A study on operators defined in infinite dimensional spaces and applications

• Grant number: 24/04400-8
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: July 01, 2024
• End date: August 31, 2025
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Marta Cilene Gadotti
• Grantee: Laisa Marafon Silva
• Host Institution: Instituto de Geociências e Ciências Exatas (IGCE). Universidade Estadual Paulista (UNESP). Campus de Rio Claro. Rio Claro , SP, Brazil
• Associated scholarship(s): 24/17211-9 - Topological degree and study of nonlinear differential equations, BE.EP.IC

Abstract

Operators defined in infinite-dimensional spaces (Banach and Hilbert) and spectral theory on Hermitian operators will be studied. To do this, we will see some results on continuous n-linear applications, the adjoint and the unitary, normal and Hermitian operators. We will apply this theory to the Fredholm Alternative. Furthermore, using the theory of compact Hermitian linear operators, we want to introduce the Sturm-Liouville Problem, which consists of solving a differential system. Finally, we will study the optimization of functions defined in infinite-dimensional spaces. Here we will need the definition of the derivative for this type of function, we will address the Fréchet and Gâteaux derivatives.