Ethan Hawthorne

Current research project

Preservers on Operator Systems

An isometry between two Banach spaces is an operator that preserves the distance between every two elements of its domain. Through this project we seek to determine the canonical forms of all surjective linear isometries between two specified Banach spaces. Obtaining such a canonical form gives us a clearer picture of the interplay between algebraic and geometrical structures. Our aim is to establish results similar to the Banach-Stone theorem and Kadison’s theorem, but for the tensor products of two Banach spaces. That is, we wish to determine the canonical form of all surjective linear isometries whose domain and codomain are the tensor product of two operator algebras. Thus, this research aims to bridge the gap between the classical study of preservers and the more modern study of operator algebras, which will open up new avenues of research. Once we have established such results we will then look at other closely related preservers to generalise the results.

Biography

I started by PhD in Mathematics at Queen’s University Belfast in October 2019. In June 2019, I graduated from Queen’s University Belfast, obtaining an MSci Mathematics with First Class Honours, and received the A C Dixon Prize for achieving the highest results in Pure Mathematics in my year. My MSci thesis was entitled ‘Isometries’ and as part of it, I studied the Banach-Stone Theorem and Kadison’s Theorem. In October 2019, I was awarded the prestigious Dunville Scholarship from Queen’s for my PhD project based on my passion and motivation for the subject. During summer 2017, I was awarded the Hamilton Prize in Mathematics from the Royal Irish Academy, for being the best Level 2 mathematics student at Queen’s. Finally, in October 2015, I received the STEM Scholarship from Queen’s.

Research interests

Broadly speaking my research interests are in the areas of

• Functional Analysis

• Operator Theory

• Operator Algebra.

  To be more specific, my research focuses on linear preservers, such as surjective linear isometries, on operator algebras.

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