NUMERICAL RANGES AND COMPRESSIONS OF S-n-MATRICES

Abstract

Let A be an n-by-n (n >= 2) S-n-matrix, that is, A is a contraction with eigenvalues in the open unit disc and with rank (I-n - A* A) = 1, and let W(A) denote its numerical range. We show that (1) if B is a k-by-k (1 <= k < n) compression of A, then W(B) not subset of W (A), (2) if A is in the standard upper-triangular form and B is a k-by-k (1 <= k < n) principal submatrix of A, then partial derivative W(B) boolean AND partial derivative W(A) = empty set, and (3) the maximum value of k for which there is a k-by-k compression of A with all its diagonal entries in partial derivative W(A) is equal to 2 if n = 2, and [n/2] if n >= 3.