Decomposable bilinear numerical radii

Abstract

Let A be an n-square normal matrix over C , and Q m, n be the set of strictly increasing integer sequences of length m chosen from 1,…, n. For α, β∈ Q m, n denote by A[ α| β] the submatrix obtained from A by using rows numbered α and columns numbered β. For k∈{0,1,…, m} write z. sfnc; α∩ β|= k if there exists a rearrangement of 1,…, m, say i 1,…, i k , i k +1,…, i m , such that α( i j )= β( i j ), j=1,…, k, and { α( i k+1 ),…, α( i m )};∩{ β( i k+1 ),…, β( i m )}= ø. Let ▪ be the group of n-square unitary matrices. Define the nonnegative number ϱ k(A)= max U∈ ▪ | det(U ∗AU) [α|β]| , where | α∩ β|= k. Theorem 1 establishes a bound for ϱ k ( A), 0⩽ k< m−1, in terms of a classical variational inequality due to Fermat. Let A be positive semidefinite Hermitian, n⩾2 m. Theorem 2 leads to an interlacing inequality which, in the case n=4, m=2, resolves in the affirmative the conjecture that ϱ m(A)⩾ϱ m−1(A)⩾⋯⩾ϱ 0(A) .