A co-isometry on a Hilbert
space ℋ is a bounded operator having an isometric adjoint. If V is a co-isometry on
ℋ and ℳ is an invariant subspace for V , then every bounded operator on ℳ that
commutes with V on ℳ can be extended to an operator on ℋ that commutes with
V , and the extension can be made without increasing the norm of the operator. This
paper is concerned with unilateral shifts. The questions asked are these: (1) Do shifts
enjoy the above property shared by co-isometries and self-adjoint operators? (The
answer to this question is “rarely”.) (2) Why not? (3) If S is a shift, ℳ is
an invariant subspace for S, S0 is the restriction of S to ℳ, and T is a
bounded operator on ℳ satisfying TS0= S0T, how tame do T and ℳ have to
be in order that T can be extended (without increasing the norm) to an
operator in the commutant of S? Extension is possible in a large number of
cases.
