## $C^\ast$-algebras generated by a subnormal operator

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- by Kit C. Chan and Z̆eljko C̆uc̆ković
- Trans. Amer. Math. Soc.
**351**(1999), 1445-1460 - DOI: https://doi.org/10.1090/S0002-9947-99-02389-2
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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 $T$ is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under $T$ cannot be orthogonal to each other. Then we show that the $C^*$-algebra generated by $T$ and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that $C^*$-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.## References

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## Bibliographic Information

**Kit C. Chan**- Affiliation: Department of Mathematics and Statistics, Bowling Green State University, Bowling Green, Ohio 43403-0221
- Email: kchan@bgnet.bgsu.edu
**Z̆eljko C̆uc̆ković**- Affiliation: Department of Mathematics, University of Toledo, Toledo, Ohio 43606-3390
- MR Author ID: 294593
- Email: zcuckovi@math.utoledo.edu
- Received by editor(s): December 4, 1995
- Received by editor(s) in revised form: March 4, 1998
- © Copyright 1999 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**351**(1999), 1445-1460 - MSC (1991): Primary 47B20, 32A37, 46L05; Secondary 46E20, 47A13, 47B37
- DOI: https://doi.org/10.1090/S0002-9947-99-02389-2
- MathSciNet review: 1624093