Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and equivalence after extension · arXivDesk
2012.00654Dec 1, 2020Added an extra section with applications of matrix-valued truncated Toeplitz operators to integral equations. To be published in Integral Equations and Operator Theory
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
Nearby in the stack
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.