# Research Articles

My main research area is Banach space theory, however, I have some papers in real analysis and know some descriptive set theory as it applies to Banach space theory. In particular, I study the geometry of infinite dimensional Banach spaces and bounded linear operators between Banach spaces. Most of my work has focused on constructing Banach spaces with very rigid structure and using methods from descriptive set theory to study operators between Banach spaces.

Since I moved to W&L in 2013, I have made an effort to get undergraduate students involved in research projects and have had some luck solving a few problems related to Tsirelson’s space and Schreier’s space. I’ve used an (*) below to indicate when my coauthor is an undergraduate student.

28. Banach spaces having a complex lattice of closed operator ideals, (with Tomasz Kania and Niels Laustsen), in preparation.

27. The $\lambda$-property in higher order Schreier spaces, (with Hung Chu*), submitted.

26. Genericity and Universality for Operator Ideals, (with Ryan M Causey), submitted.

25. The algebras of bounded operators on the Tsirelson and Baernstein spaces are not Grothendieck spaces, (with Tomasz Kania and Niels Laustsen), to appear in Houston Journal of Mathematics.

24. Time stopping for Tsirelson’s Norm, (with Michael Holt*, Noah Duncan*), Involve 11 (2018), no.5 857–866.

Summary and notes: I initiated this project in the summer of 2015 working with my undergraduate coauthors M. Holt and N. Duncan. This result is contained in Michael’s undergraduate thesis written under my direction (Michael and Noah graduated in 2016). The seminal book on Tsirelson’s space (Casazza-Shura) contains a computer code that computes the Tsirelson norm of finitely supported vectors. This is a very complex computation. For his thesis, Michael wrote Python code that also performs this computation. We were interested in rewriting this code in order to study two problems that are stated in the book. The first has to do with explicitly defining extreme points of the ball of Tsirelson’s space and the second is related to the so-called stopping time for Tsirelson’s norm. Tsirelson’s norm of a vector $x$ is defined to be the maximum of the values of $\|x\|_k$ where $\|\cdot\|_k$ is an increasing sequence of norms. For each n we want to find an upper bound on the quantity $j(n)$ defined to be the minimum $k$ so that for every vector $x$ with max support at most $n$ the Tsirelson norm is given by $\|x\|_{j(n)}$. In this paper we prove that this $j(n)$ is on the order of $O(n^{1/2})$. We do not know if this is sharp but it improves the bound in the Casazza Shura (which was $O(n)$).

23. Extreme points for Combinatorial Banach space, (with Michael Holt*, Noah Duncan* and James Quigley*) submitted.

22. Quantitative Factorization of weakly compact, Rosenthal, and ξ-Banach-Saks operators, (with Ryan M Causey), to appear in Mathematica Scandinavica.

21. Classes of operators determined by ordinal indices, (with Ryan M Causey, Daniel Freeman, and Ben Wallis),  J. Functional Analysis, 271 (2016), no. 6, 1691–1746.

20. On a generalization of Bourgain’s tree index, (with Ryan M Causey), to appear in Houston J. Math.

19. On Scottish Book Problem 157, (with Paul Humke and Trevor Richards), Real Analysis Exchange, 41 (2016), no. 2, 331-346.

18. Arbitrarily distortable Banach spaces of higher order, (with Ryan Causey and Pavlos Motakis), Israel J. Math. 214 (2016), no. 2, 553-581.

17. The stabilized set of p’s in Krivine’s theorem can be disconnected, (with Daniel Freeman and Pavlos Motakis), Adv. Math 281 (2015), 553-577.

16. Upper and lower estimates for Schauder frames and atomic decompositions, (with Daniel Freeman and R. Liu) Fund. Math. 231 (2015), 161-188

15. Uniformly factoring weakly compact operators, (with Daniel Freeman) J. Functional Anal. 266 (2014), no. 5, 2921-2943.

14. Operators on $\mathcal{L}_\infty$ spaces, (with Lon Mitchell) J. Math. Anal. Appl. 413 (2014), no. 2, 616-621.

13. The strictly singular operators in Tsirelson like reflexive spaces, (with Spiros A. Argyros and Pavlos Motakis) Illinois J. Math. 57 (2013), no. 4, 1173–1217.

12. On spaces admitting no $\ell_p$ or $c_0$ spreading model, (with Spiros A. Argyros) Positivity 17 (2013), no. 2, 265-282.

11. An extremely non-homogeneous weak Hilbert space, (with Spiros A. Argyros and Th. Raikoftsalis) Trans. Amer. Math. Soc. 364 (2012), 5015-5033.

10. Ordinal ranks on weakly compact and Rosenthal operators, (with Daniel Freeman) Extracta Math. 26(2) (2011), 173-194

9. A weak Hilbert space with few symmetries, (with Spiros A. Argyros and Th. Raikoftsalis), C.R. Math. Acad. Sci. Paris. 348:1293-1296, 2010.

8. Operators on the spaces of Bourgain and Delbaen, (with Lon Mitchell) Quaestiones Mathematicae 33 (2010), 443–448.

Summary and history: In this short paper, we construct a single operator. Seems a bit silly to write a paper constructing one operator, so let me explain. At the time I started working with Lon Mitchell (at VCU at the time), the construction of a scalar-plus-compact space had just been announced by Argyros and Haydon. This construction reinforced the importance of the construction of J. Bourgain and F. Delbaen from the early 1980s. Indeed, there were relatively few papers written that actually deal with this details of this construction. Since this space (or rather these classes of spaces) do not have an unconditional basis it’s tricky to define operators on the whole space. Therefore, I figured that constructing such an operator would be a good way to understand the spaces. It turned out to be about as complicated as I imagined. Luckily, after some work, we were able to construct a single shift-like operator on the space that could also be defined on modifications of the BD-spaces. The longer term goal (suggested by George when I was a student) was to use this construction to show that $\ell_\infty$ embedds in the space of operators or show that the space of operators is non-separable. We eventually decided to publish this short note in case it this idea could be used in some way to produce futher generalizations of the Arygros-Haydon example. As it happens, a few years later, M. Tarbard found our paper and was able to produce an Arygros-Haydon space on which a (highly non-trivial) modification of our operator was defined. His construction solved several problems left open by the Arygros-Haydon example and was important component of Tarbard’s Ph.D. thesis (written with Haydon at Oxford).

7. On strictly singular operator between separable Banach spaces, (with Pandelis Dodos) Mathematika, 56 (2010), no. 2, 285-304.

6. An ordinal indexing of the space of strictly singular operators, Israel J. of Math., 181 (2011), 47-60.

Summary and history: When I visited Athens, Greece for the first time in 2009, I presented the work on consingular operators (with George) and was asked whether if X and Y are totally incomparable Banach spaces there is a countable $\xi$ so that every strictly singular opertors between X and Y was $\mathcal{S}_\xi$ strictly singular. In the process of solving this problem, I realized it was helpful to realize the space of operators between separable Banach spaces as a Standard Borel space endowed with the topology of point-wise convergence; this observation is also contained in Kechris’ book. This allowed me to answer the above question easily and, in addition, show that if a space has few operators (every operator on the space is a multiple of the identity plus a strictly singular operator) then the same conclusion holds. The outstanding question, that I was unable to answer at the time, was whether the rank induced was a $\Pi_1^1$-rank (see above). This paper is relatively short and the proofs are not too hard, so I think the real contribution introducing the ideas of using descriptive set theory to study classes of operators between spaces instead of just classes of spaces. This is a theme in several of the papers above.

5. Operators on asymptotic $\ell_p$ spaces which are not compact perturbation of a multiple of the identity, Illinois J. Math, 52 (2009), no. 2, 515-532.

Summary and history: This is my first (out of only two) solo authored paper. I wrote this paper during my year at Amherst College as a visitor. After the construction from my thesis, there was the outstanding problem as to whether every operator on the space constructed in my thesis (and the space of Deliyanni-Manoussakis) admitted a strictly singular non-compact operator. A few years earlier, George and Thomas Schlumprecht constructed such an operator on the Gowers-Maurey space and I. Gasparis had done so on the certain asymptotic-$\ell_1$ HI spaces of Argyros and Deliyanni. In this paper I verify that such constructions of operators are robust in the sense that it can be carried out on the asymptotic $\latex \ell_p$ HI spaces. So this is the expected negative result in this class.

4. Modifications of Thomae’s function and differentiability, (with James Roberts and Craig Stevenson) Amer. Math. Monthly, 116 (2009), no. 6, 531-535.

Summary and history: This paper has an interesting story. The standard real function that is continuous on the irrational but not on the rationals is called the Thomae function. An natural problem that Craig (my friend from St. Mary’s Colllege and coauthor) ask me is if Thomae’s function is differentiable on the irrationals. I didn’t know but soon after he asked me, he came up with a cute argument to show that it was not differentiable on any irrational. A bit later, Iwas at a happy hour and asked Jim Roberts if you could modify Thomae’s function to make a function that is differentiable on the irrationals but still not continuous on the rationals. He made an off-handed comment that it must be easy (send m/n to 1/n^2). I nodded, later thought about it, but couldn’t figure it out. I decided to stop by his office and ask him to prove it for me. He couldn’t. A bit later he come to find me in my office to show me what became the main result of the paper. Turns out that any modification of the type we were thinking would have a uncountable dense set on which it is non-differentiable. Cool stuff. I just filed this result away for awhile. A few years later I was watching a discrete math talk at VCU and the speaker mentioned something well-known but that I didn’t know about called the irrationality measure of a number. The definition turned out to be exactly what was needed to determine the irrational numbers at which certain modificaitons of the Thomae function would be differentiable. So we decided to write up this short note with these observations. After the paper appear in the Monthly, I started receiving many emails about the paper and learned that most of the results had appearred before (and in the Monthly!). Well, that is a bit embarrassing but it seems to happen more often than one would think. We eventually published an Editor’s Note with some additional references. The curious thing after we started digging was that we were probably the 3rd or 4th paper to rediscover this connection in the Monthly. Maybe one day soon there will be another rediscovery. Lesson learned.

3. Descriptive set-theoretic methods applied to strictly singular and strictly cosingular operators, (with George Androulakis) Quaestiones Mathematicae, 31 (2008), 151-161.

Summary and history: This was my first paper that uses some methods from descriptive set theory. In work with Dodos, Sirotken and Troitsky, George defined and studied a subclass of the strictly singular operators called the $\mathcal{S}_{\xi}$-strictly singular operators for a countable ordinal $\latex \xi$. This paper has now been cited numerous times and the idea of refining operators ideals in this way has been used in many subsequent papers. In this original paper, the authors wanted to generalize an old result by V. Milman regarding products of strictly singular operators and, in particular, giving conditions that will imply certain products are compact. This idea was eventually used by Argyros and Motakis to construct a reflexive Banach space on which every operator had a non-trivial invariant subspace. The above paper is much more modest. At a conference in Kent State, Joe Diestel asked George and I if there was a corresponding subclass of operators for the strictly cosingular operators (Pelczynski operators). This is the dual class of strictly singular operators. This problem turns out to be much harder than the strictly singular question. We obtained a partial result, but really our main result is an adaptation of the strictly singular operators definition. The main outstanding question that I can’t solve is whether a strictly cosingular operator between separable Banach spaces has an adjoint that is strictly singular.

2. A hereditarily indecomposable asymptotic $\ell_2$ Banach space, (with George Androulakis) Glasgow Math Journal, 48 (2006) 503-532.

Summary and history: This paper contains the main result of my Ph.D. thesis. With the long-term goal of constructing a weak Hilbert HI space, I started reading the literature on HI spaces in 2004 after the paper below was finished. I read the seminal paper of Gowers and Maurey and tried to modify this example to construct a Gowers-Maurey type space with an upper $\ell_2$ estimate. This example would be a sort of $2$-convexified Gowers-Maurey space. While working on this problem, I found that a similar space was constructed by N. Dew in his Ph.D. thesis from Oxford written under Richard Haydon. He had not published this result and it could only be found in his thesis. After reading more of the surrounding literature, I noticed there was an outstanding problem (due to both Gowers and Maurey) as to whether there was an asymptotic $\ell_2$ HI space. The original (deep) example of an asymptotic $\ell_1$ HI space had been constructed in the mid-1990s by Argyros and Deliyanni. Later, I. Gasparis wrote a paper simplifying this construction and producing uncountably many totally incomparable asymptotic $\ell_1$ HI spaces. George and I were able to read and make the necessary modifications to the $\ell_2$ setting. There are many unexpected technical difficulties in this construction especially in adding the required `special functionals.’ After our paper was posted we learned of another construction that had not been circulated, satisfying the same property and due to Deliyanni and Manoussakis. Both papers are now published.

1. Embedding $\ell_\infty$ into the space of operators on certain Banach spaces, (with George Androulakis, Stephen Dilworth, and Frank Sanacory) Bull. Lond. Math. Soc., 38 (2006) 979-990.

Summary and discussion: This was my first research project. At the time it was an open question as to whether the algebra of bounded linear operators on a Banach space could be separable. In this paper we introduce a condition called quasi-subsymmetric (qss; weaker than subsymmetric) and prove that if a space $X$ has a basis that is qss then there is an unconditional sequence $(x_n)$ in $X$ and a non-compact map from the qss basis to $\latex (x_n)$. From previous work this implies that the bounded linear operators on $X$ contains $\ell_\infty$ isomorphically (and is therefore non-separable). An outstanding problem from this paper is whether one can show that $\ell_\infty$ embedds in $\mathcal{L}(X)$ if $X$ has a basis isomorphic to its subsequences.