An Operator Identities Approach to Operator Bezoutians . A General Scheme and Examples ∗

In this paper we study Bezoutians using a general method known as the method of operator identities in the integral equations literature [S76b, S97], and under the name displacement structure method in the engineering [K99] and numerical literature [HR84, O03]. The latter approach allows us to introduce a generalized concept of the operator Bezoutian and to carry over to it the classical results of Darboux (on common roots of scalar polynomials [D1876]), and of Hermite (on polynomial stability [H1856]). Several other known results scattered in the mathematical and engineering literature (Schur-Cohn [C22], Krein [K], Sakhnovich [S76a], Anderon-Jury [AJ76], Lerer-Tysmenetsky [LT82], Lerer-Rodman [LR96a, LR96b]) are shown to appear as particular instances of our general result. The unified operator identies (displacement structure) approach results in a transparent concise derivation of main results allowing us to include most of known as well as new special cases in one paper. instance ∗This work was supported in part by the NSF contracts 0242518 and 0098222. †web page: http://www.math.uconn.edu/ ̃olshevsky email: [email protected] ‡email: [email protected]