Continuity properties of vectors realizing points in the classical field of values

摘要

For an n-by-n matrix A, let f_A be its 'field of values generating function' defined as f_A: x→ x~*Ax. We consider two natural versions of the continuity, which we call strong and weak, of f_A~(-1) (which is of course multivalued) on the field of values F(A). The strong continuity holds, in particular, on the interior of F(A), and at such points z ∈?F(A) which are either corner points, belong to the relative interior of flat portions of @F(A), or whose preimage under f_A is contained in a one-dimensional set. Consequently, f_A~(-1) is continuous in this sense on the whole F(A) for all normal, 2-by-2, and unitarily irreducible 3-by-3 matrices. Nevertheless, we show by example that the strong continuity of f_A~(-1) fails at certain points of ?F(A) for some (unitarily reducible) 3-by-3 and (unitarily irreducible) 4-by-4 matrices. The weak continuity, in its turn, fails for some unitarily reducible 4-by-4 and untiarily irreducible 6-by-6 matrices.