scholarly journals Spectral properties of p-hyponormal operators

1994 ◽  
Vol 36 (1) ◽  
pp. 117-122 ◽  
Author(s):  
Muneo Chō
Keyword(s):  
Hilbert Space ◽  
Spectral Properties ◽  
Polar Decomposition ◽  
Complex Hilbert Space ◽  
Hyponormal Operator ◽  
Bounded Linear ◽  
Hyponormal Operators

Let ℋ be a complex Hilbert space and B(ℋ) be the algebra of all bounded linear opeators on ℋ. An operator T ∈ B(ℋ) is said to be p-hyponormal if (T*T)p–(TT*)p. If p = 1, T is hyponormal and if p = ½ is semi-hyponormal. It is well known that a p-hyponormal operator is p-hyponormal for q≤p. Hyponormal operators have been studied by many authors. The semi-hyponormal operator was first introduced by D. Xia in [7]. The p-hyponormal operators have been studied by A. Aluthge in [1]. Let T be a p-hyponormal operator and T=U|T| be a polar decomposition of T. If U is unitary, Aluthge in [1] proved the following properties.

scholarly journals Spectral properties of square hyponormal operators

Filomat ◽  
2019 ◽  
Vol 33 (15) ◽  
pp. 4845-4854
Author(s):  
Muneo Chō ◽  
Dijana Mosic ◽  
Biljana Nacevska-Nastovska ◽  
Taiga Saito
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Spectral Properties ◽  
Complex Hilbert Space ◽  
Hyponormal Operator ◽  
Bounded Linear ◽  
Basic Properties ◽  
Hyponormal Operators ◽  
Eigen Values

In this paper, we introduce a square hyponormal operator as a bounded linear operator T on a complex Hilbert space H such that T2 is a hyponormal operator, and we investigate some basic properties of this operator. Under the hypothesis ?(T) ? (-?(T)) ? {0}, we study spectral properties of a square hyponormal operator. In particular, we show that if z and w are distinct eigen-values of T and x,y ? H are corresponding eigen-vectors, respectively, then ?x,y? = 0. Also, we define nth hyponormal operators and present some properties of this kind of operators.

scholarly journals On the spectrum of n-tuples of p-hyponormal operators

1998 ◽  
Vol 40 (1) ◽  
pp. 123-131 ◽  
Author(s):  
B. P. Duggal
Keyword(s):  
Spectral Properties ◽  
Polar Decomposition ◽  
Partial Isometry ◽  
Hyponormal Operator ◽  
Square Root ◽  
Linear Transformations ◽  
Bounded Linear ◽  
Hyponormal Operators ◽  
Algebra Of Operators ◽  
Number Of Authors

Let B(H) denote the algebra of operators (i.e., bounded linear transformations) on the Hilbert space H. A ∈ B (H) is said to be p-hyponormal (0<p<l), if (AA*)γ < (A*A)p. (Of course, a l-hyponormal operator is hyponormal.) The p-hyponormal property is monotonic decreasing in p and a p-hyponormal operator is q-hyponormal operator for all 0<q <p. Let A have the polar decomposition A = U |A|, where U is a partial isometry and |A| denotes the (unique) positive square root of A*A.If A has equal defect and nullity, then the partial isometry U may be taken to be unitary. Let ℋU(p) denote the class of p -hyponormal operators for which U in A = U |A| is unitary. ℋU(l/2) operators were introduced by Xia and ℋU(p) operators for a general 0<p<1 were first considered by Aluthge (see [1,14]); ℋU(p) operators have since been considered by a number of authors (see [3, 4, 5, 9, 10] and the references cited in these papers). Generally speaking, ℋU(p) operators have spectral properties similar to those of hyponormal operators. Indeed, let A ε ℋU(p), (0<p <l/2), have the polar decomposition A = U|A|, and define the ℋW(p + 1/2) operator  by A = |A|1/2U |A|l/2 Let  = V |Â| Â= |Â|1/2VÂ|ÂAcirc;|1/2. Then we have the following result.

scholarly journals Generalizations of the Aluthge transform of operators

Filomat ◽  
2018 ◽  
Vol 32 (18) ◽  
pp. 6465-6474 ◽  
Author(s):  
Khalid Shebrawi ◽  
Mojtaba Bakherad
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Function ◽  
Polar Decomposition ◽  
Continuous Functions ◽  
Complex Hilbert Space ◽  
Numerical Radius ◽  
Aluthge Transform ◽  
Bounded Linear ◽  
Definition Of

Let A be an operator with the polar decomposition A = U|A|. The Aluthge transform of the operator A, denoted by ?, is defined as ? = |A|1/2U |A|1/2. In this paper, first we generalize the definition of Aluthge transformfor non-negative continuous functions f,g such that f(x)g(x) = x (x ? 0). Then, by using this definition, we get some numerical radius inequalities. Among other inequalities, it is shown that if A is bounded linear operator on a complex Hilbert space H, then h (w(A)) ? 1/4||h(g2 (|A|)) + h(f2(|A|)|| + 1/2h (w(? f,g)), where f,g are non-negative continuous functions such that f(x)g(x) = x (x ? 0), h is a non-negative and non-decreasing convex function on [0,?) and ? f,g = f (|A|)Ug(|A|).

scholarly journals On Numerical Radius Bounds Involving Generalized Aluthge Transform

2022 ◽  
Vol 2022 ◽  
pp. 1-8
Author(s):  
Tao Yan ◽  
Javariya Hyder ◽  
Muhammad Saeed Akram ◽  
Ghulam Farid ◽  
Kamsing Nonlaopon
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Polar Decomposition ◽  
Upper Bounds ◽  
Complex Hilbert Space ◽  
Numerical Radius ◽  
Aluthge Transform ◽  
Bounded Linear

In this paper, we establish some upper bounds of the numerical radius of a bounded linear operator S defined on a complex Hilbert space with polar decomposition S = U ∣ S ∣ , involving generalized Aluthge transform. These bounds generalize some bounds of the numerical radius existing in the literature. Moreover, we consider particular cases of generalized Aluthge transform and give some examples where some upper bounds of numerical radius are computed and analyzed for certain operators.

scholarly journals Generalized Weyl's theorem and hyponormal operators

2004 ◽  
Vol 76 (2) ◽  
pp. 291-302 ◽  
Author(s):  
M. Berkani ◽  
A. Arroud
Keyword(s):  
Hilbert Space ◽  
Weyl’S Theorem ◽  
Hyponormal Operator ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Bounded Linear ◽  
Generalized Weyl’S Theorem ◽  
Weyl Spectrum ◽  
Hyponormal Operators ◽  
Isolated Eigenvalues

AbstractLet T be a bounded linear operator acting on a Hilbert space H. The B-Weyl spectrum of T is the set σBW(T) of all λ ∈ Сsuch that T − λI is not a B-Fredholm operator of index 0. Let E(T) be the set of all isolated eigenvalues of T. The aim of this paper is to show that if T is a hyponormal operator, then T satisfies generalized Weyl's theorem σBW(T) = σ(T)/E(T), and the B-Weyl spectrum σBW(T) of T satisfies the spectral mapping theorem. We also consider commuting finite rank perturbations of operators satisfying generalized Weyl's theorem.

scholarly journals Spectral properties of n-normal operators

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 5063-5069 ◽  
Author(s):  
Muneo Chō ◽  
Biljana Nacevska
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Spectral Properties ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Reducing Subspace ◽  
Normal Operators ◽  
Isolated Point ◽  
Zero Number

For a bounded linear operator T on a complex Hilbert space and n ? N, T is said to be n-normal if T*Tn = TnT*. In this paper we show that if T is a 2-normal operator and satisfies ?(T) ? (-?(T)) ? {0}, then T is isoloid and ?(T) = ?a(T). Under the same assumption, we show that if z and w are distinct eigenvalues of T, then ker(T-z)? ker(T-w). And if non-zero number z ? C is an isolated point of ?(T), then we show that ker(T-z) is a reducing subspace for T. We show that if T is a 2-normal operator satisfying ?(T) ?(-?(T)) = 0, then Weyl?s theorem holds for T. Similarly, we show spectral properties of n-normal operators under similar assumption. Finally, we introduce (n,m)-normal operators and show some properties of this kind of operators.

scholarly journals Weyl's Theorem for Algebraically (𝑝, 𝑘)-Quasihyponormal Operators

2006 ◽  
Vol 13 (2) ◽  
pp. 307-313
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space ◽  
Weyl’S Theorem ◽  
Hyponormal Operator ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Bounded Linear ◽  
Generalized Weyl’S Theorem ◽  
Weyl Spectrum ◽  
Hyponormal Operators ◽  
Isolated Eigenvalues

Abstract Let 𝐴 be a bounded linear operator acting on a Hilbert space 𝐻. The 𝐵-Weyl spectrum of 𝐴 is the set σ 𝐵𝑤(𝐴) of all ⋋ ∈ ℂ such that 𝐴 – ⋋𝐼 is not a 𝐵-Fredholm operator of index 0. Let 𝐸(𝐴) be the set of all isolated eigenvalues of 𝐴. Recently, in [Berkani and Arroud, J. Aust. Math. Soc. 76: 291–302, 2004] the author showed that if 𝐴 is hyponormal, then 𝐴 satisfies the generalized Weyl's theorem σ 𝐵𝑤(𝐴) = σ(𝐴) \ 𝐸(𝐴), and the 𝐵-Weyl spectrum σ 𝐵𝑤(𝐴) of 𝐴 satisfies the spectral mapping theorem. Lee [Han, Proc. Amer. Math. Soc. 128: 2291–2296, 2000] showed that Weyl's theorem holds for algebraically hyponormal operators. In this paper the above results are generalized to an algebraically (𝑝, 𝑘)-quasihyponormal operator which includes an algebraically hyponormal operator.

Nonlinear ∗-Jordan triple derivations on von Neumann algebras

2018 ◽  
Vol 68 (1) ◽  
pp. 163-170 ◽  
Author(s):  
Fangfang Zhao ◽  
Changjing Li
Keyword(s):  
Hilbert Space ◽  
Von Neumann Algebra ◽  
Von Neumann Algebras ◽  
New Product ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Von Neumann

AbstractLetB(H) be the algebra of all bounded linear operators on a complex Hilbert spaceHand 𝓐 ⊆B(H) be a von Neumann algebra with no central summands of typeI1. ForA,B∈ 𝓐, define byA∙B=AB+BA∗a new product ofAandB. In this article, it is proved that a map Φ: 𝓐 →B(H) satisfies Φ(A∙B∙C) = Φ(A) ∙B∙C+A∙ Φ(B) ∙C+A∙B∙Φ(C) for allA,B,C∈ 𝓐 if and only if Φ is an additive *-derivation.

On Self-Adjoint Factorization of Operators

1969 ◽  
Vol 21 ◽  
pp. 1421-1426 ◽  
Author(s):  
Heydar Radjavi
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Unitary Operator ◽  
Normal Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Normal Operators ◽  
Adjoint Operators

The main result of this paper is that every normal operator on an infinitedimensional (complex) Hilbert space ℋ is the product of four self-adjoint operators; our Theorem 4 is an actually stronger result. A large class of normal operators will be given which cannot be expressed as the product of three self-adjoint operators.This work was motivated by a well-known resul t of Halmos and Kakutani (3) that every unitary operator on ℋ is the product of four symmetries, i.e., operators that are self-adjoint and unitary.1. By “operator” we shall mean bounded linear operator. The space ℋ will be infinite-dimensional (separable or non-separable) unless otherwise specified. We shall denote the class of self-adjoint operators on ℋ by and that of symmetries by .

Operators with Compact Self-Commutator

1974 ◽  
Vol 26 (1) ◽  
pp. 115-120 ◽  
Author(s):  
Carl Pearcy ◽  
Norberto Salinas
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Open Problems ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Calkin Algebra ◽  
The Ideal

Let be a fixed separable, infinite dimensional complex Hilbert space, and let () denote the algebra of all (bounded, linear) operators on . The ideal of all compact operators on will be denoted by and the canonical quotient map from () onto the Calkin algebra ()/ will be denoted by π.Some open problems in the theory of extensions of C*-algebras (cf. [1]) have recently motivated an increasing interest in the class of all operators in () whose self-commuta tor is compact.

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