DIFFERENTIAL SCHATTEN -ALGEBRAS. APPROXIMATION PROPERTY AND APPROXIMATE IDENTITIES

For symmetric operators S, we consider differential Schatten algebras Cp,qS of compact operators A from C p with SA − AS belonging to Cq. These algebras are analogues of the Sobolev W 1p,q spaces. We study their approximation property: whether every operator is approximated by finite rank operators, and the existence of approximate identities. For non-selfadjoint S, we show that Cp,qS have no bounded approximate identities and the product of any two operators is approximated by finite rank operators. For selfadjoint S, Cp,qS have approximate identities consisting of finite rank operators and hence, have the approximation property. These identities are bounded only if p = ∞. The existence of a bounded identity for C∞,1S is equivalent to 1-semidiagonality of S.

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