Rational operator functions and Bezoutian operator vessels (Q1925105)

In previous work the author has developed a theory of operator vessels associated with operators \(A_1,A_2,\dots, A_n,B_1,B_2,\dots,B_n\) on a Banach space such that the differences \(A_j-B_j\) are small (e.g., of finite rank). A special kind of vessel can be constructed when the operators \(A_j,B_j\) are obtained as rational functions of operators \(A, B\) (i.e., \(A_j=r_j(A)\) and \(B_j=r_j(B))\) with rank one difference \(A-B\). The author makes a general conjecture about sufficient conditions under which a given vessel can be identified with one of the special kind. He proves this conjecture for \(n=2\).