Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and equivalence after extension

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 $S^{\ast }$-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.

arXiv commentAdded an extra section with applications of matrix-valued truncated Toeplitz operators to integral equations. To be published in Integral Equations and Operator Theory

References in corpus (1)

• A Note on Matrix Valued Truncated Toeplitz Operators