Discrete Norms of a Matrix and the Converse to the Expander Mixing Lemma

We define the discrete norm of a complex m× n matrix A by ‖A‖∆ := max 06=ξ∈{0,1}n ‖Aξ‖ ‖ξ‖ , and show that c √ log h(A) + 1 ‖A‖ ≤ ‖A‖∆ ≤ ‖A‖, where c > 0 is an explicitly indicated absolute constant, h(A) = √ ‖A‖1‖A‖∞/‖A‖, and ‖A‖1, ‖A‖∞, and ‖A‖ = ‖A‖2 are the induced operator norms of A. Similarly, for the discrete Rayleigh norm ‖A‖P := max 06=ξ∈{0,1} 06=η∈{0,1} |ξAη| ‖ξ‖‖η‖ we prove the estimate c log h(A) + 1 ‖A‖ ≤ ‖A‖P ≤ ‖A‖. These estimates are shown to be essentially best possible. As a consequence, we obtain another proof of the (slightly sharpened and generalized version of the) converse to the expander mixing lemma by Bollobás-Nikiforov and BiluLinial. 1. Summary of results For a complex matrix A with n columns, we define the discrete norm of A by ‖A‖∆ := max 0 6=ξ∈{0,1}n ‖Aξ‖ ‖ξ‖ , where the maximum is over all non-zero n-dimensional binary vectors ξ, and ‖ · ‖ denotes the usual Euclidean vector norm. Recalling the standard definition of the induced operator L-norm ‖A‖ := sup 06=x∈Cn ‖Ax‖ ‖x‖ , we see immediately that ‖A‖∆ ≤ ‖A‖, and one can expect that, moreover, the two norms are not far from each other. 2010 Mathematics Subject Classification. Primary: 05C50; Secondary: 15A18, 15A60.