n -Power Quasi Normal Operators on the Hilbert Space

2011

Let T be a bounded linear operator on a complex Hilbert space H. In this paper we investigate some, properties of the class of n-power quasinormal operators , denoted [nQN ], satisfying Tn|T |2 − |T |2Tn = 0 and some relations between n-normal operators and n-quasinormal operators.

2021

The aim of this paper is to present certain basic properties of some classes of nonnormal operators defined on a complex separable Hilbert space. Both of the normality of their integer powers and their relations with isometries are established. The ascent of such operators as well as other important related results are also established. The decomposition of such operators, their restrictions on invariant subspaces, and some spectral properties are also presented.

DergiPark (Istanbul University), 2021

in [1]. In this paper we introduce a new classes of operators on semi-Hilbertian space (ℋ, ∥. ∥) called (,) power-(,)-normal denoted [(,) ] and (,) power-(,)-quasi-normal denoted [(,) ] associated with a Drazin invertible operator using its Drazin inverse. Some properties of [(,) ] and [(,) ] are investigated and some examples are also given. An operator ∈ ℬ (ℋ) is said to be (n, m) power-(,)normal for some positive operator and for some positive integers and if () (⋕) = (⋕) () .

In this paper we introduce n-power hypo-normal operator of order-n, n-power quasi-normal operator of order-n, quasi parahyponormal operator of order-n on a Hilbert space H. we give some properties of these operators.

arXiv (Cornell University), 2020

We show that a densely defined closable operator A such that the resolvent set of A 2 is not empty, is necessarily closed. This result is then extended to the case of a polynomial p(A). We also generalize a recent result by Sebestyén-Tarcsay concerning the converse of a result by J. von Neumann. Other interesting consequences are also given, one of them being a proof that if T is a quasinormal (unbounded) operator such that T n is normal for some n ≥ 2, then T is normal. By a recent result by Pietrzycki-Stochel, we infer that a closed subnormal operator such that T n is normal, must be normal. Another remarkable result is the fact that a hyponormal operator A, bounded or not, such that A p and A q are self-adjoint for some co-prime numbers p and q, is self-adjoint. It is also shown that an invertible operator (bounded or not) A for which A p and A q are normal for some co-prime numbers p and q, is normal. These two results are shown using Bézout's theorem in arithmetic. Notation First, we assume that readers have some familiarity with the standard notions and results in operator theory (see e.g. [17] and [25] for some background). We do recall most of the needed notions though. First, note that in this paper all operators are linear. Let H be a complex Hilbert space and let B(H) be the algebra of all bounded linear operators defined from H into H. If S and T are two linear operators with domains D(S) ⊂ H and D(T) ⊂ H respectively, then T is said to be an extension of S, written S ⊂ T , when D(S) ⊂ D(T) and S and T coincide on D(S). The product ST and the sum S + T of two operators S and T are defined in the usual fashion on the natural domains: D(ST) = {x ∈ D(T) : T x ∈ D(S)} and D(S + T) = D(S) ∩ D(T).

This paper is devoted to the study of some new classes of operators on Hilbert space called (n,m) -power D -normal [(n,m)DN] and (n,m) -power D -quasi-normal [(n,m)DQN] , associated with a Drazin invertible operator using its Drazin inverse. Some properties of [(n,m)DN] and [(n,m)DQN] are investigated and some examples are also given. Mathematics subject classification (2010): 47B15, 47B20, 47A15.

Revista Colombiana de Matemáticas, 2005

Abstract. In this paper we will investigate the normality in (WN) and (Y) classes. Keywords and phrases. Normal operators, Hilbert space, hermitian operators. 2000 Mathematics Subject Classification. Primary: 47A15. Secondary: 47B20, 47A63. ... Resumen. En este artıculo ...

2020

In this paper we investigate results on unitary equivalence of operators that include n-binormal, skew binormal and n-power-hyponormal operators acting on complex Hilbert space H. AMS subject classification 47B47, 47A30, 47B20.

2012

In this paper, we prove the following assertions: (i) Let A,B, X ∈ B(H) be such that A∗ is p-hyponormal or log-hyponormal, B is a dominant and X is invertible. If XA = BX, then there is a unitary operator U such that AU = UB and hence A and B are normal. (ii) Let T = A + iB ∈ B(H) be the cartesian decomposition of T with AB is p-hyponormal. If A or B is positive, then T is normal. (iii) Let A, V, X ∈ B(H) be such that V,X are isometries and A∗ is p-hyponormal. If V X = XA, then A is unitary. (iv) Let A,B ∈ B(H) be such that A + B ≥ ±X. Then for every paranormal operator X ∈ B(H) we have ‖AX + XB‖ ≥ ‖X‖2.

Comptes Rendus Mathematique, 2011

Let T ∈ B(H) and T = U |T | be its polar decomposition. We proved that (i) if T is log-hyponormal or p-hyponormal and U n = U * for some n, then T is normal; (ii) if the spectrum of U is contained in some open semicircle, then T is normal if and only if so is its Aluthge transform T = |T |