Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 2609.31173

Solutions to the 'moment property implies subnormality' and injectivity problems for composition operators on one-circuit graphs

Piotr Budzyński·Zenon Jan Jabłoński·Il Bong Jung·Jan Stochel
View original source ↗Read source document ↗Export metadata ↓

Abstract

We consider composition operators in $L^2(μ)$, where $μ$ is a discrete measure. Our main result shows that, for every integer $p\ge2$, there exists a non-hyponormal composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of a circuit of length $p$ and one infinite branch. In contrast, if the symbol induces a directed graph consisting of one loop and one infinite branch, then the generation of Stieltjes moment sequences does imply subnormality. These two results completely solve the ``moment property implies subnormality'' problem left open in [Adv. Math. 310 (2017), 484-556] for the entire family of one-branch directed graphs with a circuit of arbitrary length. Combining them with the examples in that paper shows that, for directed graphs consisting of one loop and a finite number of infinite branches, $2$ is the smallest number of branches for which the moment property does not imply subnormality. Finally, we construct a noninjective composition operator that generates Stieltjes moment sequences and whose symbol induces a directed graph consisting of one loop, one infinite branch, and one leaf. This operator gives a negative answer to the injectivity problem posed in the aforementioned paper. Both constructions use the Krein and Friedrichs measures of an arbitrary normalized S-indeterminate Stieltjes moment sequence.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Piotr Budzyński, Zenon Jan Jabłoński, Il Bong Jung, Jan Stochel. 2026-09-25. Solutions to the 'moment property implies subnormality' and injectivity problems for composition operators on one-circuit graphs. https://arxiv.org/abs/2609.31173

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Restriction of Fractional Derivatives of the Fourier Transform

We consider the differentiability of the Fourier transform of $f \in L^p(\mathbb{R}^n)$ at points on the graph of a function with non-vanishing Gaussian curvature. In particular, we show that if the values of $\widehat{f}$ along the graph possess $H^\ell$ Sobolev regularity, then this implies a certain degree of regularity of $\widehat{f}$ in the last coordinate direction as well. Two types of results are proved: One controls the $β$-order derivative in the last direction in terms of its $H^{-s}$ norm for suitable combinations of $(β, p, s, \ell)$. This extends the result in \cite{GoldStol}*{Theorem 1.12} to fractional derivatives and to some of its sharp endpoints. In particular, we recover the sharp range in the $s=0$ case where $\frac{\partial \widehat{f}}{\partial ξ_n}$ has a surface restriction in $L^2$. The second set of results is about $L^2(\mathbb{R}^n)$ bounds for difference quotients and fractional difference quotients of $\widehat{f}$, generalizing results in \cite{G2016}.

math.FA↗
arXiv2026

$J$-class weighted translations on locally compact groups

A bounded linear operator $T$ on a Banach space $X$ (not necessarily separable) is said to be $J$-class operator whenever the extended limit set, say $J_T(x)$ equals $X$ for some vector $x\in X$. Practically, the extended limit sets localize the dynamical behavior of operators. In this paper, using the extended limit sets we will examine the necessary and sufficient conditions for the weighted translation $T_{a,ω}$ to be $J$-class on a locally compact group $G$, within the setting of $ L^p$-spaces for $ 1 \leq p < \infty $. Precisely, we delineate the boundary between $J$-class and hypercyclic behavior for weighted translations. Then, we will show that for the torsion elements in locally compact groups, unlike the case of non-dense orbits of weighted translations, we have $J_{T_{a,ω}}(0)=L^p(G)$. Finally, we will provide some examples on which the weighted translation $ T_{a,ω}$ is $J$-class but it fails to be hypercyclic.

math.FA↗
arXiv2026

Determinant representations for Garvan formulas

In this note, we demonstrate how determinant representations for correlation functions in conformal field theory can be used to derive explicit determinant formulas for powers of the classical $η$-function, expressed via deformed elliptic functions with parameters. At genus one, comparing the fermionic and bosonic evaluations of a twisted torus partition function, together with an elliptic Frobenius/Fay determinant identity, gives for every $n\ge1$ a determinant formula for $η(τ)^{3n}$ built from a twisted Szegő kernel; at $n\equiv0,1\pmod 8$ this gives determinant identities for powers of the modular discriminant $Δ=η^{24}$ of the same shape as the classical formulas of Garvan and Milne, although those classical determinants are not recovered. At genus two, an analogous comparison for a torus formed by self-sewing, combined with a finite/infinite determinant factorization of the genus two Szegő kernel, gives a determinant formula for $η(τ)^{3κ^2}$, $-1/2<κ<1/2$, in terms of genus two data; identifying the genus two modular form to which this specializes is left open.

math.FA↗