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

2024/08/02 by Christoph Fischbacher, Fischbacher, Christoph, Fritz Gesztesy +5 · 1 citation
Engineering · Mathematics · #46B35. Secondary: 34L40 #46B70 #47B25 #47E05 #Differential Equations and Boundary Problems #Differential Equations and Numerical Methods #FOS: Mathematics #Numerical methods in engineering #Primary: 33C45 #Spectral Theory (math.SP)

paper · pdf · doi:10.48550/arxiv.2408.01514

openalex publication_date 2024/08/02 · openalex created_date 2025/01/17 · openalex updated_date 2026/07/28

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 L²(ℝ) (and hence that of the Hermite operator in L²(ℝ; e^-x2dx))) 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 L²(ℝ).

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