The angular derivative of an operator-valued analytic function
type: Item
instance of: scholarly article
title: The angular derivative of an operator-valued analytic function
zbMATH Open document ID: 0588.47018
author: Ky Fan
published in: Pacific Journal of Mathematics
publication date: 1986-01-01T00:00:00Z
review text: Let \(F: \Pi\) \(\to L(H)\) be an analytic operator function (here \(\Pi =\{z\in {\mathbb{C}}|\) Re z\(>0\}\) and H is a complex Hilbert space). Suppose that Re F(z)\(>0\) for \(z\in \Pi\) and there is a Hermitian operator A satisfying (i) (Re z)\({}^{-1}Re F(z)>A\), \(z\in \Pi\), and (ii) for any \(\epsilon >0\) there is \(z_ 0\in \Pi\) such that \(\| (Re z_ 0)^{-1}Re F(z_ 0)-A\| <\epsilon.\) Then for any \(k>0\) we have \(\lim \| z^{-1}F(z)-A\| =\lim \| (Re z)^{-1}Re F(z)-A\| =\lim \| F'(z)-A\| =0\), where \(z\in \{z\in {\mathbb{C}}|\) \(| Im z| <k Re z\}\), \(z\to \infty\).
Mathematics Subject Classification ID: 47A56
zbMATH DE Number: 3944588
zbMATH Keywords: angular derivative
zbMATH Keywords: analytic operator function
MaRDI profile type: Publication
full work available at URL: pjm.1986.121.67
OpenAlex ID: W1974119767
DOI: 10.2140/PJM.1986.121.67
description: scientific article; zbMATH DE number 3944588