Abstract
In this paper we first study the structure of the scalar and vector-valued nearly invariant subspaces with a finite defect. We then subsequently produce some fruitful applications of our new results. We produce a decomposition theorem for the vector-valued nearly invariant subspaces with a finite defect. More specifically, we show every vector-valued nearly invariant subspace with a finite defect can be written as the isometric image of a backwards shift invariant subspace. We also show that there is a link between the vector-valued nearly invariant subspaces and the scalar-valued nearly invariant subspaces with a finite defect. This is a powerful result which allows us to gain insight in to the structure of scalar subspaces of the Hardy space using vector-valued Hardy space techniques. These results have far reaching applications, in particular they allow us to develop an all encompassing approach to the study of the kernels of: the Toeplitz operator, the truncated Toeplitz operator, the truncated Toeplitz operator on the multiband space and the dual truncated Toeplitz operator.
Highlights
The purpose of this paper is to study vector and scalar-valued nearly S∗-invariant subspaces of the Hardy space defined on the unit disc
We first produce some results on the structure of nearly S∗-invariant subspaces with a finite defect, in particular we Communicated by Isabelle Chalendar
In many cases the study of Toeplitz operators becomes greatly simplified when the operator has an invertible symbol; in Sect. 2 we show that the symbol of a truncated Toeplitz operator may be chosen to be invertible in L∞
Summary
The purpose of this paper is to study vector and scalar-valued nearly S∗-invariant subspaces of the Hardy space defined on the unit disc. We first produce some results on the structure of nearly S∗-invariant subspaces with a finite defect, in particular we Communicated by Isabelle Chalendar
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