On the structure of polynomially normal operators

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.

Transactions of the American Mathematical Society, 1997

It is shown that given an essentially normal operator T T with connected spectrum, there exists a compact operator K K such that T + K T+K is strongly irreducible.

2014

Abstract. In this paper we will investigate the normality in (WN) and (Y) classes.

Israel Journal of Mathematics, 1982

Let A be a bounded linear operator in a Hilbert space. If A is normal then log[[ eA'u [I and loglleA"u II are convex functions for all u~ 0. In this paper we prove that these properties characterize normal operators.) Sponsored by the United States Army under Contract No. DAAG29-80-C-0041.

Abstract and Applied Analysis, 2012

We study some properties of -normal operators and we present various inequalities between the operator norm and the numerical radius of -normal operators on Banach algebraℬ() of all bounded linear operators , where is Hilbert space.

2008

In this paper, we prove the following: (1) If T is invertible !-hyponormal completely non-normal, then the point spectrum is empty. (2) If T1 and T2 are injective !-hyponormal and if T and S are quasisimilar, then they have the same spectra and essential spectra. (3) If T is (p,k)-quasihyponormal operator, then jp(T){ 0} = ap(T){ 0}. (4) If T ,S 2 B(H) are injective (p,k)-quasihyponormal operator, and if XT = SX, where X 2 B(H) is an invertible, then there exists a unitary operator U such that UT = SU and hence T and S are normal operators.

Journal of Mathematical Analysis and Applications, 2015

The pair (A, B) satisfies (the Putnam-Fuglede) commutativity property δ, respectively , if δ −1 AB (0) ⊆ δ −1 A * B * (0), respectively (AB − 1) −1 (0) ⊆ (A * B * − 1) −1 (0). Normaloid operators do not satisfy either of the properties δ or. This paper considers commutativity properties (δ A,λB) −1 (0) ⊆ (δ A * ,λB *) −1 (0) and (A,B − λ) −1 (0) ⊆ (A * ,B * − λ) −1 (0) for some choices of scalars λ and normaloid operators A, B. Starting with normaloid A, B ∈ B(H) such that the isolated points of their spectrum are normal eigenvalues of the operator, we prove that: (a) if (0 =)λ ∈ isoσ(L A R B) then (A,B − λ) −1 (0) ⊆ (A * ,B * − λ) −1 (0); (b) if 0 / ∈ σ p (A) ∩ σ p (B *) and 0 ∈ isoσ(L A − R λB) then (δ A,λB) −1 (0) ⊆ (δ A * ,λB *) −1 (0). Let σ π (T) denote the peripheral spectrum of the operator T. If A, B are normaloid, then: (i) either dim(B(H)/(A,B − λ)(B(H))) = ∞ for all λ ∈ σ π (A,B), or, there exists a λ ∈ σ π (A,B) ∩ σ p (A,B); (ii) if X is Hilbert-Schmidt, and AXB − λX = 0 for some λ ∈ σ π (A,B), then A * XB * − λX = 0; (iii) if V * ∈ B(H) is an isometry, λ ∈ σ π (A), A −1 (0) ⊆ A * −1 (0), and AXV − λX = 0 (or, AX − λXV = 0) for some X ∈ B(H), then A * XV * − λX = 0 (resp., A * X − λXV * = 0).

arXiv (Cornell University), 2018

The primary purpose of this paper is to show the existence of normal square and nth roots of some classes of bounded operators on Hilbert spaces. Two interesting simple results hold. Namely, under simple conditions we show that if any operator T is such that T 2 = 0, then this implies that T is normal and so T = 0. Also, we will see when the square root of an arbitrary bounded operator is normal.

International Journal of Mathematics …, 2012

In this paper, a new class of operator, k -Quasi -normal is defined as T (T * T ) k = (T * T ) k T and some properties on it are discussed. This k -Quasi -normal operator is introduced as a generalization of concept of Quasi -normal. The k -Quasi -normal operators are also characterized in terms of commutativity with the multiplication operator induced by the Radon -Nikodym derivative of the measure λ(T k ) −1 with respect to λ.

Journal of Mathematical Analysis and Applications, 2005

A Banach space operator T ∈ B(X) is said to be totally hereditarily normaloid, T ∈ THN, if every part of T is normaloid and every invertible part of T has a normaloid inverse. The operator T is said to be an H(q) operator for some integer q 1, T ∈ H(q), if the quasi-nilpotent part H 0 (T − λ) = (T − λ) −q (0) for every complex number λ. It is proved that if T is algebraically H(q), or T is algebraically THN and X is separable, then f (T) satisfies Weyl's theorem for every function f analytic in an open neighborhood of σ (T), and T * satisfies a-Weyl's theorem. If also T * has the single valued extension property, then f (T) satisfies a-Weyl's theorem for every analytic function f which is non-constant on the connected components of the open neighborhood of σ (T) on which it is defined.