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Nearly invariant subspaces of the backward shift. (English) Zbl 0687.47003
Operator theory and functional analysis, Proc. Conf., Mesa/Ariz. 1987, Oper. Theory, Adv. Appl. 35, 481-493 (1988).
[For the entire collection see Zbl 0668.00012.]
Denote by S the unilateral shift on \(H^ 2\). The author calls a subspace M of \(H^ 2\) a nearly invariant subspace for \(S^*\) if \(S^*(M\cap SH^ 2)\subset M\). Such subspaces are interesting because the kernel of a Toeplitz operator is nearly invariant for \(S^*\) (this follows immediately from the equality \(S^*TS=T\) satisfied by any Toeplitz operator T). A description of the kernels of Toeplitz operators was given by E. Hayashi, and a similar description of arbitrary nearly invariant subspaces for \(S^*\) was given by D. Hitt. In the paper under review, a new proof of Hitt’s result is provided, based on the author’s work on certain spaces of analytic functions introduced by L. de Branges and J. Rovniak. Hitt’s result says that any nearly invariant subspace M has the form \(gM'\), where g is a function of unit norm in \(H^ 2\), and \(M'\) is an invariant subspace for \(S^*\). The author improves upon Hitt’s result by characterizing the pairs \((g,M')\) which give rise in this way to a nearly invariant subspace.
Reviewer: H.Bercovici

47A15 Invariant subspaces of linear operators
47B37 Linear operators on special spaces (weighted shifts, operators on sequence spaces, etc.)