(The symbolic functional notation will be used in all that follows, regardless of whether or not the elements of H2(w) are actually functions.) The linear operator U may be extended to a closed, densely defined linear transformation U, on H2(w). It is a well known fact (see Halmos [3, p. 491) that lJ, is bounded iff +%+1/%1&0 is a bounded sequence. In this case, U,,, is unitarily equivalent to a unilateral weighted shift, and all such weighted shifts arise in this fashion. The condition that {w,+r/wS, 2 0 be bounded will be tacitly assumed in all that follows. Let Z be any Hilbert space, and let T be a bounded linear operator on 2. In Section 2, we study the operator equation ALI;, = TA. Given a solution A of the algebraic equation AU = TA, let A(1) = x. It is shown that A extends to a bounded solution of the equation AU, = TA iff ~ . Moreover, A is a Hilbert-Schmidt operator iff (1 zT)-l x is strongly square-summable relative to w-l.