A note on $p$-hyponormal operators

Huruya, Tadasi

doi:10.1090/s0002-9939-97-04004-5

Cited by 45 publications

(14 citation statements)

References 15 publications

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“…Let T = M w EM u be multiplication conditional type operator. Then T is p-hyponormal if and only if T is normal.Proof By Theorem 3.1 and [Theorem 3,[8]] we conclude that if T is p-hyponormal, then T is normal. The converse is clear.…”

mentioning

confidence: 90%

On a class of operators with normal Aluthge transformations

Estaremi

2015

Filomat

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In this paper, we show that the generalized Aluthge transformations of a large class of operators (weighted conditional type operators) are normal. As a consequence, the operator M w EM u is p-hyponormal if and only if it is normal, and under a weak condition, if M w EM u is normal, then the Holder inequality turn into equality for w, u. Also, we give some applications of p-hyponormal weighted conditional type operators, for instance, point spectrum and joint point spectrum of p-hyponormal weighted conditional type operators are equal. In the end, some examples are provided to illustrate concrete application of the main results of the paper.

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mentioning

confidence: 90%

On a class of operators with normal Aluthge transformations

Estaremi

2015

Filomat

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“…Different type of spectra and Aluthge operator fields. Some spectral results are known for ∆ 1 2 pT q :" ∆pT q where T P BpHq (see for instance [17], and Theorem 1.3 and Theorem 1.5 of [18]). In the next theorem, we extend a number of useful spectral properties of ∆pT q to the Aluthge operator field p∆ z pT qq zPS and establish some new spectral properties in this context.…”

Section: Two Possibilities Occursmentioning

confidence: 99%

Aluthge Operator Field and Its Numerical Range and Spectral Properties

Cassier

Perrin

2021

Integr. Equ. Oper. Theory

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For an arbitrary operator T acting on a Hilbert space we consider a field of operators p∆ z pT qq called the Aluthge operator field associated with T . After giving preliminary results, we establish that two fields (left and right), canonically linked to the Altuthge field p∆ z pT qq and a support subspace, are constant on each horizontal segment where they are defined. This result leads to a positive solution of a conjecture stated by Jung-Ko-Pearcy in 2000. Then we do a detailed spectral study of p∆ z pT qq and we give a Yamazaki type formula in this context.As usual, we write σ p pT q, σ app pT q, σ res pT q, σ e pT q, σ le pT q, σ re pT q and σ surj pT q for the point spectrum, the approximate point spectrum, the residual spectrum, the essential spectrum, the left essential spectrum, the right essential spectrum and the surjective spectrum of T , respectively. Recall that an operator T P BpHq is said to be Weyl if it is Fredholm of index zero and Browder if it is Fredholm of finite ascent and descent. The Weyl spectrum σ w pT q and the Browder spectrum σ b pT q of T P BpHq are defined by σ w pT q " tλ P C : λI ´T is not Weylu σ b pT q " tλ P C : λI ´T is not Browderu . Denote by S the open strip of the complex plane defined by S " tz P C; 0 ă pzq ă 1u . Let T P BpHq and let T " U |T | be its polar decomposition, we define the Aluthge field of operators associated with T by setting ∆ z pT q " |T | z U |T | 1´z Date: 20/10/2020. ˚Corresponding author.

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“…We say that A is p ‐hyponormal if

{D (| A |}^{p}) \subseteq D ({| A^{*} |}^{p})

and

false∥ false| A^{*} {false|}^{p} {f ∥ ⩽ ∥ | A |}^{p} f false∥

for every

f \in {D (| A |}^{p})

. Given

α \in (0, 1]

{normalΔ}_{α} false(A false) = {| A |}^{α} U {| A |}^{1 - α}

denotes the α‐Aluthge transform of A (see [1, 25]).…”

Section: Preliminariesmentioning

confidence: 99%

“…The Aluthge transform of a bounded operator T was introduced by Aluthge to investigate the properties of p ‐hyponormal operators. He showed that the Aluthge transform of a p ‐hyponormal operator is itself a

(() p + \frac{1}{2})

‐hyponormal operator (see [1, 2]; see also [25]). Various connections between an operator and its Aluthge transform were later on studied by Jung, Ko, and Pearcy, in particular with regard to the invariant subspace problem.…”

Section: Introductionmentioning

confidence: 99%

Aluthge transforms of unbounded weighted composition operators in L²‐spaces

Benhida

Budzyński

Trepkowski

2020

Mathematische Nachrichten

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We describe the Aluthge transform of an unbounded weighted composition operator acting in an 2-space. We show that its closure is also a weighted composition operator with the same symbol and a modified weight function. We investigate its dense definiteness. We characterize-hyponormality of unbounded weighted composition operators and provide results on how it is affected by the Aluthge transformation. We show that the only fixed points of the Aluthge transformation on weighted composition operators are quasinormal ones.

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A note on $p$-hyponormal operators

Abstract: Abstract. Let T be a p-hyponormal operator on a Hilbert space with polar decomposition T = U |T | and let T = |T | t U |T | r−t for r > 0 and r ≥ t ≥ 0. We study order and spectral properties of T . In particular we refine recent Furuta's result on p-hyponormal operators.

Cited by 45 publications

References 15 publications

On a class of operators with normal Aluthge transformations

On a class of operators with normal Aluthge transformations

Aluthge Operator Field and Its Numerical Range and Spectral Properties

Aluthge transforms of unbounded weighted composition operators in L²‐spaces

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A note on $p$-hyponormal operators

Abstract: Abstract. Let T be a p-hyponormal operator on a Hilbert space with polar decomposition T = U |T | and let T = |T | t U |T | r−t for r > 0 and r ≥ t ≥ 0. We study order and spectral properties of T . In particular we refine recent Furuta's result on p-hyponormal operators.

Cited by 45 publications

References 15 publications

On a class of operators with normal Aluthge transformations

On a class of operators with normal Aluthge transformations

Aluthge Operator Field and Its Numerical Range and Spectral Properties

Aluthge transforms of unbounded weighted composition operators in L2‐spaces

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Aluthge transforms of unbounded weighted composition operators in L²‐spaces