On non-self-conjugate operator ideals and complex structures on Banach spaces

• Grant number: 24/16659-6
• Support Opportunities: Scholarships in Brazil - Doctorate
• Start date: March 01, 2025
• End date: September 30, 2028
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: Valentin Raphael Henri Ferenczi
• Grantee: Pedro de Oliveira Emerick
• Host Institution: Instituto de Matemática e Estatística (IME). Universidade de São Paulo (USP). São Paulo , SP, Brazil
• Associated research grant: 23/12916-1 - Geometry of Banach spaces, AP.TEM

Abstract

The project is dedicated to the study of operator ideals in the sense of Pietsch (Operator Ideals, North-Holland, Amsterdam, 1980) and the theory of complex structures in Banach spaces. In particular, the aim is to address certain questions regarding non-self-conjugate operator ideals. Given an operator ideal J and its conjugate conj(J), how small can the intersection between J and conj(J) be? Could it be contained in the ideal of compact operators? Or even be a proper ideal? As a starting point, the two examples of non-self-conjugated ideals presented by Ferenczi (Math. Ann. 387: 1043-1072, 2023) described below are considered.Given F, the complex space totally incomparable with its conjugate studied by Ferenczi (Adv. Math. 231(1): 462-488, 2007), the ideal clos(Op(G)) is not self-conjugate, where Op(X) denotes the family of X-factorable operators, clos(X) the closure of X, and G is the space of 2-summable sequences with entries in F. For the second example, consider Z2(w) as defined by Kalton (Can. Math. Bull. 38(2): 218-222, 1995), which is a variation of the classical Kalton-Peck space Z2. The ideal clos(Op(Z2(w))) is also a non-self-conjugate ideal for non-zero w.The project also aims to discover new techniques for creating examples without relying on non-self-conjugate spaces, such as the spaces F and Z2(w). The study of complex structures in real spaces will be a strategy to find and understand non-self-conjugate ideals. Conversely, properties of these ideals may shed light on the theory of complex structures. Additionally, homological techniques are essential to the geometry of Banach spaces and, by extension, to this project. An exposition on this topic can be found in the book (Homological Methods in Banach Space Theory, Cambridge University Press, Cambridge, 2023).Through the analysis of the two known examples, along with any others that may be identified, the aim is to accumulate sufficient experience and knowledge to prove broader results related to the central question. The goal is to evaluate which possible properties related to the size of the intersection between an operator ideal and its conjugate can be satisfied, determining which are impossible and providing examples for the positive cases.This project is part of the FAPESP thematic project "Geometry of Banach Spaces" within the major areas of "Homological Theory in Banach Spaces" and "Operator Theory". More specifically, it connects with other ongoing research in operator ideals and complex structures, such as the recent work by Corrêa et al. (Twisted Hilbert Spaces Defined by Bi-Lipschitz Maps, 2024).