Abstract
In this paper a Jessen’s type inequality for normalized positive $C_{0}$ -semigroups is obtained. An adjoint of Jessen’s type inequality has also been derived for the corresponding adjoint semigroup, which does not give the analogous results but the behavior is still interesting. Moreover, it is followed by some results regarding positive definiteness and exponential convexity of complex structures involving operators from a semigroup.
Highlights
Introduction and preliminariesA significant theory regarding inequalities and exponential convexity for real-valued functions has been developed [, ]
The notion of Banach lattice was introduced to get a common abstract setting, within which one could talk about the ordering of elements
The phenomena related to positivity can be generalized
Summary
Let {Z(t)}t≥ be the strongly continuous positive semigroup, defined on a Banach lattice V. The theory presented is defined on such semigroups of positive linear operators defined on a Banach lattice V. For a strongly continuous positive semigroup {Z(t)}t≥ on a Banach space X, by defining Z∗(t) = (Z(t))∗ for every t, we get a corresponding adjoint semigroup {Z∗(t)}t≥ on the dual space X∗. Consider the Banach space Lip (X, Y ) of all Lipschitz continuous operators F : X → Y satisfying F(θ ) = θ , equipped with the norm [F]Lip = sup x =x. The considered dual space of the vector lattice algebra V will be denoted by V ∗, which can be the intersection of the pseudo-dual and classical dual spaces in the case of a nonlinear convex operator.
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