Fractional operators as traces of operator-valued curves

TL;DR

This paper extends the characterization of fractional powers of positive self-adjoint operators using differential operators acting on curves, generalizing previous results by Caffarelli--Silvestre and Stinga--Torrea.

Contribution

It introduces a novel approach linking fractional operators to differential operators on operator-valued curves, broadening the theoretical framework for fractional operator analysis.

Findings

Established a new representation of fractional powers as traces of operator-valued curves.

Extended classical extension problem results to a broader class of operators.

Provided a mathematical foundation for future applications in PDEs and functional analysis.

Abstract

We relate non integer powers ${\mathcal L}^{s}$, $s>0$ of a given (unbounded) positive self-adjoint operator $\mathcal L$ in a real separable Hilbert space $\mathcal H$ with a certain differential operator of order $2\lceil{s}\rceil$, acting on even curves $\mathbb R\to \mathcal H$. This extends the results by Caffarelli--Silvestre and Stinga--Torrea regarding the characterization of fractional powers of differential operators via an extension problem.