Abstract Left-Definite Theory: A Model Operator Approach, Examples, Fractional Sobolev Spaces, and Interpolation Theory

TL;DR

This paper develops a straightforward approach to abstract left-definite theory using spectral theorems, illustrating it with examples involving fractional Sobolev spaces, interpolation, and explicit domain descriptions of fractional powers of operators.

Contribution

It introduces a model operator approach to left-definite theory, providing explicit domain descriptions for fractional powers of key operators like the harmonic oscillator.

Findings

Explicit description of domains of fractional powers of the harmonic oscillator.

Application of interpolation theory to fractional Sobolev spaces.

Demonstration of a straightforward spectral theorem-based approach.

Abstract

We use a model operator approach and the spectral theorem for self-adjoint operators in a Hilbert space to derive the basic results of abstract left-definite theory in a straightforward manner. The theory is amply illustrated with a variety of concrete examples employing scales of Hilbert spaces, fractional Sobolev spaces, and domains of (strictly) positive fractional powers of operators, employing interpolation theory. In particular, we explicitly describe the domains of positive powers of the harmonic oscillator operator in $L^2(\mathbb{R})$ $\big($and hence that of the Hermite operator in $L^2\big(\mathbb{R}; e^{-x^2}dx)\big)\big)$ in terms of fractional Sobolev spaces, certain commutation techniques, and positive powers of (the absolute value of) the operator of multiplication by the independent variable in $L^2(\mathbb{R})$.