Ievgen (Eugene) Bilokopytov

14. Characterizations of the projection bands and some order properties of the space of continuous functions, preprint (2022), arXiv:2211.11192.​
ABSTRACT: We show that for an ideal H in an Archimedean vector lattice F the following conditions are equivalent:
∙ H is a projection band;
∙ Any collection of mutually disjoint vectors in H, which is order bounded in F, is order bounded in H;
∙ H is an infinite meet-distributive element of the lattice IF of all ideals in F in the sense that ⋂J∈J(H+J)=H+⋂J, for any J⊂IF.
Additionally, we show that if F is uniformly complete and H is a uniformly closed principal ideal, then H is a projection band. In the process we investigate some order properties of lattices of continuous functions on Tychonoff topological spaces.

13. Uniformly closed sublattices of finite codimension (with Vladimir G. Troitsky), preprint (2022), arXiv:2210.08805.​ABSTRACT: The paper investigates uniformly closed subspaces, sublattices, and ideals of finite codimension in Archimedean vector lattices. It is shown that every uniformly closed subspace (or sublattice) of finite codimension may be written as an intersection of uniformly closed subspaces (respectively, sublattices) of codimension one. Every uniformly closed sublattice of codimension n contains a uniformly closed ideal of codimension at most 2n. If the vector lattice is uniformly complete then every ideal of finite codimension is uniformly closed. Results of the paper extend (and are motivated by) results of [AL90a,AL90b] and , as well as Kakutani's characterization of closed sublattices of C(K) spaces.

12. ​Locally solid convergences and order continuity of positive operators, preprint (2022), arXiv:2202.02536.​ABSTRACT: We consider vector lattices endowed with locally solid convergence structures, which are not necessarily topological. We show that such a convergence is defined by the convergence to 0 on the positive cone. Some results on unbounded modification which were only available in partial cases are generalized. Order convergence is characterized as the strongest locally solid convergence in which monotone nets converge to their extremums (if they exist). We partially characterize sublattices on which the order convergence is the restriction of the order convergence of the ambient lattice. We show that homomorphism is order continuous iff it is uo-continuous. Uo convergence is characterized independently of order convergence. We show that on the space of continuous function uo convergence is weaker than the compact open convergence iff the underlying topological space contains a dense locally compact subspace. For a large class of convergences we prove that a positive operator is order continuous if and only if its restriction to a dense regular sublattice is order continuous, and that the closure of a regular sublattice is regular with the original sublattice being order dense in the closure. We also present an example of a regular sublattice of a locally solid topological vector lattice whose closure is not regular.

11. Order and uo-convergence in spaces of continuous functions (with Vladimir G. Troitsky), Topology Appl. 308 (2022), 107999, arXiv:2110.08709.​
ABSTRACT: We present several characterizations of uo-convergent nets or sequences in spaces of continuous functions C(Ω), Cb(Ω), C0(Ω), and C∞(Ω), extending results of [vdW18]. In particular, it is shown that a sequence uo-converges iff it converges pointwise on a co-meagre set. We also characterize order bounded sets in spaces of continuous functions. This leads to characterizations of order convergence.

10. Vector lattice approach to the Riesz and Stone Representation theorems, preprint (2021), arXiv:2104.01576.
ABSTRACT: We present vector-lattice-theoretic proofs of Riesz Representation Theorem and Stone Representation Theorem.

9. Order continuity and regularity on vector lattices and on lattices of continuous functions, to appear in Positivity (2021), arXiv:2103.08776.
ABSTRACT: We give several characterizations of order continuous vector lattice homomorphisms between Archimedean vector lattices. We reduce the proofs of some of the equivalences to the case of composition operators between vector lattices of continuous functions, and so we obtain a characterization of order continuity of such operators. Motivated by this, we investigate various properties of the sublattices of the space C(X), where X is a Tychonoff topological space. We also obtain several characterizations of a regular sublattice of a vector lattice, and show that the closure of a regular sublattice of a Banach lattice is also regular.

8. Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology, J. Math. Anal. Appl. 506 (2022), no. 1, arXiv:2101.06566.
ABSTRACT: In this article we investigate the disjointly non-singular (DNS) operators. Following [8] we say that an operator T from a Banach lattice F into a Banach space E is DNS, if no restriction of T to a subspace generated by a disjoint sequence is strictly singular. We partially answer a question from [8] by showing that this class of operators forms an open subset of L(F,E) as soon as F is order continuous. Moreover, we show that in this case T is DNS if and only if the norm topology is the minimal topology which is simultaneously stronger than the unbounded norm topology and the topology generated by T as a map (we say that T "complements" the unbounded norm topology in F). Since the class of DNS operators plays a similar role in the category of Banach lattices as the upper semi-Fredholm operators play in the category of Banach spaces, we investigate and indeed uncover a similar characterization of the latter class of operators, but this time they have to complement the weak topology.

7. Multiplier algebras of normed spaces of continuous functions, to appear in Mediterranean Journal of Mathematics (2020), arXiv:2008.03700.
ABSTRACT: In this article we investigate some general properties of the multiplier algebras of normed spaces of continuous functions (NSCF). In particular, we prove that the multiplier algebra inherits some of the properties of the NSCF. We show that it is often possible to construct NSCF's which only admit constant multipliers. In order to do that, using a method from [23], we prove that any separable Banach space can be realized as a NSCF over any separable metrizable space. On the other hand, we give a sufficient condition for non-separability of a multiplier algebra.

6. Maximum Modulus Principle For Multipliers and Mean Ergodic Multiplication Operators, to appear in the Journal of Operator Theory (2020), arXiv:2007.11685.
ABSTRACT: The main goal of this note is to show that (not necessarily holomorphic) multipliers of a wide class of normed spaces of continuous functions over a connected Hausdorff topological space cannot attain their multiplier norms, unless they are constants. As an application, a contractive multiplication operator is either a multiplication with a constant, or is completely non-unitary. Additionally we explore possibilities for a multiplication operator to be (weakly) compact and (uniformly) mean ergodic.

5. Which multiplication operators are surjective isometries?, J. Math. Anal. Appl. 580 (2019), no. 1, 123368.
ABSTRACT: Let F be a Banach space of continuous functions over a connected locally compact space X. We present several sufficient conditions on F guaranteeing that the only multiplication operators on F that are surjective isometries are scalar multiples of the identity. The conditions are given via the properties of the inclusion operator from F into C(X), as well as in terms of geometry of F. An important tool in our investigation is the notion of Birkhoff Orthogonality.

4. Principal Minor Assignment, Isometries of Hilbert Spaces, Volumes of Parallelepipeds and Rescaling of Sesquiolomorphic Functions, Linear Algebra Appl. 580 (2019), 37–61, see an extended version at arXiv:1808.04072.
ABSTRACT: In this article we consider the following equivalence relation on the class of all functions of two variables on a set X: we will say that L, M : X × X → C are rescalings if there are non-vanishing functions f, g on X such that M(x,y) = f(x)g(y) L(x,y), for any x,y in X. We give criteria for being rescalings when X is a topological space, and L and M are separately continuous, or when X is a domain in Cn and L and M are sesqui-holomorphic.
A special case of interest is when L and M are symmetric, and f = g only has values ±1. This relation between M and L in the case when X is finite (and so L and M are square matrices) is known to be characterized by the equality of the principal minors of these matrices. We extend this result for the case when X is infinite. As an application we characterize restrictions of isometries of Hilbert spaces on weakly connected sets as the maps that preserve the volumes of parallelepipeds spanned by finite collections of vectors.

3. Continuity and Holomorphicity of Symbols of Weighted Composition Operators, Complex Anal. Oper. Theory 13 (2019), no. 3, 1441-1464.
ABSTRACT: The main problem considered in this article is the following: if F, E are normed spaces of continuous functions over topological spaces X and Y respectively, and ω: Y → C and Ф: Y → X are such that the weighted composition operator with symbols ω and Ф is continuous from F into E, when can we guarantee that both ω and Ф are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over complex manifolds. Additionally, we consider the most basic properties of the weighted composition operators, which only have been proven before for more concrete function spaces.

2. Isometry invariant Finsler metrics on Hilbert Spaces, Archivum Mathematicum 53 (2017), no. 3, 141-153.
ABSTRACT: In this paper we study isometry-invariant Finsler metrics on inner product spaces over R or C, i.e. the Finsler metrics which do not change under the action of all isometries of the inner product space. We give a new proof of the analytic description of all such metrics. In this article the most general concept of the Finsler metric is considered without any additional assumptions that are usually built into its definition. However, we present refined versions of the described results for more specific classes of metrics, including the class of Riemannian metrics. Our main result states that for an isometry-invariant Finsler metric the only possible linear maps under which the metric is invariant are scalar multiples of isometries. Furthermore, we characterize the metrics invariant with respect to all linear maps of this type.

1. Transitive maps on topological spaces (with S. F. Kolyada), Ukrainian Math. J. 65 (2014), no. 9, 1293-1318.
ABSTRACT: In the present paper, we consider the problem of existence of nonequivalent definitions of topological transitivity, which is a classical problem of the topological dynamics. In particular, we use the fact that all available definitions of this kind imply a condition imposed on the dynamical system. The main result of our investigations is the complete classification of these dynamical systems.

15. Invariant Hermitean Metric Generated By Reproducing Kernel Hilbert Spaces, preprint (2018), arXiv:1711.05211.
ABSTRACT: For a Reproducing Kernel Hilbert Space on a complex domain we give a formula that describes the Hermitean metrics on the domain which are pull-backs of some metric on the (dual of) the RKHS via the evaluation map. Then we consider the question when such metrics are invariant with respect to the group of automorphisms of the domain. First we approach the problem by considering a stronger property, demanding that the original metric on the (dual of) the RKHS is invariant with respect to all (adjoints of) composition operators, induced by automorphisms. However, we show that only the trivial metric satisfies this property. Then we characterise RKHS's for which the Bergman metric analogue is automorphism-invariant.

16. Non-Euclidean Universal Approximation (with Anastasis Kratsios), NeurIPS -Conference on Neural Information Processing Systems 2020 (paper on Machine Learning), arXiv:2006.02341.
ABSTRACT: Modifications to a neural network's input and output layers are often required to accommodate the specificities of most practical learning tasks. However, the impact of such changes on architecture's approximation capabilities is largely not understood. We present general conditions describing feature and readout maps that preserve an architecture's ability to approximate any continuous functions uniformly on compacts. As an application, we show that if an architecture is capable of universal approximation, then modifying its final layer to produce binary values creates a new architecture capable of deterministically approximating any classifier. In particular, we obtain guarantees for deep CNNs, deep ffNN, and universal Gaussian processes. Our results also have consequences within the scope of geometric deep learning. Specifically, when the input and output spaces are Hadamard manifolds, we obtain geometrically meaningful feature and readout maps satisfying our criteria. Consequently, commonly used non-Euclidean regression models between spaces of symmetric positive definite matrices are extended to universal DNNs. The same result allows us to show that the hyperbolic feed-forward networks, used for hierarchical learning, are universal. Our result is also used to show that the common practice of randomizing all but the last two layers of a DNN produces a universal family of functions with probability one.

17. Ideal Normed Spaces of Continuous Functions, in preparation.

18. Geometry of Normed Spaces of Continuous Functions, and Intrinsic Operators, in preparation.

19. Locally solid convergences and the center of a vector lattice, in preparation.

20. Horizontal convergence on a vector lattice, in preparation.