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Journal of Operator Theory

Volume 45, Issue 2, Spring 2001  pp. 303-334.

Differential Schatten $*$-algebras. Approximation property and approximate identities

Authors Edward Kissin (1), and Victor S. Shulman (2)
Author institution: (1) School of Mathematical Sciences, University of North London, Holloway, London N7 8DB, G.B.
(2) Department of Mathematics, Vologda State Technical Uni versity, Vologda, Russia

Summary:  For symmetric operators $ S $, we consider differential Schatten algebras $ C_{S }^{p,q} $ of compact operators $A$ from $ C^{p} $ with $ SA-AS$ belonging to $ C^{q} $. Thes e algebras are analogues of the Sobolev $ W_{p,q}^{1} $ spaces. We study their approximation pr operty: whether every operator is approximated by finite rank operators, and the existence of approximate identities. For non-selfadjoint $ S $, we show that $ C _{S}^{p,q} $ have no bounded approximate identities and the product of any two operators is approximated by finite rank operators. For selfadjoint $ S$, $C_{S}^{p,q} $ have approximate identities consisting of finite rank operators and hence, have the approximation property. These identities are bounded only if $ p = \infty $. The existence of a bounded identity for $ C_{S}^{\infty,1} $ is equivalent to $ 1 $- semidiagonality of $ S $.

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