Reducing subspaces of contractions with no isometric part

Abstract:

Let $T$ be a contraction on a Hilbert space $H$ and suppose that there is no nonzero vector $f$ in $H$ such that $||{T^n}f|| = ||f||$ for every $n = 1,2, \cdots$. In this paper, the reducing subspaces of $T$ are characterized in terms of the range of $1 - {T^ \ast }T$. As a corollary, it is shown that $T$ is irreducible if $1 - {T^ \ast }T$ has $1$-dimensional range. In particular, if $U$ is the simple unilateral shift, then the restriction of ${U^ \ast }$ to any invariant subspace for ${U^ \ast }$ is irreducible.