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 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|).

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 .

scholarly journals Spectral radius inequalities for functions of operators defined by power series

Filomat ◽  
2016 ◽  
Vol 30 (10) ◽  
pp. 2847-2856
Author(s):  
S.S. Dragomir
Keyword(s):  
Hilbert Space ◽  
Power Series ◽  
Linear Operator ◽  
Spectral Radius ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Commuting Operators ◽  
Functions Of Operators ◽  
The Absolute

By the help of power series f(z)=??,n=0 anzn we can naturally construct another power series that has as coefficients the absolute values of the coefficients of f , namely fa(z):= ??,n=0 |an|zn. Utilising these functions we show among others that r[f(T)] ? fa [r(T)] where r (T) denotes the spectral radius of the bounded linear operator T on a complex Hilbert space while ||T|| is its norm. When we have A and B two commuting operators, then r2[f(AB)]? fa(r2(A)) fa(r2(B)) and r[f(AB)]?1/2[fa(||AB||)+fa(||A2||1/2||B2||1/2)].

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 Remarks on n-normal operators

Filomat ◽  
2018 ◽  
Vol 32 (15) ◽  
pp. 5441-5451 ◽  
Author(s):  
Muneo Chō ◽  
Ji Lee ◽  
Kotaro Tanahashi ◽  
Atsushi Uchiyama
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Bounded Linear ◽  
Normal Operators

Let T be a bounded linear operator on a complex Hilbert space and n,m ? N. Then T is said to be n-normal if T+Tn = TnT+ and (n,m)-normal if T+mTn = TnT+m. In this paper, we study several properties of n-normal, (n,m)-normal operators. In particular, we prove that if T is 2-normal with ?(T) ? (-?(T)) ? {0}, then T is polarloid. Moreover, we study subscalarity of n-normal operators. Also, we prove that if T is (n,m)-normal, then T is decomposable and Weyl?s theorem holds for f (T), where f is an analytic function on ?(T) which is not constant on each of the components of its domain.

Remarks on Invariant Subspace Lattices

1969 ◽  
Vol 12 (5) ◽  
pp. 639-643 ◽  
Author(s):  
Peter Rosenthal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Invariant Subspace ◽  
Complete Lattice ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Subspace Lattice ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Subspace Lattices

If A is a bounded linear operator on an infinite-dimensional complex Hilbert space H, let lat A denote the collection of all subspaces of H that are invariant under A; i.e., all closed linear subspaces M such that x ∈ M implies (Ax) ∈ M. There is very little known about the question: which families F of subspaces are invariant subspace lattices in the sense that they satisfy F = lat A for some A? (See [5] for a summary of most of what is known in answer to this question.) Clearly, if F is an invariant subspace lattice, then {0} ∈ F, H ∈ F and F is closed under arbitrary intersections and spans. Thus, every invariant subspace lattice is a complete lattice.

scholarly journals Factoring a Quadratic Operator as a Product of Two Positive Contractions

2016 ◽  
Vol 59 (2) ◽  
pp. 354-362 ◽  
Author(s):  
Chi-Kwong Li ◽  
Ming-Cheng Tsai
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Positive Operator ◽  
Bounded Linear Operator ◽  
Complex Hilbert Space ◽  
Necessary Condition ◽  
Operator Matrix ◽  
Quadratic Operator ◽  
Bounded Linear

AbstractLet T be a quadratic operator on a complex Hilbert space H. We show that T can be written as a product of two positive contractions if and only if T is of the formfor some a, b ∊ [0, 1] and strictly positive operator P with . Also, we give a necessary condition for a bounded linear operator T with operator matrix on H⊕K that can be written as a product of two positive contractions.

Products of positive operators

1974 ◽  
Vol 76 (2) ◽  
pp. 415-416 ◽  
Author(s):  
S. J. Bernau
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Positive Definite ◽  
Complex Hilbert Space ◽  
Positive Operators ◽  
Bounded Linear

Let H be a complex Hilbert space. Recall that a bounded linear operator A, on H, is positive if (Ax, x) ≥ 0 (x ∈ H) (so that A = A* necessarily) and positive definite if A is positive and invertible.

scholarly journals Reverse inequalities for the numerical radius of linear operators in Hilbert spaces

2006 ◽  
Vol 73 (2) ◽  
pp. 255-262 ◽  
Author(s):  
S. S. Dragomir
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Bounded Linear Operator ◽  
Upper Bounds ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear ◽  
The Difference ◽  
Reverse Inequalities

Some elementary inequalities providing upper bounds for the difference of the norm and the numerical radius of a bounded linear operator on Hilbert spaces under appropriate conditions are given.

scholarly journals Additive perturbations and multiplicative perturbations for the core inverse of bounded linear operator in Hilbert space

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6131-6144
Author(s):  
Fapeng Du ◽  
Zuhair Nashed
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Hilbert Spaces ◽  
Upper Bound ◽  
Bounded Linear Operator ◽  
Upper Bounds ◽  
Bounded Linear ◽  
The Core ◽  
Core Inverse

In this paper, we present some characteristics and expressions of the core inverse A# of bounded linear operator A in Hilbert spaces. Additive perturbations of core inverse are investigated under the condition R( ?)?N(A#) = {0} and an upper bound of ||?#-A#|| is obtained. We also discuss the multiplicative perturbations. The expressions of core inverse of perturbed operator T = EAF and the upper bounds of ||T#-A#|| are obtained too.

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