PhD. Dissertation Defense: Umutcan Erdur

A PLURIPOTENTIAL THEORY FOR BANACH LATTICE VALUED FUNCTIONS

Umutcan Erdur
Mathematics, PhD Dissertation, 2025

Thesis Jury

Prof. Dr Nihat Gökhan Göğüş(Thesis Advisor), Prof. Dr. Özgür Martin, Prof. Dr. Mert Çağlar, Assoc. Prof. Kağan Kurşungöz, Asst. Prof. Nilay Duruk Mutlubaş

Date & Time: June 13th , 2025 –  10.30 AM

Place: FENS 2019

Keywords : Vector valued Functions, Jensen Measures, Edwards’ Theorem,
Plurisubharmonic Functions, Dirichlet Problem

Abstract

In this thesis, we establish a foundation of a pluripotential theory for functions attaining values in a Banach lattice. For this purpose, we generalize notions such as semi-continuity, subharmonicity, plurisubharmonicity and maximality for vector valued functions,  study their properties and the cones of plurisubharmonic functions. Moreover, we investigate operator valued Jensen measures for a given cone of vector valued functions, and upper, lower envelopes of a given function with respect to the given cone.  With these at our disposal, we prove Edwards' Theorem in  Banach lattice settings. As an result our findings, we provide a Perron method of solution for a Dirichlet Problem for vector valued harmonic/maximal plurisubharmonic functions with continuous boundary data.