arXiv subjects

Janko Bračič

Publications and source records attributed to Janko Bračič.

9 recordsLinked to original sources

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

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.

math.FA↗

arXiv2024

Reflexive subspace lattices in Banach spaces

In this survey paper we present known results about reflexive subspace lattices. We show that every nest and every atomic Boolean subspace lattice in a complex Banach space is reflexive, even strongly reflexive. Our main tool is Ringrose's Lemma about presence of rank-one operators in a reflexive algebra of operators. We consider realizations of small lattices as subspace lattices in Banach spaces. We exhibit some realizations of the pentagon and prove that some of these realizations are reflexive. On the other hand, we show that double triangle subspace lattices in finite-dimensional Banach spaces are non-reflexive.

math.FA↗

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↗

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.

math.FA↗

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 $\Col(\bdJ_2\oplus \bdJ_2)$.

math.FA↗

arXiv2021

Local commutants and ultrainvariant subspaces

For an operator $A$ on a complex Banach space $X$ and a closed subspace $M\subseteq X$, the local commutant of $A$ at $M$ is the set $C(A;M)$ of all operators $T$ on $X$ such that $TAx=ATx$ for every $x\in M$. It is clear that $ C(A;M)$ is a closed linear space of operators, however it is not an algebra, in general. For a given $A$, we show that $C(A;M)$ is an algebra if and only if the largest subspace $M_A$ such that $C(A;M)=C(A;M_A)$ is invariant for every operator in $C(A;M)$. We say that these are ultrainvariant subspaces of $A$. For several types of operators we prove that there exist non-trivial ultrainvariant subspaces. For a normal operator on a Hilbert space, every hyperinvariant subspace is ultrainvariant. On the other hand, the lattice of all ultrainvariant subspaces of a non-zero nilpotent operator can be strictly smaller than the lattice of all hyperinvariant subspaces.

math.FA↗

arXiv2015

A characterization of reflexive spaces of operators

We show that for a linear space of operators ${\mathcal M}\subseteq {\mathcal B}(H_1,H_2)$ the following assertions are equivalent. (i) ${\mathcal M} $ is reflexive in the sense of Loginov--Shulman. (ii) There exists an order-preserving map $Ψ=(ψ_1,ψ_2)$ on a bilattice $Bil({\mathcal M})$ of subspaces determined by ${\mathcal M}$, with $P\leq ψ_1(P,Q)$ and $Q\leq ψ_2(P,Q)$, for any pair $(P,Q)\in Bil({\mathcal M})$, and such that an operator $T\in {\mathcal B}(H_1,H_2)$ lies in ${\mathcal M}$ if and only if $ψ_2(P,Q)Tψ_1(P,Q)=0$ for all $(P,Q)\in Bil( {\mathcal M})$. This extends to reflexive spaces the Erdos--Power type characterization of weakly closed bimodules over a nest algebra.

math.OA↗

arXiv2013

On the invertibility of elementary operators

Let $\mathscr{X}$ be a complex Banach space and $\mathcal{L}(\mathscr{X})$ be the algebra of all bounded linear operators on $\mathscr{X}$. For a given elementary operator $Φ$ of length $2$ on $\mathcal{L}(\mathscr{X})$, we determine necessary and sufficient conditions for the existence of a solution of the equation ${\rm X} Φ=0$ in the algebra of all elementary operators on $\mathcal{L}(\mathscr{X})$. Our approach allows us to characterize some invertible elementary operators of length $2$ whose inverses are elementary operators.

math.FA↗