scholarly journals Lattice Trace Operators

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
Brian Jefferies
Keyword(s):  
Hilbert Space ◽  
Banach Function Space ◽  
Integral Kernel ◽  
Finite Measure ◽  
Singular Values ◽  
Trace Class ◽  
Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Trace Class Operators

A bounded linear operator T on a Hilbert space ℋ is trace class if its singular values are summable. The trace class operators on ℋ form an operator ideal and in the case that ℋ is finite-dimensional, the trace tr(T) of T is given by ∑jajj for any matrix representation {aij} of T. In applications of trace class operators to scattering theory and representation theory, the subject is complicated by the fact that if k is an integral kernel of the operator T on the Hilbert space L2(μ) with μ a σ-finite measure, then k(x,x) may not be defined, because the diagonal {(x,x)} may be a set of (μ⊗μ)-measure zero. The present note describes a class of linear operators acting on a Banach function space X which forms a lattice ideal of operators on X, rather than an operator ideal, but coincides with the collection of hermitian positive trace class operators in the case of X=L2(μ).

scholarly journals Infinite dimensional generalizations of Choi’s Theorem

2019 ◽  
Vol 7 (1) ◽  
pp. 67-77
Author(s):  
Shmuel Friedland
Keyword(s):  
Hilbert Spaces ◽  
Quantum Channel ◽  
Trace Class ◽  
Natural Generalization ◽  
Linear Operators ◽  
Completely Positive Map ◽  
Completely Positive ◽  
Bounded Linear ◽  
Positive Map ◽  
Finite Dimensional

Abstract In this paper we give a simple sequence of necessary and sufficient finite dimensional conditions for a positive map between certain subspaces of bounded linear operators on separable Hilbert spaces to be completely positive. These criterions are natural generalization of Choi’s characterization for completely positive maps between pairs of linear operators on finite dimensional Hilbert spaces. We apply our conditions to a completely positive map between two trace class operators on separable Hilbert spaces. A completely positive map μ is called a quantum channel, if it is trace preserving, and μ is called a quantum subchannel if it decreases the trace of a positive operator.We give simple neccesary and sufficient condtions for μ to be a quantum subchannel.We show that μ is a quantum subchannel if and only if it hasHellwig-Kraus representation. The last result extends the classical results of Kraus and the recent result of Holevo for characterization of a quantum channel.

scholarly journals 2-локальные дифференцирования на алгебрах матрично-значных функций на компакте

2018 ◽  
Author(s):  
S.A. Ayupov ◽  
F.N. Arzikulov
Keyword(s):  
Hilbert Space ◽  
Banach Algebras ◽  
Algebraic Approach ◽  
Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Matrix Algebras ◽  
Dimensional Matrix ◽  
Finite Dimensional ◽  
Local Derivation

The present paper is devoted to 2-local derivations. In 1997, P. Semrl introduced the notion of 2-local derivations and described 2-local derivations on the algebra B(H) of all bounded linear operators on the infinite-dimensional separable Hilbert space H. After this, a number of paper were devoted to 2-local maps on different types of rings, algebras, Banach algebras and Banach spaces. A similar description for the finite-dimensional case appeared later in the paper of S. O. Kim and J. S. Kim. Y. Lin and T. Wong described 2-local derivations on matrix algebras over a finite-dimensional division ring. Sh. A. Ayupov and K. K. Kudaybergenov suggested a new technique and have generalized the above mentioned results for arbitrary Hilbert spaces. Namely they considered 2-local derivations on the algebra B(H) of all linear bounded operators on an arbitrary Hilbert space H and proved that every 2-local derivation on B(H) is a derivation. Then there appeared several papers dealing with 2-local derivations on associative algebras. In the present paper 2-lo\-cal derivations on various algebras of infinite dimensional matrix-valued functions on a compactum are described. We develop an algebraic approach to investigation of derivations and \mbox{2-local} derivations on algebras of infinite dimensional matrix-valued functions on a compactum and prove that every such 2-local derivation is a derivation. As the main result of the paper it is established that every \mbox{2-local} derivation on a ∗-algebra C(Q,Mn(F)) or C(Q,Nn(F)), where Q is a compactum, Mn(F) is the ∗-algebra of infinite dimensional matrices over complex numbers (real numbers or quaternoins) defined in section 1, Nn(F) is the ∗-subalgebra of Mn(F) defined in section 2, is a derivation. Also we explain that the method developed in the paper can be applied to Jordan and Lie algebras of infinite dimensional matrix-valued functions on a compactum.

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 A Counterexample in the theory of derivations

1989 ◽  
Vol 31 (2) ◽  
pp. 161-163
Author(s):  
Feng Wenying ◽  
Ji Guoxing
Keyword(s):  
Hilbert Space ◽  
Trace Class ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Inner Product ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Schmidt Norm ◽  
Schmidt Operator ◽  
Image Position

Let B(H) be the algebra of all bounded linear operators on a separable, infinite dimensional complex Hilbert space H. Let C2 and C1 denote respectively, the Hilbert–Schmidt class and the trace class operators in B(H). It is known that C2 and C1 are two-sided*-ideals in B(H) and C2 is a Hilbert space with respect to the inner product(where tr denotes the trace). For any Hilbert–Schmidt operator X let ∥X∥2=(X, X)½ be the Hilbert-Schmidt norm of X.For fixed A ∈ B(H) let δA be the operator on B(H) defined byOperators of the form (1) are called inner derivations and they (as well as their restrictions have been extensively studied (for example [1–3]). In [1], Fuad Kittaneh proved the following result.

scholarly journals A spectral mapping theorem for the Weyl spectrum

1996 ◽  
Vol 38 (1) ◽  
pp. 61-64 ◽  
Author(s):  
Woo Young Lee ◽  
Sang Hoon Lee
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Spectral Mapping Theorem ◽  
Spectral Mapping ◽  
Finite Multiplicity ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Weyl Spectrum ◽  
Isolated Points ◽  
Image Position

Suppose H is a Hilbert space and write ℒ(H) for the set of all bounded linear operators on H. If T ∈ ℒ(H) we write σ(T) for the spectrum of T; π0(T) for the set of eigenvalues of T; and π00(T) for the isolated points of σ(T) that are eigenvalues of finite multiplicity. If K is a subset of C, we write iso K for the set of isolated points of K. An operator T ∈ ℒ(H) is said to be Fredholm if both T−1(0) and T(H)⊥ are finite dimensional. The index of a Fredholm operator T, denoted by index(T), is defined by

scholarly journals Schur Algebras overC*-Algebras

2007 ◽  
Vol 2007 ◽  
pp. 1-15
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong ◽  
Sheng-Wang Wang
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Trace Class ◽  
Bounded Operators ◽  
Bounded Linear ◽  
Schur Algebras ◽  
Product Operation ◽  
The Matrix ◽  
Trace Class Operators

Let𝒜be aC*-algebra with identity1, and lets(𝒜)denote the set of all states on𝒜. Forp,q,r∈[1,∞), denote by𝒮r(𝒜)the set of all infinite matricesA=[ajk]j,k=1∞over𝒜such that the matrix(ϕ[A[2]])[r]:=[(ϕ(ajk*ajk))r]j,k=1∞defines a bounded linear operator fromℓptoℓqfor allϕ∈s(𝒜). Then𝒮r(𝒜)is a Banach algebra with the Schur product operation and norm‖A‖=sup{‖(ϕ[A[2]])r‖1/(2r):ϕ∈s(𝒜)}. Analogs of Schatten's theorems on dualities among the compact operators, the trace-class operators, and all the bounded operators on a Hilbert space are proved.

scholarly journals Inequalities for the Schatten p-norm II

1987 ◽  
Vol 29 (1) ◽  
pp. 99-104 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Compact Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
The Ideal

This paper is a continuation of [3] in which some inequalities for the Schatten p-norm were considered. The purpose of the present paper is to improve some inequalities in [3] as well as to give more inequalities in the same spirit.Let H be a separable, infinite dimensional complex Hilbert space, and let B(H) denote the algebra of all bounded linear operators acting on H. Let K(H) denote the closed two-sided ideal of compact operators on H. For any compact operator A, let |A| = (A*A)½ and s1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeated according to multiplicity. A compact operator A is said to be in the Schatten p-class Cp(1 ≤ p < ∞), if Σ s1(A)p < ∞. The Schatten p-norm of A is defined by ∥A∥p = (Σ si(A)p)1/p. This norm makes Cp into a Banach space. Hence C1 is the trace class and C2 is the Hilbert-Schmidt class. It is reasonable to let C∞ denote the ideal of compact operators K(H), and ∥.∥∞ stand for the usual operator norm.

scholarly journals Inequalities for the Schatten P-norm

1985 ◽  
Vol 26 (2) ◽  
pp. 141-143 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Compact Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Linear Operators ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
The Ideal

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 K(H) denote the ideal of compact operators on H. For any compact operator A let |A|=(A*A)1,2 and S1(A), s2(A),… be the eigenvalues of |A| in decreasing order and repeatedaccording to multiplicity. If, for some 1<p<∞, si(A)p <∞, we say that A is in the Schatten p-class Cp and ∥A∥p=1/p is the p-norm of A. Hence, C1 is the trace class, C2 is the Hilbert–Schmidt class, and C∞ is the ideal of compact operators K(H).

scholarly journals A characterization of real and complex Hilbert spaces among all normed spaces

1983 ◽  
Vol 27 (3) ◽  
pp. 339-345
Author(s):  
J. Vukman
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Linear Operators ◽  
Identity Operator ◽  
Normed Spaces ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Dimensional Range

Let X be a real or complex normed space and L(X) the algebra of all bounded linear operators on X. Suppose there exists a *-algebra B(X) ⊂ L(X) which contains the identity operator I and all bounded linear operators with finite-dimensional range. The main result is: if each operator U ∈ B(X) with the property U*U = UU* = I has norm one then X is a Hilbert space.

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.

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