Displacement Structure and Tensor Algebras
by
Tiberiu Constantinescu
University of Texas at Dallas

The Schur class T_N^S of the tensor algebra over C^N consists of all upper triangular contractions T=(T_ij)_{i, j=0}^\infty on the full Fock space F(C^N), with the property that for i <= j, T_ij=T_{i-1, j-1}\oplusT_{i-1, j-1} \oplus ... T_{i-1, j-1} (N copies). We deduce that if A=I-T^*T, then A-\sum_k=1^N S_kAS_k^*=GJG^*, where

G= æ ç ç ç ç ç è 1 T*00 0 T*01 : : ö ÷ ÷ ÷ ÷ ÷ ø ,         J= æ ç ç ç è 1 0 0 -1 ö ÷ ÷ ÷ ø ,

and S_k are the so-called left creation operators. This property suggests the study of the matrices satisfying the displacement equation R-\sum_k=1^N S_kRS_k^*=GJG^* for some G=[U, V]. Our main result is the following:

The displacement equation R-\sum_k=1^NS_kRS_k^*=GJG^* admits a positive-semidefinite solution R if and only if there is S in T_N^S such that V=US, where the infinite bloc matrices

U = [ ...    S1S2U   S12U    ...    SNU    ...    S1U   U], V = [...    S1S2V   S12V    ...    SNV    ...    S1V   V]

are the associated wave operators.

This result provides another approach to interpolation and factorization in several noncommuting variables as an application of the displacement structure theory. The talk is based on joint work with T. Kailath and A.H. Sayed.

Date received: May 22, 2000

Copyright © 2000 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caeo-25.