A description of my current research on Vector and Banach lattices is coming. Here is the summary of my studies in 2011-2020.
I work in the area of Normed Spaces of Continuous Functions (NSCF's). It is an axiomatic framework that I try to build to include most of concrete function spaces (Sobolev, Bergman, etc), as well as some existing general theories (such as theory of Reproducing Kernel Hilbert Spaces). I try to relate the following aspects of a NSCF F over a topological space X:
• Topological and geometrical properties of F as a normed space (e.g. completeness, reflexivity, strict convexity, being a Hilbert space, etc);
• Properties of F as a collection of function, in particular, common properties of elements of F (e.g. boundedness, holomorphicity, Lipschitzness, etc);
• The way F "sits" in C(X) (e.g. if the closed unit ball of F is closed in C(X) );
• To which degree F "respects" the relation |f|≤|g|;
• Properties of the inclusion operator from F into C(X) (compactness, weak compactness), and also the operation of evaluation (f, x) → f (x) on F × X;
• Properties of the evaluation map of F and of the norm of the point evaluations on F.
The class of linear operators that capture the very nature of spaces of functions is the class of weighted composition operators (WCO). Indeed, the operations of multiplication and composition can be performed on any collection of functions, while there is a lot of Banach-Stone-type theorems which show that the WCO's are the only operators that preserve various kinds of structure. In my research I try to relate the operator theoretic properties of WCO's (such as invertability or isometricity) with some general properties of their symbols (such as continuity, holomorphicity, injectivity, surjectivity, being a constant).
For more details please see my papers in the "Publications" section of the site. You can also download my thesis here. I am currently working on the extended version so that it can be published as a monograph in some distant future.
Here are the slides from some of my talks (no pauses):
2021, October - "Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology", Functional Analysis Seminar, University of Alberta, Edmonton, AB, Canada.
2021, June - "Multiplication algebras, big and small", CMS summer meeting, online.
2019, December - "Maximum Modulus Principle for Multipliers", CMS Winter Meeting, Toronto, ON, Canada.
2019, October - "Maximum Modulus Principle for Multipliers", Northeastern Analysis Meeting, Syracuse, NY, USA.
2019, June - "From Principal Minor Assignment problem for matrices to characterization of the isometries on Hilbert Spaces", CMS Summer Meeting, Regina, SK, Canada.
2019, May - "From Principal Minor Assignment Problem To Characterization Of The Isometries On Hilbert Spaces, Through Volumes Of Parallelepipeds", Combinatorics Seminar, University of Manitoba.
2019, April - "Multiplication Operators Which Are Surjective Isometries and Birkhoff Orthogonality", AMS Spring Central Sectional Meeting, Hartford, CT, USA.
2019, March - "Multiplication operators which are surjective isometries and geometry of Banach spaces", Functional Analysis Seminar, University of Alberta, Edmonton, AB, Canada.
2018, December - "Multiplication Operators Which Are Surjective Isometries and Birkhoff Orthogonality", CMS Winter Meeting, Vancouver, BC, Canada.
2018, June - "Continuity and Holomorphicity of Symbols of Weighted Composition Operators", Canadian Operator Symposium, Winnipeg, MB, Canada.
2017, October - "Continuity and Holomorphicity of Symbols of Weighted Composition Operators", Northeastern Analysis Meeting, Albany, NY, USA.
2016, June - "Invariant Metrics on a Hilbert Space", PIMS Young Researchers Conference in Mathematics and Statistics, Edmonton, AB, Canada.
2014, May - "Arzela-Ascoli Theorem in different settings", Colloquium Talk, University of Manitoba, Winnipeg, MB, Canada.