Sachi Srivastava

Maximal regularity in interpolation spaces for second order Cauchy problems

We study maximal regularity in interpolation spaces for the sum of three closed linear operators on a Banach space, and we apply the abstract results to obtain Besov and Holder maximal regularity for complete secondorder Cauchy problems under natural parabolicity assumptions. We discuss applications to partial differential equations.

Applications to Partial Differential Equations

While semigroups of operators are interesting objects of study by themselves, one of the main reasons why they are studied so extensively is due to the important role they play in the study of partial differential equations.

Vector-Valued Functions

This chapter is mostly of a preliminary nature. In the first section we collect results on measurability and integrability of vector-valued functions that will be useful throughout.

Perturbation and Convergence of Semigroups

In this chapter, the stability of various classes of semigroups under suitable sets of perturbations will be studied, viz. for general (C_0)-semigroups and contraction semigroups.

Chernoff’s Theorem and its Applications

In this chapter, a very interesting theorem, due to Chernoff [4], is proven and some of its applications, viz. the Trotter-Kato Product Formula, the Feynman-Kac Formula and the Central Limit Theorem are given.

Dissipative Operators and Holomorphic Semigroups

In this chapter we continue the study of (C_0)-semigroups concentrating on contractive and holomorphic semigroups.

Stability of Quantum Dynamical Semigroups

A one parameter semigroup of maps is said to be stable if it eventually decays to zero. Generally different topologies for convergence to zero give rise to different notions of stability. Stability is also connected with absence of fixed points. We examine these concepts in the context of quantum dynamical semigroups and dilation theory.