Janet C. Tremain

The Kadison–Singer Problem in mathematics and engineering

We will see that the famous intractible 1959 Kadison–Singer Problem in C*-algebras is equivalent to fundamental open problems in a dozen different areas of research in mathematics and engineering. This work gives all these areas common ground on which to interact as well as explaining why each area has volumes of literature on their respective problems without a satisfactory resolution.

A Physical Interpretation of Tight Frames

We characterize the existence of finite tight frames whose frame elements are of predetermined length. In particular, we derive a “fundamental inequality” which completely characterizes those sequences which arise as the lengths of a tight frame’s elements. Furthermore, using concepts from classical physics, we show that this characterization has an intuitive physical interpretation.

Real equiangular frames

Real equiangular tight frames can be especially useful in practice because of their structure. The problem is that very few of them are known. We will look at recent advances on the problem of classifying the equiangular tight frames and as a consequence give a classification of this family of frames for all real Hilbert spaces of dimension less than or equal to 50.

A decomposition theorem for frames and the Feichtinger Conjecture

In this paper we study the Feichtinger Conjecture in frame theory, which was recently shown to be equivalent to the 1959 Kadison-Singer Problem in C*-Algebras. We will show that every bounded Bessel sequence can be decomposed into two subsets each of which is an arbitrarily small perturbation of a sequence with a finite orthogonal decomposition. This construction is then used to answer two open problems concerning the Feichtinger Conjecture: 1. The Feichtinger Conjecture is equivalent to the conjecture that every unit norm Bessel sequence is a finite union of frame sequences. 2. Every unit norm Bessel sequence is a finite union of sets each of which is ω-independent for l 2 -sequences.

Consequences of the Marcus/Spielman/Srivastava Solution of the Kadison-Singer Problem

It is known that the famous, intractable 1959 Kadison-Singer problem in C∗-algebras is equivalent to fundamental unsolved problems in a dozen areas of research in pure mathematics, applied mathematics and Engineering. The recent surprising solution to this problem by Marcus, Spielman and Srivastava was a significant achievement and a significant advance for all these areas of research. We will look at many of the known equivalent forms of the Kadison-Singer Problem and see what are the best new theorems available in each area of research as a consequence of the work of Marcus, Spielman and Srivastava. In the cases where constants are important for the theorem, we will give the best constants available in terms of a generic constant taken from (A. Marcus, D. Spielman and N. Srivastava, Interlacing families II: Mixed Characteristic Polynomials and the Kadison-Singer Problem, arXiv 1306.3969v4). Thus, if better constants eventually become available, it will be simple to adapt these new constants to the theorems.

Norm Retrieval and Phase Retrieval by Projections

We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans W 1 , W 2 and W 1 ∩ W 2 = { 0 } , then W 1 ⊥ W 2 .

Physical laws governing finite tight frames

We give a physical interpretation for finite tight frames along the lines of Columbs Law in Physics. This allows us to use results from classical mechanics to anticipate results in frame theory. As a consequence, we are able to classify those frames for an N-dimensional Hilbert space which are the closest to being tight (in the sense of minimizing potential energy) while having the norms of the frame vectors prescribed in advance. This also yields a fundamental inequality that all finite tight frames must satisfy.

Unconditional convergence constants of Hilbert space frame expansions

We will prove some new, fundamental results in frame theory by computing the unconditional constant (for all definitions of unconditional) for the frame expansion of a vector in a Hilbert space and see that it is √B/A, where A, B are the frame bounds of the frame. It follows that tight frames have unconditional constant one. We then generalize this to a classification of such frames by showing that for Bessel sequences whose frame operator can be diagonalized, the frame expansions have unconditional constant one if and only if the Bessel sequence is an orthogonal sum of tight frames. We then prove similar results for cross frame expansions but here the results are no longer a classification. We also give examples to show that our results are best possible. These results should have been done 20 years ago but somehow we overlooked this topic.

Fusion Frames and Unbiased Basic Sequences

The construction of Parseval frames with special, rigid geometric properties has left many open problems even after decades of efforts. The construction of similar types of fusion frames is even less developed. We construct a large family of equi-isoclinic Parseval fusion frames by taking the Naimark complement of the union of orthonormal bases. If these bases are chosen to be mutually unbiased, then the resulting fusion frame subspaces are spanned by mutually unbiased basic sequences. By giving an explicit representation for Naimark complements, we are able to construct concrete fusion frames in their respective Hilbert spaces.

Real phase retrieval by orthogonal complements and hyperplanes

The fundamental question concerning phase retrieval by projections in Rm is what is the least number of projections needed and what dimensions can be used. We will look at recent advances concerning phase retrieval by orthogonal complements and phase retrieval by hyperplanes which raise a number of problems which would give a complete answer to this fundamental problem.