Spectral mapping theorem and Weyl’s theorem for (m,n)-paranormal operators (original) (raw)

Weyl?s Theorem for Algebraically Paranormal Operators

Integral Equations and Operator Theory, 2003

Let T be an algebraically paranormal operator acting on Hilbert space. We prove : (i) Weyl's theorem holds for f (T ) for every f ∈ H(σ(T )); (ii) 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 Weyl spectrum of T and for the essential approximate point spectrum of T .

Weyl's theorem for perturbations of paranormal operators

Proceedings of the American Mathematical Society, 2007

A bounded linear operator T ∈ L(X) on a Banach space X is said to satisfy "Weyl's theorem" if the complement in the spectrum of the Weyl spectrum is the set of all isolated points of the spectrum which are eigenvalues of finite multiplicity. In this paper we show that if T is a paranormal operator on a Hilbert space, then T + K satisfies Weyl's theorem for every algebraic operator K which commutes with T .

Some spectral mapping theorems through local spectral theory

Rendiconti del Circolo Matematico di Palermo, 2004

The spectral mapping theorems for Browder spectrum and for semi-Browder spectra have been proved by several authors , [29] and [33], by using different methods. We shall employ a local spectral argument to establish these spectral mapping theorems, as well as, the spectral mapping theorem relative to some other classical spectra.

On polynomially *-paranormal operators

2018

Let T be a bounded linear operator on a complex Hilbert space H. T is called a -paranormal operator T if kTxk2 kT2xkkxk for all x 2 H. ”-paranormal” is a generalization of hyponormal (TT TT), and it is known that a -paranormal operator has several interesting properties. In this paper, we prove that if T is polynomially -paranormal, i.e., there exists a nonconstant polynomial q(z) such that q(T) is -paranormal, then T is isoloid and the spectral mapping theorem holds for the essential approximate point spectrum of T. Also, we prove related results.

II . Weyl ' s Theorem For Algebraically K-Quasi-Paranormal Operators

2014

An operator ) (H B T  is said to be k quasi paranormal operator if x T x T x T k k k 2 2 1    for every H x , k is a natural number. This class of operators contains the class of paranormal operators and the class of quasi class A operators. Let T or T be an algebraically k quasi paranormal operator acting on Hilbert space. Using Local Spectral Theory, we prove (i)Weyl's theorem holds for f(T) for every )) ( ( T H f   ; (ii) a-Browder's theorem holds for f (S) for every T S  and )) ( ( S H f   ; (iii) the spectral mapping theorem holds for the Weyl spectrum of T and for the essential approximate point spectrum of T.

On a new class of operators and Weyl type theorems

Filomat, 2013

In the present article, we introduce a new class of operators which will be called the class of k-quasi *-paranormal operators that includes '-paranormal operators. A part from other results, we show that following results hold for a k-quasi *-paranormal operator T: (i) T has the SVEP. (ii) Every non-zero isolated point in the spectrum of T is a simple pole of the resolvent of T. (iii) All Weyl type theorems hold for T. (iv) Comments and some open problems are also presented.

Generalized Weyl’s Theorems for Perturbations Of Algebraically k - Quasi - Paranormal Operators

An operator T is called k - quasi - paranormal if kTk+1xk2  kTk+2xkkTkxk for all x 2 H where k is a natural number. A 1 - quasi - paranormal operator is quasi paranormal. In this paper, we prove that continuity of the set theoretic functions spectrum, Weyl spectrum, Browder spectrum and essential surjectivity spectrum on the classes consisting of (p, k) - quasihyponormal operators and k - quasi - paranormal operators.

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.

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 )).

Some properties of paranormal and hyponormal operators

In this article we will give some properties of paranormal and hyponormal operators. Exactly we will give some conditions which are generalization of concepts of paranormal, hyponormal, N-paranormal, N-hyponormal operators.