Some Conditions on Non-Normal Operators which Imply Normality (original) (raw)

Some conditions implying normality of operators

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 |

Commutativity theorems for normaloid Hilbert space 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).

On the Spectra of Some Non-Normal Operators

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.

On the normality of operators

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 ...

Unbounded operators having self-adjoint or normal powers and some related results

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).

A Criterion for the Normality of Unbounded Operators and Applications to Self-adjointness

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.

More on (𝜶,𝜷)-Normal Operators in Hilbert Spaces

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.

A characterization of 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.

Some properties of paranormal and hyponormal operators

In this article we will give some properties of paranormal and hyponormal operators. Exactly we will give some conditions which are generalization of concepts of paranormal, hyponormal, N-paranormal, N-hyponormal operators.

On the binary relation on self-adjoint Hilbert space operators

Comptes Rendus Mathematique, 2012

Given self-adjoint operators A, B ∈ B(H) it is said A ≤ u B whenever A ≤ U * BU for some unitary operator U. We show that A ≤ u B if and only if f (g(A) r) ≤ u f (g(B) r) for any increasing operator convex function f , any operator monotone function g and any positive number r. We present some sufficient conditions under which if B ≤ A ≤ U * BU , then B = A = U * BU. Finally we prove that if A n ≤ U * A n U for all n ∈ N, then A = U * AU. A ≤ u B ⇒ e A ≤ u e B. (1) Okayasu and Ueta [7] gave a sufficient condition for a triple of operators (A, B, U) with A, B ∈ B h (H) and U ∈ U(H) under which B ≤ A ≤ U * BU implies B = A = U * BU. In this note we use their idea and prove a similar result. In fact we present some sufficient conditions on an operator U ∈ U(H) for which B ≤ A ≤ U * BU ensures B = A = U * BU when A, B ∈ B h (H). It is known that ≤ u satisfies the reflexive and transitive laws but not the antisymmetric law in general; cf. [7]. The antisymmetric law states that A ≤ u B and B ≤ u A ⇒ A, B are unitarily equivalent. We, among other things, study some cases in which the antisymmetric law holds for the relation ≤ u. We refer the reader to [4] for general information on operators acting on