A characterization of idempotent Schur multipliers

Speaker: Daniel Naylor (Cambridge)
Series: Combinatorics Seminar

The Schur multiplier norm of a matrix A is the smallest real number K such that ‖A ∘ M‖ ⩽ K‖M‖ holds for all matrices M, where A ∘ M denotes the Hadamard product and ‖·‖ denotes the operator norm. A {0,1}-matrix is blocky if its support is a union of rectangular blocks, with no two blocks sharing a row or column. We show that every matrix with integer entries and Schur multiplier norm at most K can be written as a signed sum of at most exp(O(K⁴)) blocky matrices, resolving a conjecture of Katavolos and Paulsen. I will also discuss connections with the quantitative idempotent theorem of Green and Sanders. Joint work with Csongor Beke, Marcel Goh, Hamed Hatami and Sean Jaffe.