𝐶*-algebras generated by a subnormal operator

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 T is an injective subnormal weighted shift, then any two nontrivial subspaces invariant under T T cannot be orthogonal to each other. Then we show that the C ∗ C^* -algebra generated by T T and the identity operator contains all the compact operators as its commutator ideal, and we give a characterization of that C ∗ 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.