P. R. Halmos and L. J. Wallen
characterized partial isometries all of whose positive integral powers are partial
isometries on Hilbert space as unique direct sums of unitary operators, pure
isometries, pure co-isometries and truncated shifts, with each type of summand
occurring at most once. In the present paper, this structure is extended to partial
isometries T with no isometric part whose first N + 1 powers are partial isometries.
Moreover, if T = Σ ⊕ Tj⊕ V , it is shown that every projection P commuting
with T is of the form P = Σ ⊕ Pj⊕ Q where Pj and Q are projections
commuting with Tj and V respectively. The canonical model of L. de Branges
and J. Rovnyak is used to explicitly describe the structure of the reducing
subspaces of T in terms of the characteristic operator-function of T∗, from which
this result follows. A direct proof is also obtained using a general reducing
subspace structure theorem for arbitrary contractions with no isometric
part.
