Extensions of symmetric operators that are invariant under scaling and applications to indicial operators

TL;DR

This paper studies invariant extensions of symmetric indicial operators related to elliptic operators near corners, providing a unified framework for their analysis and explicit descriptions of their extensions and signatures.

Contribution

It introduces a general abstract framework for invariant symmetric operators and applies it to differential operators of Fuchs type, offering new extension and spectral flow formulas.

Findings

Describes extensions via generalized eigenspaces and asymptotic expansions.

Provides a Green formula and algebraic signature formula for invariant operators.

Connects the signature to spectral flow of the indicial family.

Abstract

Indicial operators are model operators associated to an elliptic differential operator near a corner singularity on a stratified manifold. These model operators are defined on generalized tangent cone configurations and exhibit a natural scaling invariance property with respect to dilations of the radial variable. In this paper we discuss extensions of symmetric indicial operators from a functional analytic point of view. In the first, purely abstract part of this paper, we consider a general unbounded symmetric operator that exhibits invariance with respect to an abstract scaling action on a Hilbert space, and we describe its extensions in terms of generalized eigenspaces of the infinitesimal generator of this action. Among others, we obtain a Green formula for the adjoint pairing, an algebraic formula for the signature, and in the semibounded case explicit descriptions of the Friedrichs and Krein extensions. In the second part we consider differential operators of Fuchs type on the half axis with unbounded operator coefficients that are invariant under dilation, and show that under suitable ellipticity assumptions on the indicial family these operators fit into the abstract framework of the first part, which in this case furnishes a description of extensions in terms of polyhomogeneous asymptotic expansions. We also obtain an analytic formula for the signature of the adjoint pairing in terms of the spectral flow of the indicial family for such operators.