The Feichtinger Conjecture and Reproducing Kernel Hilbert Spaces

Abstract

In this dissertation, we study the Feichtinger Conjecture(FC), which has been shown to be equivalent to the celebrated Kadison-Singer Problem. The FC states that every norm-bounded below Bessel sequence in a Hilbert space can be partitioned into finitely many Riesz basic sequences. This study is divided into two parts. In the first part, we explore the FC in the setting of reproducing kernel Hilbert spaces. The second part of this study introduces two new directions to explore the FC further, which are based on a factorization of positive operators in B(`). The results presented in the later part have a mixed flavor in the sense that some of them point in the direction of finding a negative answer to the FC, whereas others prove the FC for some special cases. In the first part of the thesis, we show that in order to prove the FC it is enough to prove that in every Hilbert space, contractively contained in the Hardy space H, each Bessel sequence of normalized kernel functions can be partitioned into finitely many Riesz basic sequences. In addition, we examine some of these spaces and show that the above holds in them. We also look at products and tensor products of kernels, where using Schur products we obtain some interesting results. These results allows us to prove that in the Bargmann-Fock spaces on the n-dimensional complex plane and the weighted Bergman spaces on the unit ball, the Bessel sequences of normalized kernel functions split into finitely many Riesz basic sequences. We also prove that the same result holds in the H α,β spaces as well.