arXiv subjects

Eugene Bilokopytov

Publications and source records attributed to Eugene Bilokopytov.

27 records · Page 2Linked to original sources

arXiv2021

Disjointly non-singular operators on order continuous Banach lattices complement the unbounded norm topology

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 $L\left(F,E\right)$ 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.

math.FA↗

arXiv2020

Non-Euclidean Universal Approximation

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 and deep feed-forward networks. Our results also have consequences within the scope of geometric deep learning. Specifically, when the input and output spaces are Cartan-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. We also provide conditions on a DNN's first (resp. last) few layer's connections and activation function which guarantee that these layers can have a width equal to the input (resp. output) space's dimension while not negatively affecting the architecture's approximation capabilities.

cs.LG↗

arXiv2020

Multiplier algebras of normed spaces of continuous functions

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.

math.FA↗

arXiv2020

Maximum Modulus Principle For Multipliers and Mean Ergodic Multiplication Operators

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.

math.FA↗

arXiv2019

Principal Minor Assignment, Isometries of Hilbert Spaces, Volumes of Parallelepipeds and Rescalling of Sesqui-holomorphic Functions

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 $L,M: X\times X\to \mathbb{C}$ are rescalings if there are non-vanishing functions $f,g$ on $X$ such that $M\left(x,y\right)=f\left(x\right)g\left(y\right) L\left(x,y\right)$, 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 $\mathbb{C}^{n}$ 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 $\pm 1$. 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 to 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.

math.FA↗

arXiv2019

Continuity and Holomorphicity of Symbols of Weighted Composition Operators

The main problem considered in this article is the following: if $\mathbf{F}$, $\mathbf{E}$ are normed spaces of continuous functions over topological spaces $X$ and $Y$ respectively, and $ω:Y\to\mathbb{C}$ and $Φ:Y\to X$ are such that the weighted composition operator $W_{Φ,ω}$ is continuous, when can we guarantee that both $Φ$ and $ω$ are continuous? An analogous problem is also considered in the context of spaces of holomorphic functions over (connected) 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.

math.FA↗

arXiv2019

Which multiplication operators are surjective isometries?

Let $\mathbf{F}$ be a Banach space of continuous functions over a connected locally compact space $X$. We present several sufficient conditions on $\mathbf{F}$ guaranteeing that the only multiplication operators on $\mathbf{F}$ that are surjective isometries are scalar multiples of the identity. The conditions are given via the properties of the inclusion operator from $\mathbf{F}$ into $\mathcal{C}\left(X\right)$, as well as in terms of geometry of $\mathbf{F}$. An important tool in our investigation is the notion of Birkhoff Orthogonality.

math.FA↗

arXiv2018

Invariant Hermitean Metric Generated By Reproducing Kernel Hilbert Spaces

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 studied in \cite{cd} and \cite{arsw} is automorphism-invariant.

math.FA↗

arXiv2017

Isometry invariant Finsler metrics on Hilbert Spaces

In this paper we study isometry-invariant Finsler metrics on inner product spaces over $\mathbb{R}$ or $\mathbb{C}$, 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 characterise the metrics invariant with respect to all linear maps of this type.

math.DG↗