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Bojan Kuzma

Publications and source records attributed to Bojan Kuzma.

At least 19 recordsLinked to original sources
arXiv2026

Additive preservers of permanent rank

The permanent rank of a matrix $A$ is the size of the maximal square submatrix in $A$ which has a nonzero permanent. In this paper we characterize additive transformations $Φ$ which preserve matrices of per-rank-one. Under an additional assumption that $Φ$ is surjective or that it preserves per-rank-one in both directions we prove that $Φ$ is a composition of a multiplication with diagonal matrices and permutation matrices from both sides, transposition, and injective endomorphism of the base field.

math.RA↗
arXiv2026

On the ascent and the angle between the null space and the range of elementary operators

We study the angle between the null space and the range of elementary operators of length one or two acting on $\mathcal{B}(\mathscr{X})$, the Banach algebra of all bounded linear operators on a complex Banach space $\mathscr{X}$. For the multiplication operator $μ_{A,B}(X) = AXB$, we characterize positivity of this angle in terms of the corresponding angles for $A$ and $B^*$. For elementary operators of length two $Δ_{\boldsymbol{A},\boldsymbol{B}} = μ_{A_1,B_1} - μ_{A_2,B_2}$, we establish conditions under which the angle is positive, and the ascent of $Δ_{\boldsymbol{A},\boldsymbol{B}}$ equals one. Finally, for a generalized derivation $δ_{A,B}$ and an injective holomorphic function $f$ on a neighborhood of $σ(A)\cupσ(B)$, we show that the angle between the null space and the range of $δ_{f(A),f(B)}$ is positive whenever the angle between the null space and the range of $δ_{A,B}$ is positive.

math.FA↗
arXiv2025

On preservers of strong Birkhoff-James orthogonality between $C^*$-algebras

It is shown that every linear strong Birkhoff-James isomorphism between unital $C^*$-algebras is a $*$-isomorphism followed by a unitary multiplication. Moreover, as a partial extension of this result to the non-unital case, the form of (possibly nonlinear) strong Birkhoff-James isomorphisms between compact $C^*$-algebras are determined. A nonlinear characterization of compact $C^*$-algebras in terms of strong Birkhoff-James orthogonality is also given.

math.OA↗
arXiv2025

(Real)linear preservers of multiples of unitaries and matrix pairs with some extremal norm properties

We determine the structure of linear maps on complex (real) square matrices sending unitary (orthogonal) matrices to multiples of unitary (orthogonal) matrices. The result is used to determine the linear preservers of matrix pairs satisfying the extremal norm properties $\|AB\| = \|A\| \|B\|$, $\|A^*B\| = \|A\| \|B\|$, or $\|AB^*\| = \|A\| \|B\|$, for the spectral norm $\|\cdot\|$.

math.FA↗
arXiv2024

Classification of abelian finite-dimensional $C^*$-algebras by orthogonality

The main goal of the article is to prove that if $\mathcal A_1$ and $\mathcal A_2$ are Birkhoff-James isomorphic $C^*$-algebras over the fields $\mathbb F_1$ and $\mathbb F_2$, respectively and if $\mathcal A_1$ finite-dimensional, abelian of dimension greater than one, then $\mathbb F_1=\mathbb F_2$ and $\mathcal A_1$ and $\mathcal A_2$ are (isometrically) $\ast$-isomorphic $C^*$-algebras. Furthermore, it is also proved that for a finite-dimensional $C^*$-algebra $\mathcal A$, we have $\mathcal L_{\mathcal A}^\bot$ is the sum of minimal ideals which are not skew-fields and $\mathcal L_{\mathcal A}^{\bot\bot}$ is the sum of minimal ideals which are skew-fields, where $\mathcal L_{\mathcal A}$ denotes the set of all left-symmetric elements in $\mathcal A$ and for any subset $\mathcal S\subseteq \mathcal A$, the set $\mathcal S^\bot$ represents the set of all elements of $\mathcal A$ which are Birkhoff-James orthogonal to $\mathcal S$. A procedure to extract the minimal ideals which are (commutative) fields is also given.

math.FA↗
arXiv2024

Linear preservers of parallel matrix pairs with respect to the $k$-numerical radius

Let $1 \leq k < n$ be integers. Two $n \times n$ matrices $A$ and $B$ form a parallel pair with respect to the $k$-numerical radius $w_k$ if $w_k(A + μB) = w_k(A) + w_k(B)$ for some scalar $μ$ with $|μ| = 1$; they form a TEA (triangle equality attaining) pair if the preceding equation holds for $μ= 1$. We classify linear bijections on $\mathbb M_n$ and on $\mathbb H_n$ which preserve parallel pairs or TEA pairs. Such preservers are scalar multiples of $w_k$-isometries, except for some exceptional maps on $\mathbb H_n$ when $n=2k$.

math.FA↗
arXiv2024

Non-linear classification of finite-dimensional simple $C^*$-algebras

A Banach space characterization of simple real or complex $C^*$-algebras is given which even characterizes the underlying field. As an application, it is shown that if $\mathfrak A_1$ and $\mathfrak A_2$ are Birkhoff-James isomorphic simple $C^*$-algebras over the fields $\mathbb F_1$ and $\mathbb F_2$, respectively and if $\mathfrak A_1$ is finite-dimensional with dimension greater than one, then $\mathbb F_1=\mathbb F_2$ and $\mathfrak A_1$ and $\mathfrak A_2$ are (isometrically) $\ast$-isomorphic $C^*$-algebras.

math.OA↗
arXiv2024

Birkhoff-James classification of norm's properties

For an arbitrary normed space $\mathcal X$ over a field $\mathbb F \in \{ \mathbb R, \mathbb C \}$, we define the directed graph $Γ(\mathcal X)$ induced by Birkhoff-James orthogonality on the projective space $\mathbb P(\mathcal X)$, and also its nonprojective counterpart $Γ_0(\mathcal X)$. We show that, in finite-dimensional normed spaces, $Γ(\mathcal X)$ carries all the information about the dimension, smooth points, and norm's maximal faces. It also allows to determine whether the norm is a supremum norm or not, and thus classifies finite-dimensional abelian $C^\ast$-algebras among other normed spaces. We further establish the necessary and sufficient conditions under which the graph $Γ_0(\mathcal{R})$ of a (real or complex) Radon plane $\mathcal{R}$ is isomorphic to the graph $Γ_0(\mathbb F^2, \|\cdot\|_2)$ of the two-dimensional Hilbert space and construct examples of such nonsmooth Radon planes.

math.FA↗
arXiv2024

Vanishing Immanants

We classify all the irreducible characters of a symmetric group such that the induced immanant function $d_χ$ vanishes identically on alternate matrices with the entries in the complex field.

math.RT↗
arXiv2024

On the diameter of a super-order-commuting graph

We answer a question about the diameter of an order-super-commuting graph on a symmetric group by studying the number-theoretical concept of $d$-complete sequences of primes in arithmetic progression.

math.CO↗
arXiv2023

Quasi-copulas as linear combinations of copulas

We prove that every quasi-copula can be written as a uniformly converging infinite sum of multiples of copulas. Furthermore, we characterize those quasi-copulas which can be written as a finite sum of multiples of copulas, i.e., that are a linear combination of two copulas. This generalizes a recent result of Fernández-Sánchez, Quesada-Molina, and Úbeda-Flores who considered linear combinations of discrete copulas.

math.ST↗
arXiv2023

On approximate and actual reducibility of matrix groups

We introduce the notions of $\varepsilon$-approximate fixed point and weak $\varepsilon$-approximate fixed point. We show that for a group of unitary matrices even the existence of a nontrivial weak $\varepsilon$-approximate fixed point for sufficiently small $\varepsilon$ gives an actual nontrivial common eigenvector. We give estimates for $\varepsilon$ in terms of the size $n$ of matrices and prove that the dependence is polynomial. Moreover, we show that the common eigenvector is polynomially close to the starting weak approximate fixed point.

math.GR↗
arXiv2023

Linear maps preserving parallel matrix pairs with respect to the Ky-Fan $k$-norm

Two bounded linear operators $A$ and $B$ are parallel with respect to a norm $\|\cdot\|$ if $\|A+μB\| = \|A\| + \|B\|$ for some scalar $μ$ with $|μ| = 1$. Characterization is obtained for bijective linear maps sending parallel bounded linear operators to parallel bounded linear operators with respect to the Ky-Fan $k$-norms.

math.FA↗