Bezoutian operator vessels in Banach space (Q1911964)

Operator colligations and operator vessels were introduced by M. Livsic as tools for the study of \(n\)-tuples of operators on Hilbert space with small (e.g., finite rank) imaginary parts. These notions constitute a generalization of the Livsic characteristic function for tuples of operators. The author of this paper constructed a theory of vessels for \(n\)-tuples \(A=(A_1,A_2,\dots,A_n)\) on a Banach space with the property that there is another \(n\)-tuple \(B=(B_1,B_2,\dots,B_n)\) which is a good approximation of \(A\). An example where the new theory applies occurs in the case of \(A\) being a tuple of integral operators, and \(B\) being the tuple obtained by transposing the kernels of these integral operators. The purpose of this paper is to study the interplay between invariant subspaces of the operators on the one hand, and operations on the corresponding vessels on the other. The relevant vessel notions are those of decomposition and coupling. The technical results are too involved to be included in this review.