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Vladimir G. Troitsky

Publications and source records attributed to Vladimir G. Troitsky.

At least 19 recordsLinked to original sources
arXiv2026

Relative uniform completion of a vector lattice

In the paper, we revisit several approaches to the concept of uniform completion $X^{\mathrm{ru}}$ of a vector lattice $X$. We show that many of these approaches yield the same result. In particular, if $X$ is a sublattice of a uniformly complete vector lattice $Z$ then $X^{\mathrm{ru}}$ may be viewed as the intersection of all uniformly complete sublattices of $Z$ containing $X$. $X^{\mathrm{ru}}$ may also be constructed via a transfinite process of taking uniform adherences in $Z$ with regulators coming from the previous adherences. If, in addition, $X$ is majorizing in $Z$ then $X^{\mathrm{ru}}$ may be viewed as the uniform closure of $X$ in $Z$. We show that $X^{\mathrm{ru}}$ may also be characterized via a universal property: every positive operator from $X$ to a uniformly complete vector lattice extends uniquely to $X^{\mathrm{ru}}$. Moreover, the class of positive operators here may be replaced with several other important classes of operators (e.g., lattice homomorphisms). We also discuss conditions when the uniform adherence of a sublattice equals its uniform closure, and present an example (based on a construction by R.N. Ball and A.W. Hager) where this fails.

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arXiv2024

Uniformly closed sublattices of finite codimension

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.

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arXiv2024

Atomicity of Boolean algebras and vector lattices in terms of order convergence

We prove that order convergence on a Boolean algebra turns it into a compact convergence space if and only if this Boolean algebra is complete and atomic. We also show that on an Archimedean vector lattice, order intervals are compact with respect to order convergence if and only the vector lattice is complete and atomic. Additionally we provide a direct proof of the fact that uo convergence on an Archimedean vector lattice is induced by a topology if and only if the vector lattice is atomic.

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arXiv2023

A representation of sup-completion

It was showed by Donner in 1982 that every order complete vector lattice $X$ may be embedded into a cone $X^s$, called the sup-completion of $X$. We show that if one represents the universal completion of $X$ as $C^\infty(K)$, then $X^s$ is the set of all continuous functions from $K$ to $[-\infty,\infty]$ that dominate some element of $X$. This provides a functional representation of $X^s$, as well as an easy alternative proof of its existence.

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arXiv2021

Order and uo-convergence in spaces of continuous functions

We present several characterizations of uo-convergent nets or sequences in spaces of continuous functions $C(Ω)$, $C_b(Ω)$, $C_0(Ω)$, and $C^\infty(Ω)$, 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.

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arXiv2021

Free Banach lattices under convexity conditions

We prove the existence of free objects in certain subcategories of Banach lattices, including $p$-convex Banach lattices, Banach lattices with upper $p$-estimates, and AM-spaces. From this we immediately deduce that projectively universal objects exist in each of these subcategories, extending results of Leung, Li, Oikhberg and Tursi (\emph{Israel J.\ Math.}~2019). In the $p$-con\-vex and AM-space cases, we are able to explicitly identify the norms of the free Banach lattices, and we conclude by investigating the structure of these norms in connection with nonlinear $p$-summing maps.

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arXiv2019

Vector lattices admitting a positively homogeneous continuous function calculus

We characterize the Archimedean vector lattices that admit a positively homogeneous continuous function calculus by showing that the following two conditions are equivalent for each $n$-tuple $\boldsymbol{x} = (x_1,\ldots,x_n)\in X^n$, where $X$ is an Archimedean vector lattice and $n\in\mathbb N$: - there is a vector lattice homomorphism $Φ_{\boldsymbol{x}}\colon H_n\to X$ such that $Φ_{\boldsymbol{x}}(π_i^{(n)})=x_i$ $(i\in\{1,\ldots,n\})$, where $H_n$ denotes the vector lattice of positively homogeneous, continuous, real-valued functions defined on $\mathbb R^n$ and $π_i^{(n)}\colon\mathbb R^n\to\mathbb R$ is the $i^{\text{th}}$ coordinate projection; - there is a positive element $e\in X$ such that $e\geqslant\lvert x_1\rvert\vee\cdots\vee\lvert x_n\rvert$ and the norm $\lVert x\rVert_e = \inf\bigl\{λ\in[0,\infty)\:\colon\:\lvert x\rvert\leλe\bigr\}$, defined for each $x$ in the order ideal $I_e$ of $X$ generated by $e$, is complete when restricted to the closed sublattice of $I_e$ generated by $x_1,\ldots,x_n$. Moreover, we show that a vector space which admits a `sufficiently strong' $H_n$-function calculus for each $n\in\mathbb N$ is automatically a vector lattice, and we explore the situation in the non-Archimedean case by showing that some non-Archimedean vector lattices admit a positively homogeneous continuous function calculus, while others do not.

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arXiv2016

Locally piecewise affine functions and their order structure

Piecewise affine functions on subsets of $\mathbb R^m$ were studied in \cite{Ovchinnikov:02,Aliprantis:06a,Aliprantis:07a,Aliprantis:07}. In this paper we study a more general concept of a locally piecewise affine function. We characterize locally piecewise affine functions in terms of components and regions. We prove that a positive function is locally piecewise affine iff it is the supremum of a locally finite sequence of piecewise affine functions. We prove that locally piecewise affine functions are uniformly dense in $C(\mathbb R^m)$, while piecewise affine functions are sequentially order dense in $C(\mathbb R^m)$. This paper is partially based on \cite{Adeeb:14}.

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arXiv2015

Uo-convergence and its applications to Cesàro means in Banach lattices

A net $(x_α)$ in a vector lattice $X$ is said to uo-converge to $x$ if $|x_α-x|\wedge u\xrightarrow{\rm o}0$ for every $u\ge 0$. In the first part of this paper, we study some functional-analytic aspects of uo-convergence. We prove that uo-convergence is stable under passing to and from regular sublattices. This fact leads to numerous applications presented throughout the paper. In particular, it allows us to improve several results in [26,27]. In the second part, we use uo-convergence to study convergence of Cesàro means in Banach lattices. In particular, we establish an intrinsic version of Komlós' Theorem, which extends the main results of [35,16,31] in a uniform way. We also develop a new and unified approach to Banach-Saks properties and Banach-Saks operators based on uo-convergence. This approach yields, in particular, short direct proofs of several results in [21,24,25].

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arXiv2015

Spaces of regular abstract martingales

In \cite{Troitsky:05,Korostenski:08}, the authors introduced and studied the space $\mathcal M_r$ of regular martingales on a vector lattice and the space $M_r$ of bounded regular martingales on a Banach lattice. In this note, we study these two spaces from the vector lattice point of view. We show, in particular, that these spaces need not be vector lattices. However, if the underlying space is order complete then $\mathcal M_r$ is a vector lattice and $M_r$ is a Banach lattice under the regular norm.

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arXiv2013

Fremlin Tensor Products of Concavifications of Banach Lattices

Suppose that $E$ is a uniformly complete vector lattice and $p_1,..., p_n$ are positive reals. We prove that the diagonal of the Fremlin projective tensor product of $E_{(p_1)},..., E_{(p_n)}$ can be identified with $E_{(p)}$ where $p = p_1+...+p_n$ and $E_{(p)}$ stands for the $p$-concavification of $E$. We also provide a variant of this result for Banach lattices. This extends the main result of [BBPTT].

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arXiv2012

Irreducible Semigroups of Positive Operators on Banach Lattices

The classical Perron-Frobenius theory asserts that an irreducible matrix $A$ has cyclic peripheral spectrum and its spectral radius $r(A)$ is an eigenvalue corresponding to a positive eigenvector. In Radjavi (1999) and Radjavi and Rosenthal (2000), this was extended to semigroups of matrices and of compact operators on $L_p$-spaces. We extend this approach to operators on an arbitrary Banach lattice $X$. We prove, in particular, that if $\iS$ is a commutative irreducible semigroup of positive operators on $X$ containing a compact operator $T$ then there exist positive disjoint vectors $x_1,...,x_r$ in $X$ such that every operator in $\iS$ acts as a positive scalar multiple of a permutation on $x_1,...,x_r$. Compactness of $T$ may be replaced with the assumption that $T$ is peripherally Riesz, i.e., the peripheral spectrum of $T$ is separated from the rest of the spectrum and the corresponding spectral subspace $X_1$ is finite dimensional. Applying the results to the semigroup generated an irreducible peripherally Riesz operator $T$, we show that $T$ is a cyclic permutation on $x_1,...,x_r$, $X_1=\Span{x_1,...,x_r}$, and if $S=\lim_j b_jT^{n_j}$ for some $(b_j)$ in $\mathbb R_+$ and $n_j\to\infty$ then $S=c(T_{|X_1})^k\oplus 0$ for some $c\ge 0$ and $0\le k<r$. We also extend results of Abramovich et al. (1992) and Grobler (1995) about peripheral spectra of irreducible operators.

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arXiv2009

Almost invariant half-spaces of operators on Banach spaces

We introduce and study the following modified version of the Invariant Subspace Problem: whether every operator T on a Banach space has an almost invariant half-space, that is, a subspace Y of infinite dimension and infinite codimension such that Y is of finite codimension in T(Y). We solve this problem in the affirmative for a large class of operators which includes quasinilpotent weighted shift operators on l_p (1 \le p < \infty) or c_0.

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arXiv2008

A version of Lomonosov's theorem for collections of positive operators

It is known that for every Banach space X and every proper WOT-closed subalgebra A of L(X), if A contains a compact operator then it is not transitive. That is, there exist non-zero x in X and f in X* such that f(Tx)=0 for all T in A. In the case of algebras of adjoint operators on a dual Banach space, V.Lomonosov extended this as follows: without having a compact operator in the algebra, |f(Tx)| is less than or equal to the essential norm of the pre-adjoint operator T_* for all T in A. In this paper, we prove a similar extension (in case of adjoint operators) of a result of R.Drnovsek. Namely, we prove that if C is a collection of positive adjoint operators on a Banach lattice X satisfying certain conditions, then there exist non-zero positive x in X and f in X* such that f(Tx) is less than or equal to the essential norm of T_* for all T in C.

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arXiv2006

Classes of strictly singular operators and their products

V. D. Milman proved in \cite{Milman:70} that the product of two strictly singular operators on $L_p[0,1]$ ($1\le p<\infty$) or on $C[0,1]$ is compact. In this note we utilize Schreier families $§_ξ$ in order to define the class of $§_ξ$-strictly singular operators, and then we refine the technique of Milman to show that certain products of operators from this class are compact, under the assumption that the underlying Banach space has finitely many equivalence classes of Schreier-spreading sequences. Finally we define the class of $§_ξ$-hereditarily indecomposable Banach spaces and we examine the operators on them.

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arXiv2003

Minimal vectors in arbitrary Banach spaces

We extend the method of minimal vectors to arbitrary Banach spaces. It is proved, by a variant of the method, that certain quasinilpotent operators on arbitrary Banach spaces have hyperinvariant subspaces.

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