AN EXTENSION OF PENROSE’S INEQUALITY ON GENERALIZED INVERSES TO THE SCHATTEN p-CLASSES

2014 ◽  
Vol 60 (1) ◽  
pp. 77-84
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Linear Operators ◽  
Differential Analysis ◽  
Unique Minimizer ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Affine Mappings

Abstract Let B(H) be the algebra of all bounded linear operators on a complex separable infinite dimensional Hilbert space H. In this paper we minimize the Schatten Cp-norm of suitable affine mappings from B(H) to Cp, using convex and differential analysis (Gâteaux derivative) as well as input from operator theory. The mappings considered generalize Penrose’s inequality which asserts that if A+ and B+ denote the Moore-Penrose inverses of the matrices A and B, respectively, then ||AXB − C||2 ≥ ||AA+CB+B − C||2, with A+CB+ being the unique minimizer of minimal ||:||2 norm. The main results obtained characterize the best Cp-approximant of the operator AXB.

scholarly journals On the commutants modulo Cp of A2 and A3

1986 ◽  
Vol 41 (1) ◽  
pp. 47-50 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Fredholm Operator ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Subnormal Operators ◽  
Dimensional Hilbert Space

AbstractWe prove the following statements about bounded linear operators on a complex separable infinite dimensional Hilbert space. (1) Let A and B* be subnormal operators. If A2X = XB2 and A3X = XB3 for some operator X, then AX = XB. (2) Let A and B* be subnormal operators. If A2X – XB2 ∈ Cp and A3X – XB3 ∈ Cp for some operator X, then AX − XB ∈ C8p. (3) Let T be an operator such that 1 − T*T ∈ Cp for some p ≥1. If T2X − XT2 ∈ Cp and T3X – XT3 ∈ Cp for some operator X, then TX − XT ∈ Cp. (4) Let T be a semi-Fredholm operator with ind T < 0. If T2X − XT2 ∈ C2 and T3X − XT3 ∈ C2 for some operator X, then TX − XT ∈ C2.

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.

Weyl's theorem for the square of operator and perturbations

2015 ◽  
Vol 17 (05) ◽  
pp. 1450042
Author(s):  
Weijuan Shi ◽  
Xiaohong Cao
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Weyl Spectrum ◽  
The Stability

Let H be an infinite-dimensional separable complex Hilbert space and B(H) the algebra of all bounded linear operators on H. T ∈ B(H) satisfies Weyl's theorem if σ(T)\σw(T) = π00(T), where σ(T) and σw(T) denote the spectrum and the Weyl spectrum of T, respectively, π00(T) = {λ ∈ iso σ(T) : 0 < dim N(T - λI) < ∞}. T ∈ B(H) is said to have the stability of Weyl's theorem if T + K satisfies Weyl's theorem for all compact operator K ∈ B(H). In this paper, we characterize the operator T on H satisfying the stability of Weyl's theorem holds for T2.

scholarly journals States which have a trace-like property relative to a C*-subalgebra of B(H)

1976 ◽  
Vol 17 (2) ◽  
pp. 158-160
Author(s):  
Guyan Robertson
Keyword(s):  
Hilbert Space ◽  
Spectral Radius ◽  
Operator Theory ◽  
Linear Operators ◽  
Operator Norm ◽  
Identity Operator ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Image Position

In what follows, B(H) will denote the C*-algebra of all bounded linear operators on a Hilbert space H. Suppose we are given a C*-subalgebra A of B(H), which we shall suppose contains the identity operator 1. We are concerned with the existence of states f of B(H) which satisfy the following trace-like relation relative to A:Our first result shows the existence of states f satisfying (*), when A is the C*-algebra C*(x) generated by a normaloid operator × and the identity. This allows us to give simple proofs of some well-known results in operator theory. Recall that an operator × is normaloid if its operator norm equals its spectral radius.

scholarly journals On normal derivations of Hilbert–Schmidt type

1987 ◽  
Vol 29 (2) ◽  
pp. 245-248 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Trace Class ◽  
Linear Operators ◽  
Inner Product ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Schmidt Norm ◽  
Image Position

Let H denote a separable, infinite dimensional Hilbert space. Let B(H), C2 and C1 denote the algebra of all bounded linear operators acting on H, the Hilbert–Schmidt class and the trace class in B(H) respectively. It is well known that C2 and C1 each form a two-sided-ideal in B(H) and C2 is itself a Hilbert space with the inner productwhere {ei} is any orthonormal basis of H and tr(.) is the natural trace on C1. The Hilbert–Schmidt norm of X ∈ C2 is given by ⅡXⅡ2=(X, X)½.

scholarly journals Joint numerical ranges: recent advances and applications minicourse by V. Müller and Yu. Tomilov

2020 ◽  
Vol 7 (1) ◽  
pp. 133-154
Author(s):  
V. Müller ◽  
Yu. Tomilov
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Unit Vector ◽  
Linear Operators ◽  
Numerical Radius ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Single Operator ◽  
Orbits Of Operators

AbstractWe present a survey of some recent results concerning joint numerical ranges of n-tuples of Hilbert space operators, accompanied with several new observations and remarks. Thereafter, numerical ranges techniques will be applied to various problems of operator theory. In particular, we discuss problems concerning orbits of operators, diagonals of operators and their tuples, and pinching problems. Lastly, motivated by known results on the numerical radius of a single operator, we examine whether, given bounded linear operators T1, . . ., Tn on a Hilbert space H, there exists a unit vector x ∈ H such that |〈Tjx, x〉| is “large” for all j = 1, . . . , n.

Generalized Derivation and Double Operator Integrals

2005 ◽  
Vol 12 (4) ◽  
pp. 717-726
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space ◽  
Generalized Derivation ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional

Abstract Let 𝐻 be a separable infinite dimensional complex Hilbert space, and let 𝔹(𝐻) denote the algebra of all bounded linear operators on 𝐻. Let 𝐴, 𝐵 be operators in 𝔹(𝐻). We define the generalized derivation δ 𝐴, 𝐵 : 𝔹(𝐻) ↦ 𝔹(𝐻) by δ 𝐴, 𝐵(𝑋) = 𝐴𝑋 – 𝑋𝐵. In this paper we consider the question posed by Turnsek [Publ. Math. Debrecen 63: 293–304, 2003], when ? We prove that this holds in the case where 𝐴 and 𝐵 satisfy the Fuglede–Putnam theorem. Finally, we apply the obtained results to double operator integrals.

On Quasi-Class A Operators

2013 ◽  
Vol 59 (1) ◽  
pp. 163-172
Author(s):  
Salah Mecheri
Keyword(s):  
Hilbert Space ◽  
Analytic Function ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Weyl’S Theorem ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Class A ◽  
Generalized Weyl’S Theorem

Abstract Let H be a separable infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators on H. Let A;B be operators in B(H). In this paper we prove that if A is quasi-class A and B* is invertible quasi-class A and AX = XB, for some X ∈ C2 (the class of Hilbert-Schmidt operators on H), then A*X = XB*. We also prove that if A is a quasi-class A operator and f is an analytic function on a neighborhood of the spectrum of A, then f(A) satisfies generalized Weyl's theorem. Other related results are also given.

scholarly journals About the Cauchy–Bunyakovsky–Schwarz Inequality for Hilbert Space Operators

Symmetry ◽  
2021 ◽  
Vol 13 (2) ◽  
pp. 305
Author(s):  
Nicuşor Minculete
Keyword(s):  
Hilbert Space ◽  
Operator Theory ◽  
Schwarz Inequality ◽  
Linear Operators ◽  
Real Numbers ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Hilbert Space Operators ◽  
Bohr’S Inequality ◽  
Symmetric Shape

The symmetric shape of some inequalities between two sequences of real numbers generates inequalities of the same shape in operator theory. In this paper, we study a new refinement of the Cauchy–Bunyakovsky–Schwarz inequality for Euclidean spaces and several inequalities for two bounded linear operators on a Hilbert space, where we mention Bohr’s inequality and Bergström’s inequality for operators. We present an inequality of the Cauchy–Bunyakovsky–Schwarz type for bounded linear operators, by the technique of the monotony of a sequence. We also prove a refinement of the Aczél inequality for bounded linear operators on a Hilbert space. Finally, we present several applications of some identities for Hermitian operators.

Sign in / Sign up

Export Citation Format

Share Document