scholarly journals A Hilbert algebra of Hilbert-Schmidt quadratic operators

1990 ◽  
Vol 41 (1) ◽  
pp. 123-134 ◽  
Author(s):  
J.C. Amson ◽  
N. Gopal Reddy
Keyword(s):  
Separable Hilbert Space ◽  
Trace Class ◽  
Linear Operators ◽  
Inner Product ◽  
Hilbert Algebra ◽  
Block Diagonal Matrix ◽  
Quadratic Operator ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Real Separable Hilbert Space

A quadratic operator Q of Hilbert-Schmidt class on a real separable Hilbert space H is shown to be uniquely representable as a sequence of self-adjoint linear operators of Hilbert-Schmidt class on H, such that Q(x) = Σk〈Lkx, x〉uk with respect to a Hilbert basis . It is shown that with the norm | ‖Q‖ | = (Σk ‖Lk‖2)½ and inner-product 〈〈〈Q, P〉〉〉 = Σk 〈〈Lk, Mk〉〉, together with a multiplication defined componentwise through the composition of the linear components, the vector space of all Hilbert-Schmidt quadratic operators Q on H becomes a linear H*-algebra containing an ideal of nuclear (trace class) quadratic operators. In the finite dimensional case, each Q is also shown to have another representation as a block-diagonal matrix of Hilbert-Schmidt class which simplifies the practical computation and manipulation of quadratic operators.

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.

Representation of certain Linear Operators in Hilbert Space

1971 ◽  
Vol 23 (1) ◽  
pp. 132-150 ◽  
Author(s):  
Bernard Niel Harvey
Keyword(s):  
Hilbert Space ◽  
Linear Operator ◽  
Bounded Linear Operator ◽  
Separable Hilbert Space ◽  
Finite Dimension ◽  
Linear Operators ◽  
Inner Product ◽  
Indefinite Metric ◽  
Bounded Linear ◽  
Bilinear Functional

In this paper we represent certain linear operators in a space with indefinite metric. Such a space may be a pair (H, B), where H is a separable Hilbert space, B is a bilinear functional on H given by B(x, y) = [Jx, y], [, ] is the Hilbert inner product in H, and J is a bounded linear operator such that J = J* and J2 = I. If T is a linear operator in H, then ‖T‖ is the usual operator norm. The operator J above has two eigenspaces corresponding to the eigenvalues + 1 and –1.In case the eigenspace in which J induces a positive operator has finite dimension k, a general spectral theory is known and has been developed principally by Pontrjagin [25], Iohvidov and Kreĭn [13], Naĭmark [20], and others.

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 Stochastic Integrals Based on Martingales Taking Values in Hilbert Space

1970 ◽  
Vol 38 ◽  
pp. 41-52 ◽  
Author(s):  
Hiroshi Kunita
Keyword(s):  
Hilbert Space ◽  
Probability Space ◽  
Separable Hilbert Space ◽  
Linear Operators ◽  
Inner Product ◽  
Stochastic Integrals

Let H be a separable Hilbert space with inner product (,) and norm ║ ║. We denote by K the set of all linear operators on H. Let be a probability space and suppose we are given a family of σ-fields t≥O such that for O ≤ s ≤ t and .

scholarly journals H-joint numerical ranges

2002 ◽  
Vol 66 (1) ◽  
pp. 105-117
Author(s):  
Chi-Kwong Li ◽  
Leiba Rodman
Keyword(s):  
Numerical Range ◽  
Linear Operators ◽  
Dimensional Case ◽  
Geometrical Properties ◽  
Finite Dimensional ◽  
Sesquilinear Form ◽  
Algebraic Properties ◽  
Finite Dimensional Case ◽  
Joint Numerical Range

The notion of the joint numerical range of several linear operators with respect to a sesquilinear form is introduced. Geometrical properties of the joint numerical range are studied, in particular, convexity and angle points, in connection with the algebraic properties of the operators. The main focus is on the finite dimensional case.

scholarly journals Characterizations of Operator Birkhoff–James Orthogonality

2017 ◽  
Vol 60 (4) ◽  
pp. 816-829 ◽  
Author(s):  
Mohammad Sal Moslehian ◽  
Ali Zamani
Keyword(s):  
Unit Vector ◽  
Dimensional Subspace ◽  
Linear Operators ◽  
Inner Product ◽  
Finite Dimensional Subspace ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
James Orthogonality ◽  
New Type ◽  
Norm Attaining

AbstractIn this paper, we obtain some characterizations of the (strong) Birkhoff–James orthogonality for elements of Hilbert C*-modules and certain elements of . Moreover, we obtain a kind of Pythagorean relation for bounded linear operators. In addition, for we prove that if the norm attaining set is a unit sphere of some finite dimensional subspace of and , then for every , T is the strong Birkhoff–James orthogonal to S if and only if there exists a unit vector such that . Finally, we introduce a new type of approximate orthogonality and investigate this notion in the setting of inner product C*-modules.

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.

Order of singularity applied to linear operators

1987 ◽  
Vol 106 (1-2) ◽  
pp. 103-111
Author(s):  
James E. Scroggs
Keyword(s):  
Linear Operator ◽  
Upper Bound ◽  
Jordan Block ◽  
Maximum Modulus ◽  
Linear Operators ◽  
Rate Of Growth ◽  
Finite Dimensional ◽  
Finite Dimensional Case ◽  
Norm Of The Resolvent ◽  
Complex Valued

SynopsisWe extend the notion of order of singularity for complex-valued functions, as originally set forth by Hadamard, to Banach algebra-valued functions. Restricting our attention to linear operators whose spectral radius is one, we obtain a connection between the rate of growth of the norm of iterates of a linear operator and the rate of growth of the norm of the resolvent of the operator near the spectrum of the operator. In the finite dimensional case, we obtain an upper bound on the size of the Jordan block corresponding to an eigenvalue of maximum modulus.

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 Real parts of quasi-nilpotent operators

1979 ◽  
Vol 22 (3) ◽  
pp. 263-269 ◽  
Author(s):  
P. A. Fillmore ◽  
C. K. Fong ◽  
A. R. Sourour
Keyword(s):  
Separable Hilbert Space ◽  
Adjoint Operator ◽  
Dimensional Case ◽  
Nilpotent Operator ◽  
The Real ◽  
Infinite Dimensional ◽  
Finite Dimensional ◽  
Nilpotent Operators ◽  
Finite Dimensional Case ◽  
Adjoint Operators

The purpose of this paper is to answer the question: which self-adjoint operators on a separable Hilbert space are the real parts of quasi-nilpotent operators? In the finite-dimensional case the answer is: self-adjoint operators with trace zero. In the infinite dimensional case, we show that a self-adjoint operator is the real part of a quasi-nilpotent operator if and only if the convex hull of its essential spectrum contains zero. We begin by considering the finite dimensional case.

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