Vol. 104, No. 2, 1983

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Compact operators and derivations induced by weighted shifts

C. Ray Rosentrater

Vol. 104 (1983), No. 2, 465–470

In this paper we study the question: which compact operators are contained in (δS), the norm closure of the range of the derivation δS(X) = SX XS induced by a weighted shift S? We find that (δS) always contains the lower triangular (with respect to the basis (ei) on which S is a shift) compact operators. Further, (δS) contains the n-lower triangular (operators T satisfying (Tei,ej) = 0 for i j > n) compact operators if and only if e1 en+1 ∈ℛ(δS). We also find necessary and sufficient conditions on the weights of S in order that e1 en+1 ∈ℛ(δS) and that 𝒦, the algebra of compact operators, be contained in (δS). These results completely answer the question: which essentially normal weighted shifts are d-symmetric?

Mathematical Subject Classification 2000
Primary: 47B05, 47B05
Secondary: 47B37, 47B47
Received: 9 March 1981
Revised: 9 September 1981
Published: 1 February 1983
C. Ray Rosentrater