C*-algebras generated by a subnormal operator

Using the functional calculus for a normal operator, we provide a result fo r generalized Toeplitz operators, analogous to the theorem of Axler and Shi elds on harmonic extensions of the disc algebra. Besides that result, we pr ove that if T is an injective subnormal weighted shift, then any two nontri vial subspaces invariant under T cannot be orthogonal to each other. Then w e show that the C*-algebra generated by T and the identity operator contain s all the compact operators as its commutator ideal, and we give a characte rization of that C*-algebra in terms of generalized Toeplitz operators. Mot ivated 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.