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Introduction to Finite Frame Theory Read chapter Read first chapter

2013 | OriginalPaper | Chapter

11. The Kadison–Singer and Paulsen Problems in Finite Frame Theory

Author : Peter G. Casazza

Published in: Finite Frames

Publisher: Birkhäuser Boston

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Abstract

We now know that some of the basic open problems in frame theory are equivalent to fundamental open problems in a dozen areas of research in both pure and applied mathematics, engineering, and others. These problems include the 1959 Kadison–Singer problem in C ∗-algebras, the paving conjecture in operator theory, the Bourgain–Tzafriri conjecture in Banach space theory, the Feichtinger conjecture and the R ϵ -conjecture in frame theory, and many more. In this chapter we will show these equivalences among others. We will also consider a slight weakening of the Kadison–Singer problem called the Sundberg problem. Then we will look at the recent advances on another deep problem in frame theory called the Paulsen problem. In particular, we will see that this problem is also equivalent to a fundamental open problem in operator theory. Namely, if a projection on a finite dimensional Hilbert space has a nearly constant diagonal, how close is it to a constant diagonal projection?

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previous chapter Finite Frames and Filter Banks
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Metadata
Title
The Kadison–Singer and Paulsen Problems in Finite Frame Theory
Author
Peter G. Casazza
Copyright Year
2013
Publisher
Birkhäuser Boston
Book
Finite Frames

Print ISBN: 978-0-8176-8372-6

Electronic ISBN: 978-0-8176-8373-3

Copyright Year: 2013

DOI
https://doi.org/10.1007/978-0-8176-8373-3_11

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