Constrained approximation and invariant subspaces

Abstract

We begin by formulating and solving a general constrained approximation problem. Two special cases are of particular interest. The first includes a problem studied in [9], which has several applications in the theories of systems identification, signal processing and inverse problems (see [5] for a survey of this area). The second special case is an application to the construction of (backward) minimal vectors. This has recently been introduced by Ansari and Enflo [2] as a more explicit technique for constructing hyperinvariant subspaces of bounded linear operators on Hilbert spaces. Their technique is particularly interesting since it provides a new unified method for showing that every compact operator and every normal operator has a hyperinvariant subspace. By investigating the further possibilities of their method, we extract a somewhat more general theorem, which we use to find hyperinvariant subspaces for bounded linear operators which are neither quasinilpotent nor polynomially compact. We illustrate the main result by applying it to weighted shift operators, which are not covered by the existing theorems. Another situation in which minimal vectors have been considered is in the case of operators T of multiplication by outer functions on the Hardy space H 2, as in the work of Spalsbury [14]. Here we use the theory of Toeplitz operators and the Fejer–Riesz theorem to provide explicit expressions for the backward minimal vectors yn for rational outer functions, and to provide a detailed analysis of the convergence of the sequence (T yn), which is the key to the techniques introduced in [2].