The angular derivative of an operator-valued analytic function
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 \(z\in \{z\in {\mathbb{C}}|\) \(| Im z| <k Re z\}\), \(z\to \infty\).
scientific article; zbMATH DE number 3944588