Bezout equation, Vandermonde operators, and common multiples for operator polynomials (Q1110059)

Consider the Bezout equation \[ (I)\quad X_ 1(\lambda)A_ 1(\lambda)+X_ 2(\lambda)A_ 2(\lambda)=1,\quad \lambda \in {\mathbb{C}}, \] where \(A_ 1,A_ 2,X_ 1\) and \(X_ 2\) are (Hilbert-space) operator polynomials. The paper reveals certain important connections between solvability of (I), properties of Vandermonde operators, and common multiples of \(A_ 1\) and \(A_ 2\). Especially, solvability of (I) is characterized by the Vandermonde operators \(V(A_ 1,A_ 2;p_ 1,p_ 2,q)\) with indices \(p_ 1\), \(p_ 2\) and q, in the case that both of \(A_ 1\) and \(A_ 2\) have compact spectra and inverse polynomial growth, and \(A_ 1\) (or \(A_ 2)\) is monic.