Operator Inequalities of the Jensen, Cebysev and Grüss Type

The main aim of Inequalities of the Jensen, Cebysev and Grüss Type for Continuous Functions of Selfadjoint Operators on Hilbert Spaces is to present recent results concerning inequalities of Jensen, Cebysev and Grüss type for continuous functions of bounded selfadjoint operators on complex Hilbert spaces. §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 & Statistics and other fields.§The first chapter portrays 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. This chapter also introduces and explores the fundamental results for polynomials in a linear operator, continuous functions of selfadjoint operators as well as the step functions of selfadjoint operators. By the use of these results we then introduce the spectral decomposition of selfadjoint operators which 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 which are of bounded variation, Lipschitzian, monotonic or absolutely continuous.§The second chapter illustrates some recent results obtained by the author that deal with different aspects of this well research inequality than those recently reported. They include but are not restricted to the operator version of the Dragomir-Ionescu inequality, Slater's type inequalities for operators and its inverses, Jensen's inequality for twice differentiable functions whose second derivatives satisfy some upper and lower bounds conditions, Jensen's type inequalities for log-convex functions and for differentiable log-convex functions. Finally, some Hermite-Hadamard's type inequalities for convex functions and Hermite-Hadamard's type inequalities for operator convex functions are presented as well.§The Cebysev, or in a different spelling - Chebyshev, inequality which compares the integral/discrete mean of the product with the product of the integral/discrete means is famous in the literature devoted to Mathematical Inequalities. The sister inequality due to Grüss which provides error bounds for the magnitude of the difference between the integral mean of the product and the product of the integral means has also attracted much interest since it has been discovered in 1935 with more than 180 papers published. §Inequalities of the Jensen, Cebysev and Grüss Type for Continuous Functions of Selfadjoint Operators on Hilbert Spaces 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.