On the multiplication operator by an independent variable in matrix Sobolev spaces

Abstract

We study the operator \(\mathcal {A}\) of multiplication by an independent variable in a matrix Sobolev space \(W^2(M)\). In the cases of finite measures on [a, b] with \((2\times 2)\) and \((3\times 3)\) real continuous matrix weights of full rank it is shown that the operator \(\mathcal {A}\) is symmetrizable. Namely, there exist two symmetric operators \(\mathcal {B}\) and \(\mathcal {C}\) in a larger space such that \(\mathcal {A} f = \mathcal {C} \mathcal {B}^{-1} f\), \(f\in D(\mathcal {A})\). As a corollary, we obtain some new orthogonality conditions for the associated Sobolev orthogonal polynomials. These conditions involve two symmetric operators in an indefinite metric space.