Compressed Resolvents, Schur Functions, Nevanlinna Families and Their Transformations
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Complex Analysis and Operator Theory
Compressed Resolvents, Schur Functions, Nevanlinna Families and Their Transformations Yu. M. Arlinski˘ı1 Received: 7 September 2019 / Accepted: 29 July 2020 / Published online: 9 August 2020 © Springer Nature Switzerland AG 2020
Abstract We study certain transformations of Nevanlinna families and Schur class operatorvalued functions, construct their realizations by means of compressed resolvents and by the transfer functions of conservative systems, and find fixed points of such transformations. In particular, we characterize some automorphic Schur functions and corresponding them Nevanlinna families, including periodic and scale invariant. Keywords Passive system · Transfer function · Nevanlinna function · Schur function · Redheffer product · Automorphic function Mathematics Subject Classification Primary 47A48 · 93B28 · 93C25; Secondary 47A56 · 93B20
1 Introduction Notations We use the symbols dom T , ran T , ker T for the domain, the range, and the null-subspace of a linear operator T . The closures of dom T , ran T are denoted by dom T , ran T , respectively. The identity operator in a Hilbert space H is denoted by I and sometimes by IH . If L is a subspace, i.e., a closed linear manifold in H, the orthogonal projection in H onto L is denoted by PL . The notation T L means the restriction of a linear operator T on the set L ⊂ dom T . The resolvent set of T is denoted by ρ(T ). The linear space of bounded operators acting between Hilbert spaces H and K is denoted by B(H, K) and the Banach algebra B(H, H) by B(H). Throughout
Communicated by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz. This article is part of the topical collection “Linear Operators and Linear Systems” edited by Sanne ter Horst, Dmitry S. Kaliuzhnyi-Verbovetskyi and Izchak Lewkowicz.
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Yu. M. Arlinski˘ı [email protected] Stuttgart, Germany
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this paper we consider separable Hilbert spaces over the field C of complex numbers. C+ /C− denotes the open upper/lower half-plane of C, R denotes the set of real numbers, R+ := [0, +∞), Z and N are the sets of integers and natural numbers, N0 := N ∪ {0}, D = {z ∈ C : |z| < 1} is the unit disk, T = {ζ ∈ C : |ζ | = 1} is the unit circle. By S(H1 , H2 ) we denote the Schur class (the set of all holomorphic and contractive B(H1 , H2 )-valued functions on the unit disk) and S(H) := S(H, H). For a contraction T ∈ B(H, K) the defect operator (I − T ∗ T )1/2 is denoted by DT and DT := ran DT . For defect operators one has the commutation relations T DT = DT ∗ T , T ∗ DT ∗ = DT T ∗ . For general treatments and various standard properties of linear relations used in this paper we refer to [3,26,27,37]. Let H be a complex Hilbert space and let M be a proper subspace of H. If U is a unitary operator in H, then the compressed resolvent PM (IH − zU )−1 M takes the form (1.1) PM (IH − zU )−1 M = (IM − z(z))−1 , z ∈ D, where belongs to the Schur class S(M) of B(M)-valued contractive operatorfunction
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