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Roman Drnovšek

Publications and source records attributed to Roman Drnovšek.

At least 19 recordsLinked to original sources
arXiv2026

On diagonal sections of Markov products of copulas

Given a bivariate copula $C$, the bivariate copula $Ψ(C)$ is defined as the Markov product of $C^T$ and $C$. In this paper we consider the diagonal section of $Ψ(C)$ under the additional assumption that its value at a single point is known. We determine the exact upper and lower bounds for the diagonal section of $Ψ(C)$, given that $Ψ(C)(a,a)=d \in [a^2,a]$, where $a \in [0,1]$ is a fixed point. Furthermore, we describe some additional properties of the transformation $Ψ$, such as its behavior under ordinal sums and the survival operation. We also study checkerboard copulas in the range of $Ψ$ and matrices associated to them.

math.ST↗
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.

math.FA↗
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.

math.FA↗
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.

math.FA↗
arXiv2023

On commutators of idempotents

Let $T$ be an operator on Banach space $X$ that is similar to $- T$ via an involution $U$. Then $U$ decomposes the Banach space $X$ as $X = X_1 \oplus X_2$ with respect to which decomposition we have $U = \left(\begin{matrix} I_1 & 0 \\ 0 & -I_2 \end{matrix} \right)$, where $I_i$ is the identity operator on the closed subspace $X_i$ ($i=1, 2$). Furthermore, $T$ has necessarily the form $T = \left(\begin{matrix} 0 & * \\ * & 0 \end{matrix} \right) $ with respect to the same decomposition. In this note we consider the question when $T$ is a commutator of the idempotent $P = \left(\begin{matrix} I_1 & 0 \\ 0 & 0 \end{matrix} \right)$ and some idempotent $Q$ on $X$. We also determine which scalar multiples of unilateral shifts on $l^p$ spaces ($1 \le p \le \infty$) are commutators of idempotent operators.

math.FA↗
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.

math.FA↗
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.

math.FA↗
arXiv2019

Operator norm and numerical radius analogues of Cohen's inequality

Let $D$ be an invertible multiplication operator on $L^2(X, μ)$, and let $A$ be a bounded operator on $L^2(X, μ)$. In this note we prove that $\|A\|^2 \le \|D A\| \, \|D^{-1} A\|$, where $\|\cdot\|$ denotes the operator norm. If, in addition, the operators $A$ and $D$ are positive, we also have $w(A)^2 \le w(D A) \, w(D^{-1} A)$, where $w$ denotes the numerical radius.

math.FA↗
arXiv2019

Sequences of bounds for the spectral radius of a positive operator

In 1992, Szyld provided a sequence of lower bounds for the spectral radius of a nonnegative matrix $A$, based on the geometric symmetrization of powers of $A$. In 1998, Taşçi and Kirkland proved a companion result by giving a sequence of upper bounds for the spectral radius of $A$, based on the arithmetic symmetrization of powers of $A$. In this note, we extend both results to positive operators on $L^2$-spaces.

math.FA↗
arXiv2018

A sharp inequality involving hyperbolic and inverse hyperbolic functions

We prove that the inequality $$\cosh \left( \mathrm{arcosh}(2 \cosh u) \cdot \tanh u \right) < \exp \left( u \cdot \tanh u \right)$$ holds for all $u > 0$. We check with the computation program Mathematica that the ratio between the left-hand and the right-hand side is greater than 0,97 for all $u \ge 0$, so this is a quite sharp inequality. It is also equivalent to any of the two inequalities: $$ \cosh \left( \sqrt{1 - \frac{1}{t^2}} \cdot \mathrm{arcosh}\,{2t} \right) < \exp \left( \sqrt{1 - \frac{1}{t^2}} \cdot \mathrm{arcosh}\,{t} \right) $$ for all $t > 1$, and $$ \cosh \left( c \cdot \mathrm{arcosh}{\frac{2}{\sqrt{1-c^2}}} \right) < \exp \left( c \cdot \mathrm{arcosh}{\frac{1}{\sqrt{1-c^2}}} \right)$$ for all $c \in (0,1)$.

math.CA↗
arXiv2018

On order automorphisms of the effect algebra

We give short proofs of two Šemrl's descriptions of order automorphisms of the effect algebra. This sheds new light on both formulas that look quite complicated. Our proofs rely on Molnár's characterization of order automorphisms of the cone of all positive operators.

math.FA↗
arXiv2017

On algebras generated by positive operators

We study algebras generated by positive matrices, i.e., matrices with nonnegative entries. Some of our results hold in more general setting of vector lattices. We reprove and extend some theorems that have been recently shown by Kandić and Šivic. In particular, we give a more transparent proof of their result that the unital algebra generated by positive idempotent matrices $E$ and $F$ such that $E F \ge F E$ is equal to the linear span of the set $\{I, E, F, E F, F E, E F E, F E F, (E F)^2, (F E)^2\}$, and so its dimension is at most $9$. We give examples of two positive idempotent matrices that generate unital algebra of dimension $2n$ if $n$ is even, and of dimension $(2n - 1)$ if $n$ is odd. We also prove that the algebra generated by positive matrices $B_1$, $B_2$, $\ldots$, $B_k$ is triangularizable if $A B_i \ge B_i A$ ($i=1,2, \ldots, k$) for some positive matrix $A$ with distinct eigenvalues.

math.FA↗
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 $L^p(μ)$ $(1<p<\infty)$.

math.FA↗
arXiv2016

Triangularizability of trace-class operators with increasing spectrum

For any measurable set $E$ of a measure space $(X, μ)$, let $P_E$ be the (orthogonal) projection on the Hilbert space $L^2(X, μ)$ with the range $ran \, P_E = \{f \in L^2(X, μ) : f = 0 \ \ a.e. \ on \ E^c\}$ that is called a standard subspace of $L^2(X, μ)$. Let $T$ be an operator on $L^2(X, μ)$ having increasing spectrum relative to standard compressions, that is, for any measurable sets $E$ and $F$ with $E \subseteq F$, the spectrum of the operator $P_E T|_{ran \, P_E}$ is contained in the spectrum of the operator $P_F T|_{ran \, P_F}$. In 2009, Marcoux, Mastnak and Radjavi asked whether the operator $T$ has a non-trivial invariant standard subspace. They answered this question affirmatively when either the measure space $(X, μ)$ is discrete or the operator $T$ has finite rank. We study this problem in the case of trace-class kernel operators. We also slightly strengthen the above-mentioned result for finite-rank operators.

math.FA↗
arXiv2016

Triangularizability of Families of Polynomially Compact Operators

A recent paper of Shemesh shows triangularizability of a pair $\{A, B\}$ of complex matrices satisfying the condition $A [A,B]=[A,B] B=0$, or equivalently, the matrices $A$ and $B$ commute with their product $A B$. In this paper we extend this result to polynomially compact operators on Banach spaces. The case when the underlying space is Hilbert and one of operators is normal is also studied. Furthermore, we consider families of polynomially compact operators whose iterated commutators of some fixed length are zero. We also obtain a structure result in the special case of a finite family of algebraic operators.

math.FA↗
arXiv2015

Inequalities on the spectral radius and the operator norm of Hadamard products of positive operators on sequence spaces

Relatively recently, K.M.R. Audenaert (2010), R.A. Horn and F. Zhang (2010), Z. Huang (2011), A.R. Schep (2011), A. Peperko (2012), D. Chen and Y. Zhang (2015) have proved inequalities on the spectral radius and the operator norm of Hadamard products and ordinary matrix products of finite and infinite non-negative matrices that define operators on sequence spaces. In the current paper we extend and refine several of these results and also prove some analogues for the numerical radius. Some inequalities seem to be new even in the case of $n\times n$ non-negative matrices.

math.FA↗