Infinite Toeplitz and Hankel matrices with operator-valued entries

Infinite Toeplitz matrices with operator-valued entries arise, for example, when interpreting Wiener-Hopf integral operators on L(2)(0, infinity) as matrices acting on the direct sum of countably many copies of L(2)(0, 1). This paper concerns the question of asymptotically inverting such infinite Toeplitz matrices by having recourse to their finite principal sections. As expected from the corresponding theories for the scaler and matrix-valued cases, this problem leads to the investigation of compactness properties of infinite Hankel matrices. By introducing the concept of Q(n)-compact operators on spaces of square-summable sequences with values in a separable Hilbert space, criteria for the applicability of the finite section method to Toeplitz operators with symbols in C + H-infinity, in PC, or with locally sectorial symbols are established.