Ryan O'Loughlin (Leeds)

Truncated Toeplitz operators and a spectral theorem for complex symmetric operators

There is an ongoing investigation into what role truncated Toeplitz operators can play in the study of complex symmetric operators. Two notable open questions in this area are Is every complex symmetric matrix unitarily equivalent to a direct sum of truncated Toeplitz operators? Do all unitary equivalences between complex symmetric matrices and truncated Toeplitz operators arise from modified Clark basis representations? If the first question is indeed true, then this would provide a decomposition theorem for complex symmetric operators (similar to the spectral theorem for normal operators). In this talk, after a brief background on the subject, I will give a description of the 3-by-3 complex symmetric matrix representations of truncated Toeplitz operators, which will provide numerous examples showing question 2 to be negative. If time permits, I will also show how the problem of determining whether a complex symmetric matrix is a representation of a truncated Toeplitz operator with respect to a conjugation invariant basis can be rephrased into a question purely in real algebraic geometry.