Realizations of Slice Hyperholomorphic Generalized Contractive and Positive Functions

Abstract

Introduction 1 1.1. Schur analysis 2 1.2. Negative squares 2 1.3. The slice hyperholomorphic case 3 2. Preliminaries 4 2.1. Negative squares and kernels 4 2.2. Slice hyperholomorphic functions 6 3. Slice hyperholomorphic operator-valued functions 8 3.1. Realizations 13 4. The Hardy space of the half-space H + 14 5. The Schauder-Tychonoff fixed point theorem 21 5.1. The Schauder-Tychonoff fixed point theorem 21 5.2. An invariant subspace theorem 23 6. The spaces P(S) 24 7. Realization for elements in S κ (J 1 , J 2 ) 26 8. The space L(Φ) and realizations for generalized positive functions 33 8.1. The indefinite case 33 8.2. The positive case 40 References 42

Schatten class and Berezin transform of quaternionic linear operators

“…The Hardy spaces for p > 2 are considered in . The Bergman spaces are treated in and the Fock spaces in . Weighted Bergman spaces, Bloch, Besov, and Dirichlet spaces on the unit ball are studied in .…”

“…Recent works on Schur analysis in the slice hyperholomorphic setting introduced and studied the quaternionic Hardy spaces H 2 . /, where is the quaternionic unit ball D or the half space H C of quaternions with positive real part [26][27][28]. The Hardy spaces H p .B/ for p > 2 are considered in [29].…”

Wiener Algebra for the Quaternions

“…Returning to the quaternionic setting, see [1], let H + = {p ∈ H : Re p > 0} and Π +,i = H + ∩ C i for any i ∈ S. Let f i = f | Π+,i , where f is a function defined on H + . We let H 2 (Π + , i) denote the space of slice hyperholomorphic functions on H + so that…”

Positive and Generalized Positive Real Lemma for Slice Hyperholomorphic Functions

“…The following definitions first appeared in [7,8]. In the statement, and in this paper in general, H denotes the skew-field of the quaternions.…”