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

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 Lp for any p∈(2,∞], there is a wide class of matrix-valued truncated Toeplitz operators which possess a matrix symbol in Lp for some p∈(2,∞]. In the case when the matrix-valued truncated Toeplitz operator has a symbol in Lp for some p∈(2,∞], 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.