On p-quasi-hyponormal operators

Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded lin- ear operators on H. Recently H.Kim proved that Fuglede-Putnam's theorem holds for injective (p,k)-quasihyponormal and p-hyponormal operators. In this paper we prove that Fuglede-Putnam's theorem holds for injective (p,k)-quasihyponormal and log-hyponormal op- erators. We also show that this results remains true for dominant and injective (p,k)-quasihyponormal operators. Other related re- sults are also given.

2007

For Hilbert space operators A and B, let δ AB denote the generalised derivation δ AB (X) = AX − XB and let AB denote the elementary operator AB (X) = AXB−X. If A is a pk-quasihyponormal operator, A ∈ pk − QH, and B * is an either p-hyponormal or injective dominant or injective pk − QH operator (resp., B * is an either p-hyponormal or dominant or pk − QH operator), then δ AB (X) = 0 =⇒ δ A * B * (X) = 0 (resp., AB (X) = 0 =⇒ A * B * (X) = 0).

Let T be a bounded linear operator on a complex Hilbert space H. T is called (p,k)-quasihyponormal if T ((T T)p (TT )p)T 0 for 0 < p 1 and k 2 N. In this paper, we prove that Weyl type theorems for (p,k)-hyponormal operators. Espe- cially we prove that if T is (p,k)-quasihyponormal, then general- ized a-Weyl's theorem holds for T. Let B(H) denote the algebra of all bounded linear operators acting on an infinite dimensional separable Hilbert space H. If T 2 B(H), we shall write N(T) and R(T) for the null space and the range of T, respectively. Also, let (T) and a(T) denote the spectrum and the approximate point spectrum of T, respectively. An operator T is called Fredholm if R(T) is closed, (T) = dimN(T) < 1 and (T) = dimH/R(T) < 1. Moreover if i(T) = (T) (T) = 0, then T is called Weyl. The essential spectrum e(T) and the Weyl spectrum W(T) are defined by e(T) = { 2 C : T is not Fredholm}

Annals of the Alexandru Ioan Cuza University - Mathematics, 2013

Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A, B be operators in B(H). In this paper we prove that if A is quasi-class A and B * is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A * X = XB * . We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f (A) satisfies generalized Weyl's theorem. Other related results are also given.

Scientiae Mathematicae Japonicae, 2005

The equation AX = XB implies A * X = XB * when A and B are normal operators is known as the familiar Fuglede-Putnam theorem. In this paper, the hypothesis on A and B can be relaxed by using a Hilbert-Schmidt operator X: Let A be a (p, k)-quasihyponormal operator and B * be an invertible (p, k)-quasihyponormal operator such that AX = XB for a Hilbert Schmidt operators X, then A * X = XB * . As a consequence of this result, we obtain that the range of the generalized derivation induced by this class of operators is orthogonal to its kernel.

Filomat, 2007

For Hilbert space operators A and B, let ?AB denote the generalized derivation ?AB(X) = AX - XB and let /\AB denote the elementary operator rAB(X) = AXB-X. If A is a pk-quasihyponormal operator, A ? pk - QH, and B*is an either p-hyponormal or injective dominant or injective pk - QH operator (resp., B*is an either p-hyponormal or dominant or pk - QH operator), then ?AB(X) = 0 =? SA*B*(X) = 0 (resp., rAB(X) = 0 =? rA*B*(X) = 0). .

Journal of Al-Nahrain University Science, 2009

In this paper,we prove that, if is the quotient of a decomposable on a separable Banach space (M-hyponormal operator on a real Hilbert space), then is hypercyclic operators. We also show that these classes of operators are supercyclic operators.

Advances in Pure Mathematics, 2012

T B H  to be * p-paranormal and the monotonicity of A p q. We also present an alternative proof of a result of M. Fujii, et al. [1, Theorem 3.4].

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 .

A Banach space operator T satisfies generalized a-Weyl's theorem if the complement of its upper semi B-Weyl spectrum in its approximate point spectrum is the set of eigenvalues of T which are isolated in the approximate spectrum of T. In this note we characterize hypecyclic and supercyclic operators satisfying generalized a-Weyl's theorem.