A weaker condition for normality

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 |

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 ...

arXiv (Cornell University), 2013

In this paper we give and prove a criterion for the normality of unbounded closed operators, which is a sort of a maximality result which will be called "double maximality". As applications, we show, under some assumptions, that the sum of two symmetric operators is essentially self-adjoint; and that the sum of two unbounded normal operators is essentially normal. Some other important results are also established.

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.

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.

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).

2014

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

2010

We discuss the content and significance of John von Neumann's quantum ergodic theorem (QET) of 1929, a strong result arising from the mere mathematical structure of quantum mechanics. The QET is a precise formulation of what we call normal typicality, i.e., the statement that, for typical large systems, every initial wave function ψ 0 from an energy shell is "normal": it evolves in such a way that |ψ t ψ t | is, for most t, macroscopically equivalent to the micro-canonical density matrix. The QET has been mostly forgotten after it was criticized as a dynamically vacuous statement in several papers in the 1950s. However, we point out that this criticism does not apply to the actual QET, a correct statement of which does not appear in these papers, but to a different (indeed weaker) statement. Furthermore, we formulate a stronger statement of normal typicality, based on the observation that the bound on the deviations from the average specified by von Neumann is unnecessarily coarse and a much tighter (and more relevant) bound actually follows from his proof.