On a new class of operators and Weyl type theorems (original) (raw)

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.

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.

A note on k-quasi-∗-paranormal operators

Journal of Inequalities and Applications, 2013

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we introduce a new class of operators satisfying T * T k x 2 ≤ T k+2 x T k x for all x ∈ H, where k is a natural number. This class includes the classes of *-paranormal and k-quasi-*-class A. We prove some of the properties of these operators.

On m-quasi class A(k ) and absolute-(k , m)-paranormal operators

Hacettepe Journal of Mathematics and Statistics, 2018

In this paper, we introduce a new class of operators, called m-quasi class A(k *) operators, which is a superclass of hyponormal operators and a subclass of absolute-(k * , m)-paranormal operators. We will show basic structural properties and some spectral properties of this class of operators. We show that if T is m-quasi class A(k *), then σnp(T) \ {0} = σp(T) \ {0}, σna(T) \ {0} = σa(T) \ {0} and T − µ has nite ascent for all µ ∈ C. Also, we consider the tensor product of m-quasi class A(k *) operators. Dedicated to the memory of Professor Takayuki Furuta with deep gratitude.

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 .

SVEP and Bishop's property for k *-paranormal operators

Operators and Matrices, 2011

A bounded linear operator T on a complex Hilbert space H is said to be k *-paranormal if T * x k ≤ T k x for every unit vector x ∈ H. This class of operators is an extension of hyponormal operators and have many interesting properties. We show that k *-paranormal operators have Bishop's property (β), i.e., if f n (λ) is an analytic function on some open set D ⊂ C such that (T − z)f n (z) → 0 uniformly on every compact subset K ⊂ D, then f n (z) → 0 uniformly on K. In case of k = 2, this means that *-paranormal operators have Bishop's property (β).