Discontinuity of the Lempert Function and the Kobayashi-Royden Metric of the Spectral Ball

Abstract

Some results on the discontinuity properties of the Lempert function and the Kobayashi pseudometric in the spectral ball are given. (2000). Primary 32F45; Secondary 32A07. Mathematics Subject Classification

A local form for the automorphisms of the spectral unit ball

“…Without the fruitful discussions I had with him on this topic, this paper wouldn't have been written. Many equivalent definitions of this notion can be found, for instance in [3,4], or [5,Proposition 3]. We point one out: M is cyclic if and only if for any λ ∈ C, dim Ker(M − λI n ) ≤ 1.…”

Spectral Nevanlinna-Pick and Caratheodory-Fejer problems for $n\geq3$

“…Having in mind Proposition 3 and the equality l Ωn (A 1 , 0) = r(A 1 ) (cf. [10]), it is enough to show that if A 1 ∈ Ω n and r(A 1 ) = inf{|α| : ∃ψ ∈ O(D, G n ) : ψ(0) = 0, ψ(α) = σ(A 1 ), ψ ′ n (0) = 0}, then A 1 has equal eigenvalues. If the characteristic polynomial of A 1 does not have the form x n + a, then this follows as in [10, Proposition 10 (iii)] with an obvious modification in the proof of [10, Lemma 11].…”

Separate continuity of the Lempert function of the spectral ball

“…Theorem 1 below says that the scalar matrices and the matrices satisfying (iii) are the only cases when l Ωn (A, ·) is a continuous function. Then the mentioned above result [5,Proposition 4] shows that (i) implies (iii) and hence the conditions (i), (ii) and (iii) are equivalent.…”

“…In particular, it showed that when both A and B are cyclic (or non-derogatory) matrices, i.e. they admit a cyclic vector (see other equivalent properties in [5]), then equality holds in (1). It follows that l Ωn is continuous on C n × C n , where (see [6,Proposition 1.2]).…”

“…is continuous at B for any B (or simply, continuous on the whole Ω n ). By [5,Proposition 4], for any matrix A ∈ C n with at least two different eigenvalues, the function l Ωn (A, ·) is not continuous at any scalar matrix. On the other hand, l Ωn (0, B) = r(B) and hence l Ωn (A, ·) is a continuous function for any scalar matrix A (since the automorphism Φ λ (X) = (X − λI)(I − λX) −1 of Ω n maps λI n to 0, where λ ∈ D).…”