Hostname: page-component-586b7cd67f-vdxz6 Total loading time: 0 Render date: 2024-11-23T22:30:32.012Z Has data issue: false hasContentIssue false
  • English
  • Français

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.
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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

REFERENCES

1.Halmos, P. R., A Hilbert space problem book (Springer-Verlag, 1982).CrossRefGoogle Scholar
2.van Hemmen, J. L. and Ando, T., An inequality for trace ideals, Comm. Math. Phys. 76 (1980), 143–148.CrossRefGoogle Scholar
3.Kittaneh, F., Inequalities for the Schatten p-norm, Glasgow Math. J. 26 (1985), 141–143.CrossRefGoogle Scholar
4.Ringrose, J. R., Compact non-self-adjoint operators (Van Nostrand Reinhold Co., 1971).Google Scholar
5.Simon, B., Trace ideals and their applications (Cambridge University Press, 1979).Google Scholar
6.Weiss, G., The Fuglede commutativity theorem modulo the Hilbert-Schmidt class and generating functions for matrix operators. I, Trans. Amer. Math. Soc. 246 (1978), 193–209.Google Scholar