Multidimensional Toeplitz and Truncated Toeplitz Operators

In this thesis we extend previous studies of Toeplitz and truncated Toeplitz operators by studying both Toeplitz and truncated Toeplitz operators with matrix symbols. We address the question of whether there is a smallest (matricial) Toeplitz kernel containing a given element or subspace of the Hardy space. This will in turn show how Toeplitz kernels can often be completely described by a fixed number of vectors, called maximal functions. We also discover an interesting and fundamental link between this topic and cyclic vectors for the backward shift. We 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 powerful observation allows us to develop an all-encompassing approach to the study of the kernels of the Toeplitz operator, the truncated Toeplitz operator, the matrix-valued truncated Toeplitz operator and the dual truncated Toeplitz operator. We study matrix-valued truncated Toeplitz operators with symbols having each entry in Lp for some p ∈ (2, ∞]. We develop an approach which bypasses the technical difficulties which arise when dealing with problems concerning matrix-valued truncated Toeplitz operators with unbounded symbols. Using this new approach we express the kernel of the matrix-valued truncated Toeplitz operator as an isometric image of an S∗-invariant subspace. Also, we construct a Toeplitz operator which is equivalent after extension to the matrix-valued truncated Toeplitz operator. We characterise the dual, and in some cases the predual, of the backward shift invariant subspaces of the Hardy space H1. We then use our duality results to show that under certain conditions on the inner function I, every bounded truncated Toeplitz operator on the model space corresponding to I has a bounded symbol if and only if every compact truncated Toeplitz operator on the model space has a symbol which is of the form I multiplied by a continuous function.