Truncated Hankel operators with matrix-valued symbols: Basic properties

Abstract In this paper, we study truncated Hankel operators with matrix-valued symbols which are compressions of Hankel operators on the model space corresponding to a square inner matrix. An operator equation characterization of such operators is given and some basic questions such as which symbols give rise to a zero truncated Hankel operator is answered. Additionally, we show that truncated Hankel operators with matrix-valued symbols exhibit a specific type of shift symmetry. We also note that by considering truncated Hankel operators with matrix-valued symbols, we can give a unified treatment of asymmetric truncated Hankel operators.