ) is not closed . We also give the necessary and sufficient conditions for which M C is Weyl or M C is lower semi-Fredholm with nonnegative index for some C ∈ B ( K , H ) . In addition, we explore 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.

$a$-Weyl’s theorem for operator matrices

If M C = ( A 0 C B ) is a 2 × 2 upper triangular matrix on the Hilbert space H ○+ K, then a-Weyl's theorem for A and B need not imply a-Weyl's theorem for M C , even when C = 0. In this note we explore how a-Weyl's theorem and a-Browder's theorem survive for 2 x 2 operator matrices on the Hilbert space.

Browder's theorems and spectral continuity

Let X be a complex infinite dimensional Banach space. We use σ_a(T) andσ_{ea}(T) , respectively, to denote the approximate point spectrum and the essential approximate point spectrum of a bounded operator T onX . Also, \pi _a(T) denotes the set &lt;$&gt;{\rm{iso} σ_a(T)\backslash σ_{ea}(T)}&lt;$&gt;. An operator T onX obeys the a-Browder's theorem provided that&lt;&gt; . We investigate connections between the Browder's theorems, the spectral mapping theorem and spectral continuity.