Abstract
This paper studies matrix-valued truncated Toeplitz operators, which are a vectorial generalisation of truncated Toeplitz operators. It is demonstrated that, although there exist matrix-valued truncated Toeplitz operators without a matrix symbol in L^p for any p in (2, infty ], there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in L^p for some p in (2, infty ]. In the case when the matrix-valued truncated Toeplitz operator has a symbol in L^p for some p in (2, infty ], an approach is developed which bypasses some of the technical difficulties which arise when dealing with problems concerning matrix-valued truncated Toeplitz operators with unbounded symbols. Using this new approach, two new notable results are obtained. The kernel of the matrix-valued truncated Toeplitz operator is expressed as an isometric image of an S^*-invariant subspace. Also, a Toeplitz operator is constructed which is equivalent after extension to the matrix-valued truncated Toeplitz operator. In a different yet overlapping vein, it is also shown that multidimensional analogues of the truncated Wiener–Hopf operators are unitarily equivalent to certain matrix-valued truncated Toeplitz operators.
Highlights
The purpose of this paper is to study the matrix-valued truncated Toeplitz operator
In particular for a matrix-valued truncated Toeplitz operator (MTTO) which has a matrix symbol with each entry lying in Lp for p ∈
We first find a Toeplitz operator which is equivalent after extension to the modified MTTO, and we change the codomain of this Toeplitz operator to produce an operator which is equivalent after extension to a MTTO which has a symbol in L(p,n×n) for p ∈ (2, ∞)
Summary
The purpose of this paper is to study the matrix-valued truncated Toeplitz operator (abbreviated to MTTO). The MTTO is a vectorial generalisation of the truncated Toeplitz operator. We make a powerful observation, that when studying a given property of a MTTO it is often convenient to initially modify the MTTO by changing its codomain (in a natural way), one can deduce results about the MTTO from the modified MTTO.
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