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