# Journal of Operator Theory

Volume 89, Issue 1, Winter 2023 pp. 205-248.

Dual pairs of operators, harmonic analysis of singular nonatomic
measures and Krein-Feller diffusion

**Authors**:
Palle E.T. Jorgensen (1), James Tian (2)

**Author institution:** (1) Department of Mathematics, The University of Iowa, Iowa City, IA 52242-1419, U.S.A.

(2) Mathematical Reviews, 416 4th Str. Ann Arbor, MI 48103-4816, U.S.A.

**Summary: **We show that a Krein--Feller operator is naturally associated to a
fixed measure $\mu$, (positive, $\sigma$-finite, and nonatomic).
Dual pairs of operators are introduced, carried by the two Hilbert
spaces, $L^{2} (\mu )$ and $L^{2} (\lambda )$,
where $\lambda$ denotes Lebesgue measure. An associated operator
pair consists of two densely defined operators,
each one contained in the adjoint of the other. This yields a
rigorous analysis of the corresponding $\mu$-Krein--Feller operator.
As an application, including the case of fractal measures, we compute the associated
diffusion, semigroup, Dirichlet forms, and $\mu$-generalized heat
equation.

**DOI: **http://dx.doi.org/10.7900/jot.2021may30.2359

**Keywords: **reproducing kernel Hilbert space, Gaussian free fields, generalized Ito integration, Krein--Feller operators, dual pairs, iterated function systems, selfadjoint extensions, Dirichlet forms

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