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Inequalities for the Schatten p-norm II

Published online by Cambridge University Press:  18 May 2009

Fuad Kittaneh
Fuad Kittaneh
Affiliation:
Department of Mathematics, United Arab Emirates University, P.O. Box 15551, Al-Ain, U.A.E.
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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.

Type
Research Article
Information
Glasgow Mathematical Journal , Volume 29 , Issue 1 , January 1987 , pp. 99 - 104
Copyright
Copyright © Glasgow Mathematical Journal Trust 1987

References

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