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.