Effect of change in inner product on spectral analysis of certain linear operators in Hilbert Spaces arising from orthogonal polynomials

TL;DR

This paper investigates how the spectral properties of certain linear operators in Hilbert spaces change when considering different inner products, focusing on differential and matrix operators related to orthogonal polynomials.

Contribution

It provides conditions for when formal differential operators have polynomial eigenfunctions and analyzes the impact of inner product changes on spectral properties of these operators.

Findings

Certain differential operators lack polynomial eigenfunctions.

Spectral properties depend on the inner product structure.

Conditions identified for polynomial eigenfunctions in differential operators.

Abstract

In this paper we present how spectral properties of certain linear operators vary when operators are considered in different Hilbert spaces having common dense domain as the space of polynomials in one real variable with complex coefficients. This is done taking differential operator and matrix operator representation of dilation operator $E_{p,d}$ dilating $p$, a polynomial sequence, by $d$, a non-constant sequence of non-zero complex scalars. For the purpose of identification of $E_{p,d}$ with formal differential operator, which includes finite order differential operators as special cases, we derive conditions under which a formal differential operator has a polynomial sequence as a sequence of eigen functions corresponding to non-zero eigen values. As a consequence we get, for instance, $$ ((1/72)x^4-3x) y^{(4)}-x y^{(2)}-(1/3)x y^{(1)} + (4/3)y.$$ does not have any polynomial sequence as such a sequence of eigen functions, since no fourth order polynomial is such an eigen svector.