Vol. 104, No. 2, 1983

Recent Issues
Vol. 328: 1
Vol. 327: 1  2
Vol. 326: 1  2
Vol. 325: 1  2
Vol. 324: 1  2
Vol. 323: 1  2
Vol. 322: 1  2
Vol. 321: 1  2
Online Archive
The Journal
Editorial Board
Submission Guidelines
Submission Form
Policies for Authors
ISSN: 1945-5844 (e-only)
ISSN: 0030-8730 (print)
Special Issues
Author Index
To Appear
Other MSP Journals
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