Preface

Linear Operator Theory in Hilbert spaces plays a central role in contemporary mathematics with numerous applications for Partial Differential Equations, in Approximation Theory, Optimization Theory, Numerical Analysis, Probability Theory and Statistics and other fields. The main aim of this short book is to present recent results concerning inequalities of the Jensen, Čebyšev and Grüss type for continuous functions of bounded selfadjoint operators on complex Hilbert spaces. The book is intended for use by both researchers in various fields of Linear Operator Theory and Mathematical Inequalities, domains which have grown exponentially in the last decade, as well as by postgraduate students and scientists applying inequalities in their specific areas. In Chap. 1, we recall some fundamental facts concerning bounded selfadjoint operators on complex Hilbert spaces. The generalized Schwarz’s inequality for positive selfadjoint operators as well as some results for the spectrum of this class of operators are presented. Then we introduce and explore the fundamental results for polynomials in a linear operator, continuous functions of selfadjoint operators as well as the step functions of selfadjoint operators. Using these results, we then introduce the spectral decomposition of selfadjoint operators (the Spectral Representation Theorem) that will play a central role in the rest of the book. This result is used as a key tool in obtaining various new inequalities for continuous functions of selfadjoint operators, functions that are of bounded variation, Lipschitzian, monotonic or absolutely continuous. Another tool that will greatly simplify the error bounds provided in the book is the Total Variation Schwarz’s Inequality for which a simple proof is offered. Jensen’s type inequalities in their various settings ranging from discrete to continuous case play an important role in different branches of Modern Mathematics. A simple search in the MathSciNet database of the American Mathematical Society with the key words “jensen” and “inequality” in the title reveals more than 300 items intimately devoted to this famous result. However, the number of papers where this inequality is applied is a lot larger and far more difficult to find.