Products of Block Toeplitz Operators

Let D be the open unit disk in the complex plane and ∂D the unit circle. Let dσ(w) be the normalized Lebesgue measure on the unit circle. We denote by L(C) (L for n=1) the space of C-valued Lebesgue square integrable functions on the unit circle. The Hardy space H(C) (H for n=1) is the closed linear span of C-valued analytic polynomials. We observe that L(C) = L ⊗ C and H(C) = H ⊗ C, where ⊗ denotes the Hilbert space tensor product. Let Mn×n be the set of n× n complex matrices. Ln×n denotes the space ofMn×n-valued essentially bounded Lebesgue measurable functions on the unit circle and H n×n denotes the space of Mn×n-valued essentially bounded analytic functions in the disk. Let P be the projection of L(C) onto H(C). For F ∈ Ln×n, the block Toeplitz operator TF :H (C)→H(C) with symbol F is defined by the rule TFh=P (Fh). The Hankel operator HF :H (C)→L(C)⊖H(C) with symbol F is defined by HFh= (I−P )(Fh). The block Toeplitz operator TF has the following matrix representation: