**Completions of partial operator matrices**

We consider complex symmetric completions of a partial operator matrix whose specified part is an operator from a Hilbert space H 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 completion problems for various classes of partial operator matrices, in particular partial rectangular Toeplitz matrices and block Toeplitz operators.

The talk is based on joint work with Kamila Kliś-Garlicka and Marek Ptak.