Abstract or Additional Information
In this talk we present an extension of a probabilistic result of Marcus, Spielman, and Srivastava, which resolved the Kadison-Singer problem, for block diagonal positive semidefinite random matrices. We use this result to show several selector results, which generalize their partition counterparts. This includes a selector form of Weaver’s $KS_r$ conjecture for block diagonal trace class operators, which extends a selector result for Bessel sequences, or equivalently rank one matrices, due to Londner and the author. We also show a selector variant of Feichtinger’s conjecture for a (possibly infinite) collection of Bessel sequences, extending earlier results for a single Bessel sequence. We prove a generalization of the $R_\epsilon$ conjecture of Casazza, Tremain, and Vershynin for infinite collection of equal norm Bessel sequences. In particular, our selector result yields a conjectured asymptotically optimal bound for a single Bessel sequence in terms of Riesz sequence tightness parameter.