Numerical ranges and compressions of S_n-matrices

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) ⊂≠ 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 ∂W(B) ∩ ∂W(A) = ∅, and (3) the maximum value of k for which there is a k-by-k compression of A with all its diagonal entries in ∂W(A) is equal to 2 if n = 2, and [n/2] if n≥3.