My Research Overview
Current Research interests - Linear Multilinear Algebra, Functional Analysis, and Operator Algebras
Current Research interests - Linear Multilinear Algebra, Functional Analysis, and Operator Algebras
Recent Project Descriptions
Isomorphism and property recognition problems: How important is a given relation? The question is intentionally vague, and its answer depends very much on the context. One can measure the importance of the relation by using graphs: The vertices, typically (though not always), are all the elements of a given object and two vertices are connected by a directed edge if they are related. This approach concentrates on the relation alone and lefts aside how the relation interacts with other structures/operations inside the object. Having built the graphs on two objects within the same category, one can then examine how close the two objects are if the relation on them behaves in exactly the same way. Technically, if the two graphs are isomorphic, does it imply that the two objects are isomorphic? We call this an isomorphism problem. Currently, I am working with Bojan Kuzma on the relation of Birkhoff-James orthogonality in the category of normed spaces. Recently, we have solved the isomorphism of the real as well as complex finite-dimensional C*-algebra in the category of finite-dimensional C*-algebras ([1,2,3]). Another measure of the importance of the relation is whether it can classify elements with a particular property inside an object. We call this property recognition problem. We have already completed one project about property recognition problem with Alexander Guterman and Svetlana Zhilina which is available on DOI:10.1007/s43036-024-00321-0.
Linear maps preserving parallel pairs: Two bounded linear operators A and B are parallel with respect to a norm ||•|| if ||A+\mu B|| = ||A|| + ||B|| for some scalar \mu with |\mu| = 1. Linear preserver problems concern the characterization of linear operators on the space of bounded linear operators that leave certain functions, subsets, relations, etc., invariant. We have characterized the bijective linear maps sending parallel pairs of matrices to parallel pairs of matrices with respect to the Ky-Fan k-norms (DOI:10.1016/j.laa.2024.01.018) and with respect to k-numerical radius (Arxiv:2408.16066). This is a joint project with Bojan Kuzma; Chi-Kwong Li; and Edward Poon.