Closed Commutants of the Backward Shift Operator

We characterize the closed operators with domain contained in the Hardy space H 2 that commute with the backward shift. Also, we give necessary and suucient conditions for such an operator to be a Toeplitz operator with symbol the complex conjugate of a function in H 2. In particular, we show that this fact depends only on the domain. Introduction. Let F be a function in the Hardy space of the unit disk H 2. We can deene the unbounded Toeplitz operator T F operating on a suitable linear subman-ifold of H 2 (for instance H 1). In recent years many questions have been raised about the behavior of these operators. Most of these problems appear naturally when studying the algebra of multipliers or the backward shift invariant subspaces of the so-called de Branges-Rovnyak spaces (see 11] and 17]). If S denotes the backward shift operator on H 2 , it is not diicult to see that T F commutes with S. More generally, if Q is a closed operator on some linear submanifold of H 2 that commutes with S , then the domain of Q, D(Q) is dense in some (closed) S invariant subspace of H 2. Therefore Beurling's theorem assures that D(Q) is dense in H 2 or in (uH 2) ? = H(u), for some inner function u. If Q is a bounded operator on H 2 that commutes with S , it is easy to see that Q = T ' with ' 2 H 1. The analogous result for bounded operators on H(u) that commute with S u = S = H(u) is a well known theorem of Sarason 14]. Moreover, we can choose ' 2 H 1 so that Q = T ' = H(u) and k'k 1 = kQk. There are two natural questions appearing at this point. What are the closed operators that commute with S (or with S u)? Do the above results for bounded operators have analogous versions for closed operators? The purpose of this paper is to answer these questions in both cases, when D(Q) is dense in H 2 and in H(u), for some inner function u. In particular, we nd necessary and suucient conditions for such an operator Q to have the form