scholarly journals Norm estimates for functions of non-selfadjoint operators nonregular on the convex hull of the spectrum

2017 ◽  
Vol 50 (1) ◽  
pp. 267-277 ◽  
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Hull ◽  
Meromorphic Functions ◽  
Holomorphic Functions ◽  
Fractional Powers ◽  
Bounded Linear ◽  
Norm Estimates ◽  
Selfadjoint Operators ◽  
Finite Dimensional

Abstract We consider a bounded linear operator A in a Hilbert space with a Hilbert-Schmidt Hermitian component (A − A*)/2i. A sharp norm estimate is established for functions of A nonregular on the convex hull of the spectrum. The logarithm, fractional powers and meromorphic functions of operators are examples of such functions. Our results are based on the existence of a sequence An (n = 1, 2, ...) of finite dimensional operators strongly converging to A, whose spectra belongs to the spectrum of A. Besides, it is shown that the resolvents and holomorphic functions of An strongly converge to the resolvent and corresponding function of A.

A note on normal operators

1963 ◽  
Vol 59 (4) ◽  
pp. 727-729 ◽  
Author(s):  
S. J. Bernau ◽  
F. Smithies
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Adjoint Operator ◽  
Spectral Theorem ◽  
Unitary Operators ◽  
Unitary Space ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Normal Operators ◽  
Finite Dimensional Case

We recall that a bounded linear operator T in a Hilbert space or finite-dimensional unitary space is said to be normal if T commutes with its adjoint operator T*, i.e. TT* = T*T. Most of the proofs given in the literature for the spectral theorem for normal operators, even in the finite-dimensional case, appeal to the corresponding results for Hermitian or unitary operators.

scholarly journals Resolvents of functions of operators with Hilbert-Schmidt hermitian components

Filomat ◽  
2018 ◽  
Vol 32 (14) ◽  
pp. 4937-4947
Author(s):  
Michael Gil’
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Convex Hull ◽  
Operator Function ◽  
Separable Hilbert Space ◽  
Operator Equations ◽  
Bounded Linear ◽  
Operator Functions ◽  
Schmidt Operator ◽  
Unit Operator

Let H be a separable Hilbert space with the unit operator I. We derive a sharp norm estimate for the operator function (?I-f(A))-1 (? ? C), where A is a bounded linear operator in H whose Hermitian component (A- A*)/2i is a Hilbert-Schmidt operator and f(z) is a function holomorphic on the convex hull of the spectrum of A. Here A* is the operator adjoint to A. Applications of the obtained estimate to perturbations of operator equations, whose coefficients are operator functions and localization of spectra are also discussed.

scholarly journals Weyl type theorems for selfadjoint operators on Krein spaces

Filomat ◽  
2018 ◽  
Vol 32 (17) ◽  
pp. 6001-6016
Author(s):  
Il An ◽  
Jaeseong Heo
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Krein Space ◽  
Linear Operators ◽  
Fredholm Theory ◽  
Bounded Linear ◽  
Selfadjoint Operators ◽  
Krein Spaces ◽  
Weyl Type Theorems ◽  
Kreĭn Space

In this paper, we introduce a notion of the J-kernel of a bounded linear operator on a Krein space and study the J-Fredholm theory for Krein space operators. Using J-Fredholm theory, we discuss when (a-)J-Weyl?s theorem or (a-)J-Browder?s theorem holds for bounded linear operators on a Krein space instead of a Hilbert space.

Products of Normal Operators

1988 ◽  
Vol 40 (6) ◽  
pp. 1322-1330 ◽  
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]).

scholarly journals An observation about pseudospectra

Filomat ◽  
2021 ◽  
Vol 35 (3) ◽  
pp. 995-1000
Author(s):  
Boting Jia ◽  
Youling Feng
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Finite Dimensional

For ? > 0 and a bounded linear operator T acting on some Hilbert space, the ?-pseudospectrum of T is ??(T) = {z ? C : ||(zI-T)-1|| > ?-1}. This note provides a characterization of those operators T satisfying ??(T) = ?(T) + B(0,?) for all ? > 0. Here B(0,?) = {z ? C : |z| < ?}. In particular, such operators on finite dimensional spaces must be normal.

scholarly journals Cosine of angle and center of mass of an operator

2012 ◽  
Vol 62 (1) ◽  
Author(s):  
Kallol Paul ◽  
Gopal Das
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Center Of Mass ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Alternative Method

AbstractWe consider the notion of real center of mass and total center of mass of a bounded linear operator relative to another bounded linear operator and explore their relation with cosine and total cosine of a bounded linear operator acting on a complex Hilbert space. We give another proof of the Min-max equality and then generalize it using the notion of orthogonality of bounded linear operators. We also illustrate with examples an alternative method of calculating the antieigenvalues and total antieigenvalues for finite dimensional operators.

scholarly journals Bounded point evaluations for cyclic Hilbert space operators

2003 ◽  
Vol 4 (2) ◽  
pp. 301
Author(s):  
A. Bourhim
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Point Evaluations ◽  
Analytic Bounded Point Evaluations

<p>In this talk, to be given at a conference at Seconda Università degli Studi di Napoli in September 2001, we shall describe the set of analytic bounded point evaluations for an arbitrary cyclic bounded linear operator T on a Hilbert space H and shall answer some questions due to L. R. Williams.</p>

scholarly journals Exponential Function of a bounded Linear Operator on a Hilbert Space.

2014 ◽  
Vol 11 (3) ◽  
pp. 1267-1273
Author(s):  
Baghdad Science Journal
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Exponential Function ◽  
Bounded Linear Operator ◽  
Bounded Linear ◽  
Exponential Operator

In this paper, we introduce an exponential of an operator defined on a Hilbert space H, and we study its properties and find some of properties of T inherited to exponential operator, so we study the spectrum of exponential operator e^T according to the operator T.

The Set of Finite Operators is Nowhere Dense

1989 ◽  
Vol 32 (3) ◽  
pp. 320-326 ◽  
Author(s):  
Domingo A. Herrero
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Dense Subset ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Dimensional Hilbert Space

AbstractA bounded linear operator A on a complex, separable, infinite dimensional Hilbert space is called finite if for each . It is shown that the class of all finite operators is a closed nowhere dense subset of

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 .

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