Diagonals and numerical ranges of weighted shift matrices

Abstract

For any n-by-n matrix A, we consider the maximum number k = k(A) for which there is a k-by-k compression of A with all its diagonal entries in the boundary partial derivative W (A) of the numerical range W (A) of A. For any such compression, we give a standard model under unitary equivalence for A. This is then applied to determine the value of k(A) for A of size 3 in terms of the shape of W (A). When A is a matrix of the form (0 W-1 ... 0 ... w(n-1) w(n) 0 ), we show that k(A) = n if and only if either vertical bar w(1)vertical bar = ... = vertical bar w(n)vertical bar or n is even and vertical bar w(1)vertical bar = vertical bar w(3)vertical bar = ... = vertical bar w(n-1)vertical bar and vertical bar w(2)vertical bar = vertical bar w(4)vertical bar = ... = lwn For such matrices A with exactly one of the wi's zero, we show that any k, 2 <= k <= n - 1, can be realized as the value of k(A), and determine exactly when the equality k(A) = n - 1 holds. (C) 2012 Elsevier Inc. All rights reserved.