Subscalarity of ( p , k ) -quasihyponormal operators

Journal of Mathematical Analysis and Applications, 2011

In this paper, we show that every (p, k)-quasihyponormal operator has a scalar extension and give some spectral properties of the scalar extensions of (p, k)-quasihyponormal operators. As a corollary, we get that such an operator with rich spectrum has a nontrivial invariant subspace. Finally, we prove that the sum of a p-hyponormal operator and an algebraic operator which are commuting is subscalar.

Linear Algebra and its Applications, 2007

A Hilbert space operator A ∈ B(H) is said to be p-quasi-hyponormal for some 0 < p 1, A ∈ p − QH , if A * (|A| 2p − |A * | 2p)A 0. If H is infinite dimensional, then operators A ∈ p − QH are not supercyclic. Restricting ourselves to those A ∈ p − QH for which A −1 (0) ⊆ A * −1 (0), A ∈ p * − QH , a necessary and sufficient condition for the adjoint of a pure p * − QH operator to be supercyclic is proved. Operators in p * − QH satisfy Bishop's property (β). Each A ∈ p * − QH has the finite ascent property and the quasinilpotent part H 0 (A − λI) of A equals (A − λI) −1 (0) for all complex numbers λ; hence f (A) satisfies Weyl's theorem, and f (A *) satisfies a-Weyl's theorem, for all non-constant functions f which are analytic on a neighborhood of σ (A). It is proved that a Putnam-Fuglede type commutativity theorem holds for operators in p * − QH .

In this paper, we prove that Weyl&#39;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.

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 &lt; 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&#39;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) &lt; 1 and (T) = dimH/R(T) &lt; 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}

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). .

2000

Let A be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of A is the set æBw(A) of all ‚ 2C such that A ¡ ‚I is not a B-Fredholm operator of index 0. Let E(A) be the set of all isolated eigenvalues of A. Recently, in (3) the author showed that if A is

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&#x27;s theorem holds for injective (p,k)-quasihyponormal and p-hyponormal operators. In this paper we prove that Fuglede-Putnam&#x27;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.

m-hikari.com

In this paper we introduce a new class quasiparahyponormal operator. We give a characterization of such operators and other known classes of operators.

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.