[Paper Review] Matrix-valued truncated Toeplitz operators: unbounded symbols, kernels and equivalence after extension

Ryan O’Loughlin|White Rose Research Online (University of Leeds, The University of Sheffield, University of York)|Dec 1, 2020

Holomorphic and Operator Theory20 references4 citations

TL;DR

This paper introduces a novel approach to matrix-valued truncated Toeplitz operators (MTTOs) with unbounded symbols by modifying the codomain to bypass technical challenges. It establishes that the kernel of such MTTOs is isometrically equivalent to an $S^*$-invariant subspace and constructs a Toeplitz operator equivalent after extension, extending results to multidimensional truncated Wiener-Hopf operators via unitary equivalence.

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.

Motivation & Objective

• To address technical challenges in analyzing matrix-valued truncated Toeplitz operators (MTTOs) with unbounded symbols.
• To develop a new method that bypasses limitations arising from unbounded symbols in MTTO theory.
• To characterize the kernel of MTTOs with $L^p$-symbol entries ($p \in (2, \infty]$) as an isometric image of an $S^*$-invariant subspace.
• To construct a Toeplitz operator equivalent after extension to a given MTTO.
• To establish unitary equivalence between multidimensional truncated Wiener-Hopf operators and specific MTTOs.

Proposed method

• Introduce a codomain modification technique to transform the MTTO into a more tractable form for analysis.
• Use the Beurling-Lax theorem and orthogonal decompositions in Hardy spaces to characterize model spaces $K_\Theta$ as $S^*$-invariant subspaces.
• Define the MTTO as $A_G^\Theta(f) = P_\Theta(Gf)$ for $f \in K_\Theta \cap (H^\infty)^n$, with $G$ as the matrix symbol.
• Establish that when $G \in L^{(p,n\times n)}$ for $p \in (2, \infty]$, the modified operator allows kernel characterization via isometric image of $S^*$-invariant subspaces.
• Construct a new Toeplitz operator via unitary equivalence and extension techniques to match the spectral behavior of the MTTO.
• Apply Fourier and Cayley transforms to relate MTTOs on the unit disk to truncated Wiener-Hopf operators on the real line, proving unitary equivalence in the $n\times n$ case.

Experimental results

Research questions

• RQ1 Can the kernel of a matrix-valued truncated Toeplitz operator with unbounded symbol be characterized in terms of $S^*$-invariant subspaces?
• RQ2 Is it possible to construct a standard Toeplitz operator that is equivalent to an MTTO after extension, even when the symbol is not in $L^\infty$?
• RQ3 What is the relationship between multidimensional truncated Wiener-Hopf operators and matrix-valued truncated Toeplitz operators?
• RQ4 How does modifying the codomain of an MTTO enable analysis of operators with unbounded symbols?
• RQ5 Under what conditions is a truncated Wiener-Hopf operator unitarily equivalent to an MTTO?

Key findings

• The kernel of an MTTO with symbol $G \in L^{(p,n\times n)}$ for $p \in (2, \infty]$ is isometrically isomorphic to an $S^*$-invariant subspace.
• A new Toeplitz operator is constructed that is equivalent to the MTTO after extension, providing a spectral link.
• Multidimensional truncated Wiener-Hopf operators are unitarily equivalent to certain MTTOs via a unitary transformation involving the Fourier and Cayley transforms.
• The approach allows analysis of MTTOs with unbounded symbols by reducing them to equivalent problems in $L^p$-symbol settings.
• The results generalize to the scalar case, yielding new insights even for classical truncated Toeplitz operators.
• The equivalence between truncated Wiener-Hopf operators and MTTOs extends to $n\times n$ matrix cases, with applications in MIMO systems and integral equations.