# The Laplace transform on the cones of lattice-structured Banach spaces

Rupčić, Diana (2016) The Laplace transform on the cones of lattice-structured Banach spaces. Doctoral thesis, Faculty of Science > Department of Mathematics.

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## Abstract

The main task of this research is the characterization of positive definite functions using Laplace transform of a measure. Although mathematicians dealt with this problem at the beginning of the 20th century, early results were applicable only in very special cases. Gradually, $B(\mathcal{H})$-valued positive definite functions were observed, i.e. functions with values in the space of bounded linear operators on a complex Hilbert space $\mathcal{H}$ defined on a semigroup with involution. In this case representing measure has values in the positive cone of Hermitian operators and is defined as a natural generalization of a classic positive measure with values in a convex cone of nonnegative numbers which is allowed infinite values in a compactification of the cone. The purpose of this research was to establish Nussbaum type theorem in this more general case where domain of the function was a positive cone in a Banach space that is also a vector lattice, but not necessarily a Banach lattice. Those type of spaces also include Sobolev spaces $W^{1,p}(\Omega)$. Conditions on the initial space, convex cone and the very function were examined that enable an integral representation using the Laplace transform of a measure which is defined on the dual of the initial space. The basis was the Berg-Maserick theorem which characterizes $\alpha$-bounded positive definite functions via generalized Laplace transform. In a setting of lattice-structured Banach spaces with order unit, integral representation of those type of functions was derived where absolute value is a locally bounded function. It was shown that representing measure is Radon and concentrated on a subset of the topological dual.

Item Type: Thesis (Doctoral thesis) positive definite function, integral representation, Laplace transform, representing measure, Nussbaum theorem, convex cone, character, $\alpha$-boundedness, absolute value, ultraweak operator topology, continuity on line segments, ordered Banach space, vector lattice Šikić, Hrvoje 2016 88 NATURAL SCIENCES > MathematicsNATURAL SCIENCES > Mathematics > Mathematical Analysis Faculty of Science > Department of Mathematics Iva Prah 13 Jul 2016 12:24 13 Jul 2016 12:24 http://digre.pmf.unizg.hr/id/eprint/4972

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