On the spectral Nevanlinna–Pick problem

Abstract

Abstract. We give several characterizations of the symmetrized n-disc G n which generalize to the case n ≥ 3 the characterizations of the symmetrized bidisc that were used in order to solve the two-point spectral Nevanlinna-Pick problem in M 2 (C). Using these characterizations of the symmetrized n-disc, which give necessary and sufficient conditions for an element to belong to G n , we obtain necessary conditions of interpolation for the general spectral Nevanlinna-Pick problem. They also allow us to give a method to construct analytic functions from the open unit disc of C into G n and to obtain some of the complex geodesics on G n .

Some New Observations on Interpolation in the Spectral Unit Ball

“…In this regard, a new idea idea was introduced by Agler & Young in the paper [1]. This idea was further developed over several works -notably in [2], in the papers [7] and [8] by Costara, and in David Ogle's thesis [13]. It can be summarised in two steps as follows:…”

“…Since we would like to focus on the matricial interpolation problem, we will paraphrase the results from [13] and [8] in the language of non-derogatory matrices. Given an n × n complex matrix W , let its characteristic polynomial χ W (z) = z n + n j=1 (−1) j s j (W )z n−j , and define the rational function…”

“…Then, the most general statement that is known about (*) is: Result 1.2 (paraphrased from [13] and [8]). Let ζ 1 , .…”

“…We have opted to rely on the notation of [8]. This leads to a slight discrepancy between our definition of p n in (2.2) and that in [11].…”

The Lempert function of the symmetrized polydisc in higher dimensions is not a distance

“…(Thus, all the introduced invariant functions are continuous (in both variables) for D = G n .) Even more, G n is c G n -finitely compact (see Corollary 3.2 in [4]). …”