Entropy and Kolmogorov complexity

This thesis is dedicated to studying the theory of entropy and its relation to the Kolmogorov complexity. Originating in physics, the notion of entropy was introduced to mathematics by C. E. Shannon as a way of measuring the rate at which information is coming from a data source. There are, however, a few different ways of telling how much...

Functional Calculus via Transference, Double Operator Integrals and Applications

This thesis is dedicated to the study of several aspects of the theory of functional calculus. This theory considers the combination of an operator A and a function f(z) of a variable z, resulting in an operator f(A). One then attempts to study properties of the operator f(A) in terms of properties of the operator A and the function f. A...

Noncommutative and vector-valued Rosenthal inequalities

This thesis is dedicated to the study of a class of probabilistic inequalities, called Rosenthal inequalities. These inequalities provide two-sided estimates for the p-th moments of the sum of a sequence of independent, mean zero random variables in terms of a suitable norm on the sequence itself. Rosenthal inequalities are named after H.P....

Unconditional decompositions and multiplier theorems