A New Characterization of the Domains of Integral Powers of the Self-Adjoint Friedrichs-Legendre Operator

Abstract

Let $A$ be the self-adjoint operator in $L^{2}(-1,1)$, generated by the second-order classical Legendre differential equation% \[ \ell\lbrack y](t)=-\left( (1-t^{2})y^{\prime}(t)\right) ^{\prime}+ky(t)=\lambda y(t)\quad(t\in(-1,1)), \] which has the Legendre polynomials $\{P_{m}\}_{m=0}^{\infty}$ as a complete sequence of eigenfunctions; here $k$ is a fixed, non-negative real number. This is the Friedrichs extension of the minimal operator associated with $\ell[\cdot]$ in $L^2(-1,1)$. For each $n \in \mathbb{N}$, we show that $\mathcal{D}(A^{n})$ is characterized by \textit{one} integrability condition instead of $2n$ boundary conditions as dictated by the classical Glazman-Krein-Naimark theory. We also prove that if $f\in\mathcal{D}(A^{n})$ then $f^{(n)}\in L^{2}(-1,1).$ This smoothness result extends known results when $n=1$ and $n=2.$ Furthermore, this result is optimal in the sense that there exists $g\in\mathcal{D}(A^{n})$ with $g^{(n+1)}\notin L^{2}(-1,1)$.