Elementary operators and the Aluthge transform

Zenodo (CERN European Organization for Nuclear Research), 2023

We investigate the necessary and sufficient conditions for Toeplitz operators, tensor product of Toeplitz operators, Commutator of operators, Duggal transform and Aluthge transform of Toeplitz operators on the Bergman space to be 2-isometry. Further, some sufficient conditions for operator matrices on the vector valued Bergman space are shown to be 2isometry.

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.

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 ...

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 |

2005

A bounded linear operator T is called ∞-hyponormal if T is p-hyponormal for every p> 0. In this paper ∞-hyponormality of the Aluthge transformations of ∞-hyponormal operators is investigated. It is shown that the Aluthge transfor- mation of an ∞-hyponormal operator is not necessarily ∞-hyponormal. It is also shown that the (generalized) Aluthge transformation of an ∞-hyponormal operator T is ∞-hyponormal provided |T ||T ∗ | = |T ∗ ||T |. Moreover we give an example of an ∞-hyponormal operator T whose Aluthge transformation ˜ T is ∞-hyponormal but |T ||T ∗ | � |T ∗ ||T |.

Mathematische Nachrichten, 2014

A Hilbert space operator T ∈ L (H) is M-hyponormal if there exists a positive real number M such that (T − μ)(T − μ) * ≤ M 2 (T − μ) * (T − μ) for all μ ∈ σ (T). Let A, B * ∈ L (H) be M-hyponormal and let d AB ∈ L (L (H)) denote either the generalized derivation δ AB (X) = AX − X B or the elementary operator AB = AX B − X. We prove that if A, B * are M-hyponormal, then f (d AB) satisfies the generalized Weyl's theorem and f (d * AB) satisfies the generalized a-Weyl's theorem for every f that is analytic on a neighborhood of σ (d AB).

Integral Equations and Operator Theory, 2005

Let B(H) denote the algebra of all bounded linear operators on a separable infinite dimensional complex Hilbert space H into itself. Let A = (A1, A2, .., An) and B = (B1, B2, .., Bn) be n-tuples in B(H), we define the elementary operator EA,B : B(H) → B(H) by EA,B(X) = È n i=1 AiXBi.

Integral Equations and Operator Theory, 2002

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.

2010

Let A, B * ∈ B(H) be w-hyponormal operators, and let d AB ∈ B(B(H)) denote either the generalized derivation δ AB (X) = AX -XB or the length two elementary operator AB (X) = AXB -X. We prove that d AB has the single-valued extension property, and the ) denote the space of functions which are analytic on σ (d AB ), and let H c (σ (d AB )) denote the space of f ∈ H(σ (d AB )) which are non-constant on every connected component of σ (d AB ). It is proved that, for every h ∈ H(σ (d AB )) and f , g ∈ H c (σ (d AB )), the complement of the Weyl spectrum σ w (h )) consists of isolated points in σ (h(d f (A)g(B) )) which are eigenvalues of finite multiplicity.