arXiv subjects

Marko Kandić

Publications and source records attributed to Marko Kandić.

At least 19 recordsLinked to original sources

arXiv2026

On the nonnegative rank of positive operators

In this paper we introduce the concept of a nonnegative rank of a positive operator $T\colon X\to Y$ between ordered vector spaces. In the case of nonnegative matrices, our definition agrees with the standard definition of a nonnegative rank. Under some natural and mild assumptions on the cone $Y_+$, we prove that the nonnegative rank and the rank agree whenever the rank is at most two. This can be considered as the infinite-dimensional version of \cite[Theorem 4.1]{CR93}. We also provide an example of a positive rank-three operator on the Banach lattice $C[0,1]$ with an infinite nonnegative rank.

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arXiv2026

On positive automorphisms of algebras of operators on atomic Archimedean vector lattices

Let $X$ be an Archimedean vector lattice. We investigate subalgebras of $\mathscr{L}(X)$ consisting of regular operators that contain all rank-one operators of the form $a \otimes φ_b$, where $a$ and $b$ are atoms of $X$ and $φ_b$ denotes the coordinate functional associated with $b$. Our main result shows that every positive automorphism of such a subalgebra contained in $\mathscr{L}(c_{00}(Λ))$, is necessarily spatial, meaning that it is implemented by a transformation of the form $$ T \mapsto P D\, T\, D^{-1} P^{-1}, $$ where $P$ is a permutation operator and $D$ is a positive diagonal operator. An important tool for this analysis-one that is also of independent interest-is the Kakutani representation theorem, which we use to establish that every finite-dimensional vector subspace of $X$ is order closed.

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arXiv2025

Positive commutators of positive square-zero operators

In this paper we first consider the question which nonnegative matrices are commutators of nonnegative square-zero matrices. Then, we treat infinite-dimensional analogues of these results for operators on the Banach lattices $L^p[0,1]$ and $\ell^p$ ($ 1 \leq p < \infty$). In the last setting we need to extend the notion of the nonnegative rank of a nonnegative matrix.

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arXiv2025

Similarities of subspace lattices in Banach spaces

A collineation of a subspace lattice $\fL$ in a complex Banach space $\eX$ is an invertible operator $S$ on $\eX$ with the property that the image $S\eM$ of a subspace $\eM$ belongs to $\fL$ if and and only if $\eM$ belongs to it. Hence, $S$ is a collineation of $\fL$ if and only if it implements an order automorphism of $\fL$. We study the group $\Col(\fL)$ of all collineations of $\fL$ and its subgroup $\Grp(\Alg(\fL))$ of all invertible operators that fix every subspace in $\fL$. We show that $\Grp(\Alg(\fL))$ is a normal subgroup of $\Col(\fL)$; moreover, if $\fL$ is a reflexive subspace lattice, then $\Col(\fL)$ is the normalizer of $\Grp(\Alg(\fL))$ in the group of all invertible operators on $\eX$. One of the main questions that we consider is whether $\Grp(\Alg(\fL))$ is a complemented subgroup in $\Col(\fL)$. For certain subspace lattices $\fL$, such as some realizations of the diamond or the double triangle, some nests in the space of continuous functions on $[0,1]$, and the classical Volterra nest in $L^1[0,1]$, we characterize the complement of $\Grp(\Alg(\fL))$ in $\Col(\fL)$. On the other hand, for the Volterra nests in $L^p[0,1]$, where $1<p<\infty$, a further study is needed, and we prove only some partial results.

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arXiv2025

Positive self-commutators of positive operators

We consider a positive operator $A$ on a Hilbert lattice such that its self-commutator $C = A^* A - A A^*$ is positive. If $A$ is also idempotent, then it is an orthogonal projection, and so $C = 0$. Similarly, if $A$ is power compact, then $C = 0$ as well. We prove that every positive compact central operator on a separable infinite-dimensional Hilbert lattice $\mathcal H$ is a self-commutator of a positive operator. We also show that every positive central operator on $\mathcal H$ is a sum of two positive self-commutators of positive operators.

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arXiv2024

Artinian and Noetherian vector lattices

In this paper, we study Artinian and Noetherian properties in vector lattices and provide a concrete representation of these spaces. Furthermore, we describe for which Archimedean uniformly complete vector lattices every decreasing sequence of prime ideals is stationary (a property that we refer to as prime Artinian). We also completely characterize the prime ideals in vector lattices of continuous root functions and piecewise polynomials. This is a useful space for studying how having decreasing stationary sequences of prime ideals does not imply having increasing stationary sequences of prime ideals, and vice versa.

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arXiv2023

Commutators greater than a perturbation of the identity

Let $a$ and $b$ be elements of an ordered normed algebra $\mathcal A$ with unit $e$. Suppose that the element $a$ is positive and that for some $\varepsilon>0$ there exists an element $x\in \mathcal A$ with $\|x\|\leq \varepsilon$ such that $$ ab-ba \geq e+x . $$ If the norm on $\mathcal A$ is monotone, then we show $$ \|a\|\cdot \|b\|\geq \tfrac{1}{2} \ln \tfrac{1}{\varepsilon} , $$ which can be viewed as an order analog of Popa's quantitative result for commutators of operators on Hilbert spaces. We also give a relevant example of positive operators $A$ and $B$ on the Hilbert lattice $\ell^2$ such that their commutator $A B - B A$ is greater than an arbitrarily small perturbation of the identity operator.

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arXiv2022

On the diagonal of Riesz operators on Banach lattices

This paper extends the well-known Ringrose theory for compact operators to polynomially Riesz operators on Banach spaces. The particular case of an ideal-triangularizable Riesz operator on an order continuous Banach lattice yields that the spectrum of such operator lies on its diagonal, which motivates the systematic study of an abstract diagonal of a regular operator on an order complete vector lattice $E$. We prove that the class $\mathscr D$ of regular operators for which the diagonal coincides with the atomic diagonal is always a band in $\mathcal L_r(E)$, which contains the band of abstract integral operators. If $E$ is also a Banach lattice, then $\mathscr D$ contains positive Riesz operators.

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arXiv2022

Hyperinvariant subspaces for sets of polynomially compact operators

We prove the existence of a non-trivial hyperinvariant subspace for several sets of polynomially compact operators. The main results of the paper are: (i) a non-trivial norm closed algebra $\mathcal A\subseteq \mathcal B(\mathscr X)$ which consists of polynomially compact quasinilpotent operators has a non-trivial hyperinvariant subspace; (ii) if there exists a non-zero compact operator in the norm closure of the algebra generated by an operator band $\mathcal S$, then $\mathcal S$ has a non-trivial hyperinvariant subspace.

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arXiv2022

Collineations preserving the lattice of invariant subspaces of a linear transformation

Given a linear transformation $A$ on a finite-dimensional complex vector space $\eV$, in this paper we study the group $\Col(A)$ consisting of those invertible linear transformations $S$ on $\eV$ for which the mapping $Φ_S$ defined as $Φ_S\colon \eM\mapsto S\eM$ is an automorphism of the lattice $\Lat(A)$ of all invariant subspaces of $A$. By using the primary decomposition of $A$, we first reduce the problem of characterizing $\Col(A)$ to the problem of characterizing the group $\Col(N)$ of a given nilpotent linear transformation $N$. While $\Col(N)$ always contains all invertible linear transformations of the commutant $(N)'$ of $N$, it is always contained in the reflexive cover $\Alg\Lat(N)'$ of $(N)'$. We prove that $\Col(N)$ is a proper subgroup of $(\Alg\Lat(N)')^{-1}$ if and only if at least two Jordan blocks in the Jordan decomposition of $N$ are of dimension $2$ or more. We also determine the group $\Col(\bdJ_2\oplus \bdJ_2)$.

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arXiv2021

On separability of unbounded norm topology

In this paper, we continue the investigation of topological properties of unbounded norm (un-)topology in normed lattices. We characterize separability and second countability of un-topology in terms of properties of the underlying normed lattice. We apply our results to prove that an order continuous Banach function space $X$ over a semi-finite measure space is separable if and only if it has a $σ$-finite carrier and is separable with respect to the topology of local convergence in measure. We also address the question when a normed lattice is a normal space with respect to the un-topology.

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arXiv2021

Prime ideals and Noetherian properties in vector lattices

In this paper we study the set of prime ideals in vector lattices and how the properties of the prime ideals structure the vector lattice in question. The different properties that will be considered are firstly, that all or none of the prime ideals are order dense, secondly, that there are only finitely many prime ideals, thirdly, that every prime ideal is principal, and lastly, that every ascending chain of prime ideals is stationary (a property that we refer to as prime Noetherian). We also completely characterize the prime ideals in vector lattices of piecewise polynomials, which turns out to be an interesting class of vector lattices for studying principal prime ideals and ascending chains of prime ideals.

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arXiv2018

Relatively Uniformly Continuous Semigroups on Vector Lattices

In this paper we study continuous semigroups of positive operators on general vector lattices equipped with the relative uniform topology $τ_{ru}$. We introduce the notions of strong continuity with respect to $τ_{ru}$ and relative uniform continuity for semigroups. These notions allow us to study semigroups on non-locally convex spaces such as $L^p(\mathbb{R})$ for $0<p<1$ and non-complete spaces such as $Lip(\mathbb{R})$, $UC(\mathbb{R})$, and $C_c(\mathbb{R})$. We show that the (left) translation semigroup on the real line, the heat semigroup and some Koopman semigroups are relatively uniformly continuous on a variety of spaces.

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arXiv2017

Metrizability of minimal and unbounded topologies

In 1987, I. Labuda proved a general representation theorem that, as a special case, shows that the topology of local convergence in measure is the minimal topology on Orlicz spaces and $L_{\infty}$. Minimal topologies connect with the recent, and actively studied, subject of "unbounded convergences". In fact, a Hausdorff locally solid topology $τ$ on a vector lattice $X$ is minimal iff it is Lebesgue and the $τ$ and unbounded $τ$-topologies agree. In this paper, we study metrizability, submetrizability, and local boundedness of the unbounded topology, $uτ$, associated to $τ$ on $X$. Regarding metrizability, we prove that if $τ$ is a locally solid metrizable topology then $uτ$ is metrizable iff there is a countable set $A$ with $\overline{I(A)}^τ=X$. We prove that a minimal topology is metrizable iff $X$ has the countable sup property and a countable order basis. In line with the idea that uo-convergence generalizes convergence almost everywhere, we prove relations between minimal topologies and uo-convergence that generalize classical relations between convergence almost everywhere and convergence in measure.

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arXiv2017

Positive operators as commutators of positive operators

It is known that a positive commutator $C=A B - B A$ between positive operators on a Banach lattice is quasinilpotent whenever at least one of $A$ and $B$ is compact. In this paper we study the question under which conditions a positive operator can be written as a commutator between positive operators. As a special case of our main result we obtain that positive compact operators on order continuous Banach lattices which admit order Pelczyński decomposition are commutators between positive operators. Our main result is also applied in the setting of a separable infinite-dimensional Banach lattice $L^p(μ)$ $(1<p<\infty)$.

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arXiv2017

The countable sup property for lattices of continuous functions

In this paper we find sufficient and necessary conditions under which vector lattice $C(X)$ and its sublattices $C_b(X)$, $C_0(X)$ and $C_c(X)$ have the countable sup property. It turns out that the countable sup property is tightly connected to the countable chain condition of the underlying topological space $X$. We also consider the countable sup property of $C(X\times Y)$. Even when both $C(X)$ and $C(Y)$ have the countable sup property it is possible that $C(X\times Y)$ fails to have it. For this construction one needs to assume the continuum hypothesis. In general, we present a positive result in this direction and also address the question when $C(\prod_{λ\inΛ} X_λ)$ has the countable sup property. Our results can be understood as vector lattice theoretical versions of results regarding products of spaces satisfying the countable chain condition. We also present new results for general vector lattices that are of an independent interest.

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arXiv2016

Topological aspects of order in $C(X)$

In this paper we consider the relationship between order and topology in the vector lattice $C_b(X)$ of all bounded continuous functions on a Hausdorff space $X$. We prove that the restriction of $f\in C_b(X)$ to a closed set $A$ induces an order continuous operator iff $A=\overline{\mathrm{Int} A}.$ This result enables us to easily characterize bands and projection bands in $C_0(X)$ and $C_b(X)$ through the one-point compactification and the Stone-Čech compactification of $X$, respectively. With these characterizations we describe order complete $C_0(X)$ and $C_b(X)$-spaces in terms of extremally disconnected spaces. Our results serve us to solve an open question on lifting un-convergence in the case of $C_0(X)$ and $C_b(X)$.

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arXiv2016

On the dimension of the algebra generated by two positive semi-commuting matrices

Gerstenhaber's theorem states that the dimension of the unital algebra generated by two commuting $n\times n$ matrices is at most $n$. We study the analog of this question for positive matrices with a positive commutator. We show that the dimension of the unital algebra generated by the matrices is at most $\frac{n(n+1)}{2}$ and that this bound can be attained. We also consider the corresponding question if one of the matrices is a permutation or a companion matrix or both of them are idempotents. In these cases, the upper bound for the dimension can be reduced significantly. In particular, the unital algebra generated by two semi-commuting positive idempotent matrices is at most $9$-dimensional. This upper bound can be attained.

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