Some Classes of Operators Related to p-Hyponormal Operator

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.

Integral Equations and Operator Theory, 1997

Far East Journal of Mathematical Sciences

Fuji, Izumino and Nakamoto [7] introduced p-paranormal operators for p > 0 as a generalization of paranormal operators. Fujii, Jung, S. H. Lee, M. Y. Lee and Nakamoto [9] introduced class A(p, r) as a further generalization of class A(k). An operator T ∈ classA(p, r) for p > 0 and r > 0 if (|T * | r |T | 2p |T * | r) r p+r ≥ |T * | 2r and class AI(p, r) is class of all invertible operators which belong to class A(p, r).

As a further generalization of paranormal operators, we shall introduce a new class "absolute-(p, r)-paranormal" operators for p > 0 and r > 0 such that |T | p |T * | r x r ≥ |T * | r x p+r for every unit vector x. And we shall show several properties on absolute-(p, r)-paranormal operators as generalizations of the results on absolutek-paranormal and p-paranormal operators introduced in [10] and [6], respectively.

In this paper, spectral properties of p -hyponormal composition operators acting on an L 2 -space are studied.

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.

In this paper, we prove that Weyl's theorem holds for algebraically (p, k)-quasihyponormal operators and spectral map-ping theorem holds for the Weyl spectrum of algebraically (p, k)-quasihyponormal operators. Also, we study related results.

European Journal of Pure and Applied Mathematics

In this paper we introduce a new class of operators called M−quasi paranormal operators. A bounded linear operator T in a complex Hilbert space H is said to be a M−quasi paranormal operator if it satisfies ∥T 2x∥ 2 ≤ M∥T 3x∥ · ∥T x∥, ∀x ∈ H, where M is a real positive number. We prove basic properties, the structural and spectral properties of this class of operators.

Mathematische Nachrichten, 2014

A Hilbert space operator T ∈ L (H) is M-hyponormal if there exists a positive real number M such that (T − μ)(T − μ) * ≤ M 2 (T − μ) * (T − μ) for all μ ∈ σ (T). Let A, B * ∈ L (H) be M-hyponormal and let d AB ∈ L (L (H)) denote either the generalized derivation δ AB (X) = AX − X B or the elementary operator AB = AX B − X. We prove that if A, B * are M-hyponormal, then f (d AB) satisfies the generalized Weyl's theorem and f (d * AB) satisfies the generalized a-Weyl's theorem for every f that is analytic on a neighborhood of σ (d AB).

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.

Let T be a w-hyponormal operator with the polar decomposition T U[ 7]. In this paper, we show the following:

Studia Mathematica, 2004

Weyl type theorems for p-hyponormal and M-hyponormal operators by Xiaohong Cao (Xi'an), Maozheng Guo (Beijing) and Bin Meng (Beijing) Abstract. "Generalized Weyl's theorem holds" for an operator when the complement in the spectrum of the B-Weyl spectrum coincides with the isolated points of the spectrum which are eigenvalues; and "generalized a-Weyl's theorem holds" for an operator when the complement in the approximate point spectrum of the semi-B-essential approximate point spectrum coincides with the isolated points of the approximate point spectrum which are eigenvalues. If T or T * is p-hyponormal or M-hyponormal then for every f ∈ H(σ(T)), generalized Weyl's theorem holds for f (T), so Weyl's theorem holds for f (T), where H(σ(T)) denotes the set of all analytic functions on an open neighborhood of σ(T). Moreover, if T * is p-hyponormal or M-hyponormal then for every f ∈ H(σ(T)), generalized a-Weyl's theorem holds for f (T) and hence a-Weyl's theorem holds for f (T). a (T) = C \ σ a (T). An operator T ∈ B(H) is called Fredholm if it has closed finite-codimensional range and finite-dimensional null space. The index of a Fredholm operator T ∈ B(H) is given by ind(T) = n(T) − d(T). An operator T ∈ B(H) is called Weyl if it is Fredholm of index zero, and Browder if it is Fredholm of finite ascent and descent, or equivalently, if T is Fredholm and T − λI is invertible for all sufficiently small λ = 0 in C. For T ∈ B(H), we write α(T) for the ascent of T and β(T) for the descent of T .

2010

Let A, B * ∈ B(H) be w-hyponormal operators, and let d AB ∈ B(B(H)) denote either the generalized derivation δ AB (X) = AX -XB or the length two elementary operator AB (X) = AXB -X. We prove that d AB has the single-valued extension property, and the ) denote the space of functions which are analytic on σ (d AB ), and let H c (σ (d AB )) denote the space of f ∈ H(σ (d AB )) which are non-constant on every connected component of σ (d AB ). It is proved that, for every h ∈ H(σ (d AB )) and f , g ∈ H c (σ (d AB )), the complement of the Weyl spectrum σ w (h )) consists of isolated points in σ (h(d f (A)g(B) )) which are eigenvalues of finite multiplicity.

2015

In this paper, we study power similarity of operators. In particular, we show that if $T \in \mathit{PS}(H)$ (defined below) for some hyponormal operator $H$, then $T$ is subscalar. From this result, we obtain that such an operator with rich spectrum has a nontrivial invariant subspace. Moreover, we consider invariant and hyperinvariant subspaces for $T \in \mathit{PS}(H)$.

2020

Abstract. A survey of the theory of k-hyponormal operators starts with the construction of a polynomially hyponormal operator which is not subnormal. This is achieved via a natural dictionary between positive functionals on specific convex cones of polynomials and linear bounded operators acting on a Hilbert space, with a distinguished cyclic vector. The class of unilateral weighted shifts provides an optimal framework for studying k-hyponormality. Non-trivial links with the theory of Toeplitz operators on Hardy space are also exposed in detail. A good selection of intriguing open problems, with precise references to prior works and partial solutions, is offered. Mathematics Subject Classification (2000) . Primary 47B20; Secondary 47B35, 47B37, 46A55, 30E05.