The unicellularity of contractions of class $C\sb 0$

In this paper, we shall generalize the unicellularity of operators on finite-dimensional spaces to that of the contraction of class ${C_0}$ on Hilbert spaces.

We prove:

(1) Each nilpotent operator on Hilbert space is Banach reducible (Theorem 3).

(2) A contraction $T$ of class ${C_0}$ on Hilbert space is unicellular if and only if $T$ has one-point spectrum and every invariant subspace for $T$ is cyclic (Theorem 6).

(3) A contraction $T$ of class ${C_0}$ on Hilbert space is unicellular if and only if $T$ has one-point spectrum and all invariant subspaces of $T$ are hyperinvariant subspaces of $T$ (Theorem 8).