Abstract
Graham, Hamada, Kohr and Kohr studied the normalized time \(T\) reachable families \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) of the Loewner differential equation, which are generated by the Caratheodory mappings with values in a subfamily \(\Omega \) of the Caratheodory family \({\mathcal {N}}_A\) for the Euclidean unit ball \({\mathbb {B}}^n\), where \(A\) is a linear operator with \(k_+(A)<2m(A)\) (\(k_+(A)\) is the Lyapunov index of \(A\) and \(m(A)=\min \{\mathfrak {R}\left\langle Az,z\right\rangle \big |z\in {\mathbb {C}}^n,\Vert z\Vert =1\}\)). They obtained some compactness and density results, as generalizations of related results due to Roth, and conjectured that if \(\Omega \) is compact and convex, then \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\) is compact and \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},ex\,\Omega )\) is dense in \(\widetilde{\mathcal {R}}_T(id_{{\mathbb {B}}^n},\Omega )\), where \(ex\,\Omega \) denotes the corresponding set of extreme points and \(T\in [0,\infty ]\). We confirm this, by embedding the Caratheodory mappings in a suitable Bochner space.
Talk to us
Join us for a 30 min session where you can share your feedback and ask us any queries you have
Disclaimer: All third-party content on this website/platform is and will remain the property of their respective owners and is provided on "as is" basis without any warranties, express or implied. Use of third-party content does not indicate any affiliation, sponsorship with or endorsement by them. Any references to third-party content is to identify the corresponding services and shall be considered fair use under The CopyrightLaw.