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# Revisiting the Bourgain-Tzafriri restricted invertibility theorem

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## Abstract

Wewill give somenew techniques for working with problems surrounding the Bourgain- Tzafriri Restricted Invertibility Theorem. First we show that the parameters which work in the theorem for allT � 2 √ 2 closely approximate the parameters which work for all operators. This yields a generalization of the theorem which simultaneously does restricted invertibility on a small partition of the vectors and yields a direct proof that the Bourgain-Tzafriri Conjecture is equivalent to the Feichtinger Conjecture. We also fill in two gaps in the theory involving the relationship between paving results for norm one operators with zero diagonal and restricted invertibility results.
Operators
and
Matrices
Volume 3, Number 1 (2009), 97–110
REVISITING THE BOURGAIN–TZAFRIRI
RESTRICTED INVERTIBILITY THEOREM
PETER G. CASAZZA AND JANET C. TREMAIN
Abstract. We will give some new techniques for working with problems surrounding the Bourgain-
Tzafriri Restricted Invertibility Theorem. First we show that the parameters which work in the
theorem for all T22 closely approximate the parameters which work for all operators.
This yields a generalization of the theorem which simultaneously does restricted invertibility on
a small partition of the vectors and yields a direct proof that the Bourgain-Tzafriri Conjecture
is equivalent to the Feichtinger Conjecture. We also ll in two gaps in the theory involving the
relationship between paving results for norm one operators with zero diagonal and restricted
invertibility results.
Mathematics subject classication (2000): Primary: 46B03, 46B07, 47A05.
Keywords and phrases:Restricted invertibility theorem, Bourgain-Tzafriri conjecture, Feichtinger con-
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Paper OaM-03-04
... Over the 55 year history of the Kadison-Singer Problem, a significant amount of research was generated resulting in a number of partial results as well as a large number of equivalent problems. These include the Anderson Paving Conjectures [2,3,4], the Akemann-Anderson Projection Paving Conjecture [1], the Weaver Conjectures [45], the Casazza-Tremain Conjecture [23], the Feichtinger Conjecture [14], the R ǫ -Conjecture [23], the Bourgain-Tzafriri Conjecture [23], the Sundberg Problem [20]. Many directions for approaching this problem were proposed and solutions were given for special cases: All matrices with positive coefficients are pavable [29] as are all matrices with "small" coefficients [11]. ...
... Over the 55 year history of the Kadison-Singer Problem, a significant amount of research was generated resulting in a number of partial results as well as a large number of equivalent problems. These include the Anderson Paving Conjectures [2,3,4], the Akemann-Anderson Projection Paving Conjecture [1], the Weaver Conjectures [45], the Casazza-Tremain Conjecture [23], the Feichtinger Conjecture [14], the R ǫ -Conjecture [23], the Bourgain-Tzafriri Conjecture [23], the Sundberg Problem [20]. Many directions for approaching this problem were proposed and solutions were given for special cases: All matrices with positive coefficients are pavable [29] as are all matrices with "small" coefficients [11]. ...
... Over the 55 year history of the Kadison-Singer Problem, a significant amount of research was generated resulting in a number of partial results as well as a large number of equivalent problems. These include the Anderson Paving Conjectures [2,3,4], the Akemann-Anderson Projection Paving Conjecture [1], the Weaver Conjectures [45], the Casazza-Tremain Conjecture [23], the Feichtinger Conjecture [14], the R ǫ -Conjecture [23], the Bourgain-Tzafriri Conjecture [23], the Sundberg Problem [20]. Many directions for approaching this problem were proposed and solutions were given for special cases: All matrices with positive coefficients are pavable [29] as are all matrices with "small" coefficients [11]. ...
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