Research Projects

The proposed research aims to investigate the geometric, dual, and structural aspects of the quotient lifting property (QLP) and its variants—namely the total quotient lifting property (TQLP), quotient unique lifting property (QULP), and compact quotient lifting property (CQLP)—in the context of Banach space theory. Given a Banach space X and a closed subspace J of X, the pair (X, J) is said to have the QLP if for every bounded linear operator T: E \to X/J , there exists a bounded linear lifting \tilde{T}: E \to X with \|\tilde{T}\|=\|T\| such that \pi \circ \tilde{T} = T, where \pi: X \to X/J is the quotient map. This project will undertake a comprehensive analysis of conditions under which such norm-one liftings exist, particularly in the framework of classical Banach spaces such as c_0, l_\infty, l_1, C(K), and L_1(\mu). We will explore whether proximinality or Chebyshevness implies QLP nd its variants, and whether these notions coincide in finite- and infinite-codimensional settings. Another major focus will be the stability of QLP and its variants under operations such as taking duals, quotients, subspaces, and l_p-direct sums. In connection with operator theory, we will investigate QLP in ideal pairs such as (L(X), K(X)) and their lifting behavior under compact perturbations. A key novelty of this proposal lies in establishing connections between QLP and its dual counterparts, particularly Property-(SU) and Property-(U), which govern the extension of linear functionals. Leveraging the duality between Hahn-Banach-type extension theorems and lifting problems, we aim to provide a unified framework linking these notions to structural projections in X and X^*. We also aim to explore links with Birkhoff-James orthogonality and m-heavy subspaces to understand the geometric implications of lifting properties. This work is expected to yield characterizations and structural results relevant to both geometric functional analysis and operator theory. The outcomes will contribute to a deeper understanding of the duality and approximation phenomena in Banach spaces, providing tools that may have implications for approximation theory, optimization, and mathematical models involving partial liftings.