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

TL;DR

This paper investigates the multiplication operator in matrix Sobolev spaces with finite measures and matrix weights, demonstrating its symmetrizability and deriving new orthogonality conditions for related Sobolev polynomials.

Contribution

It shows the symmetrizability of the multiplication operator in matrix Sobolev spaces with specific matrix weights, providing new orthogonality conditions involving symmetric operators.

Findings

The operator is symmetrizable for certain matrix weights.

Existence of symmetric operators in a larger space representing the multiplication operator.

New orthogonality conditions for Sobolev orthogonal polynomials involving indefinite metric 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.