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A generalization of Schur functions: applications to Nevanlinna functions, orthogonal polynomials, random walks and unitary and open quantum walks. (English) Zbl 1464.47011

Let \(H_0\) be a finite-dimensional Hilbert space. A matrix-valued analytic function \(f:\mathbb{C}\backslash\mathbb{R}\to B(H_0)\) is called a Nevanlinna function if \(f(z)^*=f(\overline{z})\) and \(\frac{\operatorname{Im} f(z)}{\operatorname{Im} z}\ge 0\) for all \(z\in \mathbb{C}\backslash\mathbb{R}\). The authors prove two integral representations and an operator representation of the form \[ f(z) = -dz^{-1} + PT(1-zQT)^{-1}P, \] with \(T\) a self-adjoint operator on a Hilbert space \(H\) that contains \(H_0\), \(P\) the orthogonal projection of \(H\) onto \(H_0\), \(Q=1-P\), and \(d\ge 0\).
More generally, for \(T:D(T)\to B\) a closed operator on a Banach space \(B\) and \(P\) a bounded projection of \(B\) onto a closed subspace \(B_0\subseteq B\), they define a first return function (FR-function) as \[ f(z)=PT(1-zQT)^{-1}P, \] where \(Q=1-P\), for those \(z\in\mathbb{C}\) for which the resolvent \((1-zQT)^{-1}\) exists as an operator defined everywhere on \(B\). If \(T\) is a self-adjoint operator on some Hilbert space, then \(f\) is a Nevanlinna function, so FR-functions generalize Nevanlinna functions. It is shown that the Schur algorithm for Nevanlinna functions has a natural analogue for FR-functions.
If the Banach space \(B\) is has a decomposition \(B=B_-\oplus B_0\oplus B_+\) into a direct sum of closed subspaces and \(T_L\), \(T_R\) are operators on \(B_L=B_-\oplus B_0\), \(B_R=B_0\oplus B_+\), resp., then (with appropriate projections \(P,P_L,P_R\)) the FR-functions \(f_L\) and \(f_R\) of \(T_L\), \(T_R\), resp., can be composed as
(a)
\(f(z)=f_L(z)+f_R(z)\), which is the FR-function of \(T=(T_L\oplus 0_+) + (0_-\oplus T_R)\),
(b)
\(f(z)=f_L(z)f_R(z)\), which is the FR-function of \(T=(T_L\oplus 1_+)(1_-\oplus T_R)\),
on appropriate domains for the operators and functions. Conversely, these can be viewed as splitting formulas for \(T\) and \(f\).
The last four sections of the paper contain applications of the theory of FR-functions developed in the first part to (i) orthogonal polynomials on the real line (Krushchev formula), (ii) recurrence of random walks, (iii) recurrence of quantum walks, (iv) recurrence of open quantum walks.

MSC:

47A56 Functions whose values are linear operators (operator- and matrix-valued functions, etc., including analytic and meromorphic ones)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47B36 Jacobi (tridiagonal) operators (matrices) and generalizations
60J10 Markov chains (discrete-time Markov processes on discrete state spaces)
81Q99 General mathematical topics and methods in quantum theory
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