Complex symmetric completions of partial operator matrices

Abstract

ABSTRACT We study complex symmetric completions of a partial operator matrix which specified part is an operator from a Hilbert space into a closed proper subspace. We give necessary and sufficient conditions for such a completion to exist, with respect to a given conjugation C, and we describe all possible completions of that type. We apply these results to various classes of partial operator matrices, in particular, partial rectangular matrices, asymmetric truncated Toeplitz operators and integral operators of Wiener–Hopf type. Moreover, we obtain necessary and sufficient conditions for a (scalar) Toeplitz operator to be C-symmetric and use them to solve completion problems for block Toeplitz operators.