Generalizations of the Aluthge transform of operators

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|).

Download Full-text

On Numerical Radius Bounds Involving Generalized Aluthge Transform

Journal of Function Spaces ◽

10.1155/2022/2897323 ◽

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.

Download Full-text

On Self-Adjoint Factorization of Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-1969-156-6 ◽

1969 ◽

Vol 21 ◽

pp. 1421-1426 ◽

Cited By ~ 4

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 .

Download Full-text

Spectral radius inequalities for functions of operators defined by power series

Filomat ◽

10.2298/fil1610847d ◽

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)].

Download Full-text

Spectral properties of square hyponormal operators

Filomat ◽

10.2298/fil1915845c ◽

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.

Download Full-text

Remarks on n-normal operators

Filomat ◽

10.2298/fil1815441c ◽

2018 ◽

Vol 32 (15) ◽

pp. 5441-5451 ◽

Cited By ~ 2

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.

Download Full-text

Remarks on Invariant Subspace Lattices

Canadian Mathematical Bulletin ◽

10.4153/cmb-1969-082-1 ◽

1969 ◽

Vol 12 (5) ◽

pp. 639-643 ◽

Cited By ~ 1

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.

Download Full-text

Factoring a Quadratic Operator as a Product of Two Positive Contractions

Canadian Mathematical Bulletin ◽

10.4153/cmb-2015-049-4 ◽

2016 ◽

Vol 59 (2) ◽

pp. 354-362 ◽

Cited By ~ 1

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.

Download Full-text

Products of positive operators

Mathematical Proceedings of the Cambridge Philosophical Society ◽

10.1017/s0305004100049100 ◽

1974 ◽

Vol 76 (2) ◽

pp. 415-416 ◽

Cited By ~ 1

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.

Download Full-text

Products of Normal Operators

Canadian Journal of Mathematics ◽

10.4153/cjm-1988-059-5 ◽

1988 ◽

Vol 40 (6) ◽

pp. 1322-1330 ◽

Cited By ~ 5

Author(s):

Pei Yuan Wu

Keyword(s):

Hilbert Space ◽

Linear Operator ◽

Separable Hilbert Space ◽

Polar Decomposition ◽

Sufficient Conditions ◽

Bounded Linear ◽

Underlying Space ◽

Finite Dimensional ◽

Normal Operators ◽

Necessary And Sufficient

Which bounded linear operator on a complex, separable Hilbert space can be expressed as the product of finitely many normal operators? What is the answer if “normal” is replaced by “Hermitian”, “nonnegative” or “positive”? Recall that an operator T is nonnegative (resp. positive) if (Tx, x) ≧ 0 (resp. (Tx, x) ≥ 0) for any x ≠ 0 in the underlying space. The purpose of this paper is to provide complete answers to these questions.If the space is finite-dimensional, then necessary and sufficient conditions for operators expressible as such are already known. For normal operators, this is easy. By the polar decomposition, every operator is the product of two normal operators. An operator is the product of Hermitian operators if and only if its determinant is real; moreover, in this case, 4 Hermitian operators suffice and 4 is the smallest such number (cf. [10]).

Download Full-text

Fuglede–Putnam type theorems via the generalized second Aluthge transform

Asian-European Journal of Mathematics ◽

10.1142/s1793557120500187 ◽

2018 ◽

Vol 13 (01) ◽

pp. 2050018

Author(s):

Manzar Maleki ◽

Ali Reza Janfada

Keyword(s):

Linear Operator ◽

Bounded Linear Operator ◽

Polar Decomposition ◽

Aluthge Transform ◽

Bounded Linear ◽

Schmidt Operator

Let [Formula: see text] and [Formula: see text] be the polar decompositions of [Formula: see text] and [Formula: see text]. A pair [Formula: see text] is said to have the FP-property if [Formula: see text] implies [Formula: see text] for any [Formula: see text]. Let [Formula: see text] denote the generalized second Aluthge transform of a bounded linear operator [Formula: see text] such that [Formula: see text] is the polar decomposition of [Formula: see text] where [Formula: see text] denotes the first Aluthge transform of operator [Formula: see text]. We show that (i) if [Formula: see text] is class [Formula: see text] and [Formula: see text] is invertible class [Formula: see text] operators with [Formula: see text] such that [Formula: see text] for some Hilbert Schmidt operator [Formula: see text], then [Formula: see text]; (ii) if [Formula: see text] for any [Formula: see text], then [Formula: see text] for any [Formula: see text], furthermore, if [Formula: see text] is invertible, then [Formula: see text]. Finally, if [Formula: see text] and [Formula: see text] and [Formula: see text] is an operator such that [Formula: see text], then we prove that [Formula: see text] for any [Formula: see text] such that [Formula: see text].

Download Full-text