Transactions of the American Mathematical Society

Author(s): Kit C. Chan; Zeljko Cuckovic
Journal: Trans. Amer. Math. Soc. 351 (1999), 1445-1460.
MSC (1991): Primary 47B20, 32A37, 46L05; Secondary 46E20, 47A13, 47B37
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Abstract: Using the functional calculus for a normal operator, we provide a result for generalized Toeplitz operators, analogous to the theorem of Axler and Shields on harmonic extensions of the disc algebra. Besides that result, we prove that if

is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under

cannot be orthogonal to each other. Then we show that the

-algebra generated by

and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that

-algebra in terms of generalized Toeplitz operators. Motivated by these results, we further obtain their several-variable analogues, which generalize and unify Coburn's theorems for the Hardy space and the Bergman space of the unit ball.

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Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 47B20, 32A37, 46L05, 46E20, 47A13, 47B37

Kit C. Chan
Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221
Email: kchan@bgnet.bgsu.edu

Zeljko Cuckovic
Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606-3390
Email: zcuckovi@math.utoledo.edu

DOI: 10.1090/S0002-9947-99-02389-2
PII: S 0002-9947(99)02389-2
Keywords: $C^*$-algebras, subnormal operators, weighted shifts, functional calculus, analytic functions, joint spectrum, Toeplitz operators
Received by editor(s): December 4, 1995
Received by editor(s) in revised form: March 4, 1998
Copyright of article: Copyright 1999, American Mathematical Society

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