On the structure of J-Potapov sequences

Abstract

Let J be an m × m signature matrix (i.e., J ∗ = J and J 2 = I m ) and let D ≔ { z ∈ C : | z | < 1 } . Denote P J , 0 ( D ) the set of all meromorphic m × m matrix-valued functions f in D which are holomorphic at 0 and take J-contractive values at all points of D at which f is holomorphic. Then it was proved in [B. Fritzsche, B. Kirstein, U. Raabe, On some interrelations between J-Potapov functions and J-Potapov sequences, OT Series, in press] that the Taylor coefficient sequences of the functions belonging to P J , 0 ( D ) are exactly the infinite J-Potapov sequences of complex m × m matrices. The main goal of this paper is to investigate the inner structure of infinite J-Potapov sequences. It turns out that such sequences have a clear geometric structure. Let ( A j ) j = 0 ∞ be a J-Potapov sequence. Then we will show that for each n ∈ N the matrix A n belongs to a certain matrix ball depending on the sequence ( A j ) j = 0 n - 1 . This observation leads us to the consideration of those J-Potapov sequences ( A j ) j = 0 ∞ for which there exists some k ∈ N such that for j ∈ { k , k + 1 , … } the matrix A j coincides with the center of the corresponding matrix ball. We will call these J-Potapov sequences J-central of order k . It turns out that J-central J-Potapov sequences have a recursive structure. We investigate connections between J-Potapov sequences and their J-Potapov–Ginzburg transforms, which are m × m Schur sequences. In particular, we derive formulas containing explicit interrelations between the parameters of the corresponding matrix balls. An essential consequence of these formulas is the observation that the concept of centrality is invariant with respect to J-Potapov–Ginzburg transform.