Colloquium Lectures

Simultaneous solutions of matrix and operator Sylvester equations
Speaker. Prof. Vũ Quốc Phóng, Ohio University
Venue: Room 301, Buildding A5
Time:: 9:30

Abstract:
We consider a system of Sylvester equations A_i X-XB_i=C_i where (A_1,…,A_k) and (B_1,...,B_k) are (commuting) k-tuples of m×m and n×n matrices, and (C_1,…,C_k) is a (compatible) k-tuple of m×n matrices, and prove that this system of Sylvester equations has a unique simultaneous solution if and only if the joint spectra of (A_1,…,A_k) and (B_1,...,B_k) don’t intersect. We relate the problem of finding efficient algorithms for evaluating (simultaneous) solutions of Sylvester equations to the same problem for evaluating functions of (several commuting) matrices. We also consider the simultaneous Sylvester equations for the case when (A_1,…,A_k) and (B_1,...,B_k) are commuting k-tuples of bounded linear operators on Banach spaces E and F, respectively, and prove that if the Taylor joint spectra of (A_1,…,A_k) and (B_1,...,B_k) don’t intersect, then the system of Sylvester equations have a unique simultaneous solutions. Some applications and open problems are also discussed.

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