Block Toeplitz operators with rational symbols and discrete singular systems

Abstract

This thesis concerns block Toeplitz operators (equations). Consider the block Toeplitz operator T = [<I> k-j ]k,j=O' where the <I>k are complex m x m matrices such that 00 (0.1) v=-oo The norm in (0.1) is the usual operator norm on an m x m matrix. The condition (0.1) means that the symbol 00 ( 0.2) v=-oo belongs t.o the Wiener class w mxm of all absolutely convergent sequences of complex m x m matrices. Let 1 ~ p ~ oo be fixed. The block Toeplitz operator T induces a bounded linear operator (also denoted by T) on l'';, namely, 00 (0.3) Yk = (Tx )k = L <I>k-vXv , k = 0, 1, 2, ... ' v=O where x = (xo, x 1,X2, ... ) Et;.