Discontinuity of the Lempert function of the spectral ball

Abstract

Abstract. We give some further criteria for continuity or discontinuity of the Lempert function of the spectral ball Ω n , with respect to one or both of its arguments, in terms of cyclicity of the matrices involved.

Green functions of the spectral ball and symmetrized polydisk

“…Now we must study the homogeneity of the functions σ i (V + A) in terms of the entries of A. This is Lemma 4.2 from [12].…”

Separate continuity of the Lempert function of the spectral ball

“…is continuous at B for any A. They conjecture that this holds for any B ∈ C n , and prove it for n ≤ 3 [6, Proposition 1.4], and the converse statement for all dimensions (see [6,Theorem 1.3]).…”

“…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]). The converse is also true, since l Ωn is an upper semicontinuous function, l Gn is a continuous function and (1) holds.…”

“…The goal of this note is to study the continuity of l Ωn separately with respect to each argument. In [6], the authors looked for matrices B such that l Ωn (A, .) is continuous at B for any A.…”

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

“…The case k = 2 and A 1 cyclic has been studied in [11]. Now we formulate the complete reduction in the case n = 2 (see also [1,2,3]).…”