Abstract. For a composition operator $C_\phi $ on the Hardy space $H^2(\mathbb{D})$ with $\phi(0)=0$, the subspaces $z^{k}H^{2}$ are invariant. In this paper, we demonstrate that certain linear fractional maps induce composition operators for which the restrictions to different subspaces $z^{k}H^{2}$ are not unitarily equivalent, even though they have the same norm and spectrum. On the other hand, we show that these restrictions are unitarily equivalent to compact perturbations of each other.

DOI: 10.14232/actasm-013-048-y

AMS Subject Classification (1991): 47B33; 47D06

Keyword(s): operator theory, composition operator, semigroup

Received August 4, 2013, and in final form November 3, 2013. (Registered under 48/2013.)