Abstract:
The strongly continuous semigroups of operators are of great significance due to the fact that
these have numerous applications in various areas of science. The qualitative understanding
of a complicated deterministic system can be acquired by analyzing the solutions of a related
differential equations in terms of these operators. Moreover, the theory of convexity, means,
Cauchy means and inequalities has a huge impact in everyday science. Therefore, the
pronounced nature of the means and inequalities defined on semigroups of operators, can
not be contradicted.
This dissertation is a staunch effort to generalize the theory of means and inequalities
to the operator semigroups. A new theory of power means is introduced on a C0-group of
continuous linear operators. A mean value theorem is proved. Moreover, the Cauchy-type
power means on a C0-group of continuous linear operators, are obtained systematically.
A Jessen’s type inequality for normalized positive C0-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 exponential convexity of complex structures involving
operators from a semigroup. Few applications of Jessen’s type inequality are also presented,
yielding the Hölder’s type and Minkowski’s type inequalities for corresponding semigroup.
Moreover, a Dresher’s type inequality for two-parameter family of means, is also proved.
A Jensen’s and Hermite-Hadamard’s type inequalities are also obtained for a semigroup of
positive linear operators and a superquadratic mapping defined on a Banach lattice algebra.
The corresponding mean value theorems conduct us to find a new sets of Cauchy’s type
means.
