scholarly journals Norm estimations, continuity, and compactness for Khatri-Rao products of Hilbert Space operators

2018 ◽  
Vol 14 (4) ◽  
pp. 382-386
Author(s):  
Arnon Ploymukda ◽  
Pattrawut Chansangiam
Keyword(s):  
Hilbert Space ◽  
Trace Class ◽  
Compact Operators ◽  
Operator Norm ◽  
Trace Norm ◽  
Block Matrices ◽  
Hilbert Space Operators ◽  
Schmidt Norm

We provide estimations for the operator norm, the trace norm, and the Hilbert-Schmidt norm for Khatri-Rao products of Hilbert space operators. It follows that the Khatri-Rao product is continuous on norm ideals of compact operators equipped with the topologies induced by such norms. Moreover, if two operators are represented by block matrices in which each block is nonzero, then their Khatri-Rao product is compact if and only if both operators are compact. The Khatri-Rao product of two operators are trace-class (Hilbert-Schmidt class) if and only if each factor is trace-class (Hilbert-Schmidt class, respectively).

scholarly journals Schatten's theorems on functionally defined Schur algebras

2005 ◽  
Vol 2005 (14) ◽  
pp. 2175-2193 ◽  
Author(s):  
Pachara Chaisuriya ◽  
Sing-Cheong Ong
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Bounded Operator ◽  
Trace Class ◽  
Compact Operators ◽  
Bounded Operators ◽  
Schur Algebras ◽  
Schur Product ◽  
Product Operation ◽  
Trace Class Operators

For each triple of positive numbersp,q,r≥1and each commutativeC*-algebraℬwith identity1and the sets(ℬ)of states onℬ, the set𝒮r(ℬ)of all matricesA=[ajk]overℬsuch thatϕ[A[r]]:=[ϕ(|ajk|r)]defines a bounded operator fromℓptoℓqfor allϕ∈s(ℬ)is shown to be a Banach algebra under the Schur product operation, and the norm‖A‖=‖|A|‖p,q,r=sup{‖ϕ[A[r]]‖1/r:ϕ∈s(ℬ)}. Schatten's theorems about the dual of the compact operators, the trace-class operators, and the decomposition of the dual of the algebra of all bounded operators on a Hilbert space are extended to the𝒮r(ℬ)setting.

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 Exact and Approximate Operator Parallelism

2015 ◽  
Vol 58 (1) ◽  
pp. 207-224 ◽  
Author(s):  
Ali Zamani ◽  
Mohammad Sal Moslehian
Keyword(s):  
Hilbert Space ◽  
Linear Operators ◽  
Operator Norm ◽  
Normed Spaces ◽  
Bounded Linear Operators ◽  
Bounded Linear ◽  
C Algebra ◽  
Hilbert Space Operators ◽  
The Relationship

AbstractExtending the notion of parallelism we introduce the concept of approximate parallelism in normed spaces and then substantially restrict ourselves to the setting of Hilbert space operators endowed with the operator norm. We present several characterizations of the exact and approximate operator parallelism in the algebra B(ℋ) of bounded linear operators acting on a Hilbert space H . Among other things, we investigate the relationship between the approximate parallelism and norm of inner derivations on B(ℋ). We also characterize the parallel elements of a C*-algebra by using states. Finally we utilize the linking algebra to give some equivalent assertions regarding parallel elements in a Hilbert C*-module.

Subadditivity Inequalities for Compact Operators

2014 ◽  
Vol 57 (1) ◽  
pp. 25-36 ◽  
Author(s):  
Jean-Christophe Bourin ◽  
Tetsuo Harada ◽  
Eun-Young Lee
Keyword(s):  
Hilbert Space ◽  
Compact Operators ◽  
Concave Functions ◽  
Residual Term ◽  
Hilbert Space Operators ◽  
Open Questions ◽  
The Matrix

AbstractSome subadditivity inequalities for matrices and concave functions also hold for Hilbert space operators, but (unfortunately!) with an additional ε term. It does not seem possible to erase this residual term. However, in case of compact operators we show that the ε term is unnecessary. Further, these inequalities are strict in a certain sense when some natural assumptions are satisfied. The discussion also emphasizes matrices and their compressions and several open questions or conjectures are considered, both in the matrix and operator settings.

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).

Simultaneous Triangularization of Algebras of Polynomially Compact Operators

1991 ◽  
Vol 34 (2) ◽  
pp. 260-264 ◽  
Author(s):  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Jacobson Radical ◽  
Compact Operators ◽  
General Algebra ◽  
Closed Algebra ◽  
Quasinilpotent Operators ◽  
Simultaneous Triangularization

AbstractIf A is a norm closed algebra of compact operators on a Hilbert space and if its Jacobson radical J(A) consists of all quasinilpotent operators in A then A/ J(A) is commutative. The result is not valid for a general algebra of polynomially compact operators.

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