scholarly journals On Relation between the Joint Essential Spectrum and the Joint Essential Numerical Range of Aluthge Transform

2021 ◽  
pp. 1-9
Author(s):  
O. S. Cyprian
Keyword(s):  
Hilbert Space ◽  
Essential Spectrum ◽  
Numerical Range ◽  
Complex Hilbert Space ◽  
Aluthge Transform

Associated with every commuting m-tuples of operators on a complex Hilbert space X is its Aluthge transform. In this paper we show that every commuting m-tuples of operators on a complex Hilbert space X and its Aluthge transform have the same joint essential spectrum. Further, it is shown that the joint essential spectrum of Aluthge transform is contained in the joint essential numerical range of Aluthge transform.

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 A characterization of spectral operators on Hilbert spaces

1982 ◽  
Vol 23 (1) ◽  
pp. 91-95 ◽  
Author(s):  
Ernst Albrecht
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Essential Spectrum ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Spectral Operator ◽  
Bounded Linear ◽  
Image Position

Let H be a complex Hilbert space and denote by B(H) the Banach algebra of all bounded linear operators on H. In [5; 6] J. Ph. Labrousse proved that every operator S∈B(H) which is spectral in the sense of N. Dunford (see [3]) is similar to a T∈B(H) with the following propertyConversely, he showed that given an operator S∈B(H) such that its essential spectrum (in the sense of [5; 6]) consists of at most one point and such that S is similar to a T∈B(H) with the property (1), then S is a spectral operator. This led him to the conjecture that an operator S∈B(H) is spectral if and only if it is similar to a T∈B(H) with property (1). The purpose of this note is to prove this conjecture in the case of operators which are decomposable in the sense of C. Foias (see [2]).

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.

Essentially Convexoid Operators

1981 ◽  
Vol 33 (2) ◽  
pp. 257-274
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Convex Hull ◽  
Numerical Range ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Quotient Mapping ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Essentially Normal

Let H be a separable complex Hilbert space and let B(H) denote the algebra of all bounded linear operators on H. Let π be the quotient mapping from B(H) onto the Calkin algebra B(H)/K(H), where K(H) denotes all compact operators on B(H). An operator T ∈ B(H) is said to be convexoid[14] if the closure of its numerical range W(T) coincides with the convex hull co σ(T) of its spectrum σ(T). T ∈ B(H) is said to be essentially normal, essentially G1, or essentially convexoid if π(T) is normal, G1 or convexoid in B(H)/K(H) respectively.

scholarly journals A Formula for the Numerical Range of Elementary Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-4
Author(s):  
M. Barraa
Keyword(s):  
Hilbert Space ◽  
Numerical Range ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Unitary Operators ◽  
Elementary Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Elementary Operators

Let BH be the algebra of bounded linear operators on a complex Hilbert space H. For k-tuples of elements of BH, A=A1,…,Ak and B=B1,…,Bk, let RA,B denote the elementary operator on BH defined by RA,BX=∑i=1kAiXBi. In this paper, we prove the following formula for the numerical range of RA,B: VRA,B,BBH=[∪U∈UHW(∑i=1kUAiU*Bi)-]-, where UH is the set of unitary operators.

Relations Between Generalized Growth Conditions and Several Classes of Convexoid Operators

1977 ◽  
Vol 29 (5) ◽  
pp. 1010-1030 ◽  
Author(s):  
Takayuki Furuta
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Numerical Range ◽  
Growth Conditions ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In this paper we shall discuss some classes of bounded linear operators on a complex Hilbert space. If T is a bounded linear operator T acting on the complex Hilbert space H, then the following two inequalities always hold:where σ(T) indicates the spectrum of T, W(T) denotes the numerical range of T defined by W(T) = {(Tx, x) : ||x|| = 1 and x ∊ H} and means the closure of W(T) respectively.

Jordan Loops and Decompositions of Operators

1977 ◽  
Vol 29 (5) ◽  
pp. 1112-1119 ◽  
Author(s):  
Arlen Brown ◽  
Carl Pearcy
Keyword(s):  
Hilbert Space ◽  
Essential Spectrum ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Compact Perturbation ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Spectrum Of An Operator ◽  
Infinite Dimensional

Let be a separable, infinite dimensional, complex Hilbert space, and let denote the algebra of all bounded linear operators on . In what follows we shall denote the spectrum, essential spectrum, and left essential spectrum of an operator T in , respectively. Furthermore, if and T1 is unitarily equivalent to a compact perturbation of an operator T2, then we write T1~ T2, and if the compact perturbation can be chosen to have norm less than e, we write T1 ~ T2(ϵ).

scholarly journals Centers of mass for operator-families

1992 ◽  
Vol 34 (1) ◽  
pp. 123-126
Author(s):  
Makoto Takaguchi
Keyword(s):  
Hilbert Space ◽  
Numerical Range ◽  
Complex Hilbert Space ◽  
Bounded Operators ◽  
Centers Of Mass ◽  
Image Position ◽  
Joint Numerical Range

Let Hbe a complex Hilbert space and let B(H) be the algebra of (bounded) operators on H. Let A =(A,…,An) be an n-tuple of operators on H. The joint numerical range of A is the subset W(A) of ℂn such that

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