Inequalities in Hilbert Spaces

Abstract

The main result in this thesis is a new generalization of Selberg's inequality in Hilbert spaces with a proof. In Chapter 1 we define Hilbert spaces and give a proof of the Cauchy-Schwarz inequality and the Bessel inequality. As an example of application of the Cauchy-Schwarz inequality and the Bessel inequality, we give an estimate for the dimension of an eigenspace of an integral operator. Next we give a proof of Selberg's inequality including the equality conditions following [Furuta]. In Chapter 2 we give selected facts on positive semidefinite matrices with proofs or references. Then we use this theory for positive semidefinite matrices to study inequalities. First we give a proof of a generalized Bessel inequality following [Akhiezer,Glazman], then we use the same technique to give a new proof of Selberg's inequality. We conclude with a new generalization of Selberg's inequality with a proof. In the last section of Chapter 2 we show how the matrix approach developed in Chapter 2.1 and Chapter 2.2 can be used to obtain optimal frame bounds. We introduce a new notation for frame bounds.