Glasnik Matematicki, Vol. 60, No. 1 (2025), 107-125. \( \)

THE LAPLACE TRANSFORM ON THE CONES OF LATTICE-STRUCTURED BANACH SPACES

Diana Hunjak

Faculty of Transport and Traffic Sciences, University of Zagreb, 10 000 Zagreb, Croatia
e-mail:diana.hunjak@fpz.unizg.hr

Abstract.   Characterizations of positive definite functions defined on convex cones using the Laplace transform of a measure are commonly referred to as Nussbaum-type theorems. This paper establishes a Nussbaum-type theorem in the context where the domain of a \(B(\mathcal{H})\)-valued positive definite function is a positive cone within a Banach space that is also a vector lattice, but not necessarily a Banach lattice. Such spaces include examples like Sobolev spaces \(W^{1,p}(\Omega)\). Utilizing the Berg-Maserick theorem, we prove that the unique representing measure is Radon measure concentrated on a subset of the topological dual.

2020 Mathematics Subject Classification.   43A35, 44A10, 46A40

Key words and phrases.   Positive definite function, integral representation, Laplace transform, \(\alpha\)-boundedness, Banach lattice.

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https://doi.org/10.3336/gm.60.1.07

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