Frames and the Kadison-singer Problem: a Report for Aim

This is an introduction to the problems connecting frame theory and the Kadison-Singer Problem. 1. Overview For an extensive introduction to the aspects of frame theory needed for an understanding of the Kadison-Singer Problem we refer to the survey paper [3], which is posted in this section of the web page “The Kadison-Singer Problem”. In the following section we will just give the basic definitions for the concepts we will be working with. In Section 3, we will then state several conjectures in frame theory, which are equivalent to the Kadison-Singer Problem. In Section 5, we will discuss the Rado-Horn Theorem as one tool to attack in particular algorithmic aspects of the Kadison-Singer Problem. 2. Basic definitions and notations Throughout let H denote a (finite or infinite dimensional) Hilbert space, and let HN denote an N -dimensional Hilbert space. 2.1. Frames and Bessel sequences. First we state the definition of a frame, which can be regarded as the most natural generalization of the concept of orthonormal bases. Frames provide robust and stable – but usually nonunique (redundant) – representations of vectors, which makes them particularly useful both in pure as well as applied mathematics. Definition 2.1. A family {fi}i∈I of elements of H is called a frame for H, if there are constants 0 < A ≤ B < ∞ (called the lower and upper frame bounds, respectively) such that, for all f ∈ H,