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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