Searcharxiv⌕ Search
Home/Archive/arXiv

arXiv · 0707.3426

Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators

Michael T. Jury
View original source ↗Read source document ↗Export metadata ↓

Abstract

We prove that the norm of a weighted composition operator on the Hardy space H^2 of the disk is controlled by the norm of the weight function in the de Branges-Rovnyak space associated to the symbol of the composition operator. As a corollary we obtain a new proof of the boundedness of composition operators on H^2, and recover the standard upper bound for the norm. Similar arguments apply to weighted Bergman spaces. We also show that the positivity of a generalized de Branges-Rovnyak kernel is sufficient for the boundedness of a given composition operator on the standard functions spaces on the unit ball.

Explore related subjects

math.FA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael T. Jury. 2007-07-23. Reproducing kernels, de Branges-Rovnyak spaces, and norms of weighted composition operators. https://arxiv.org/abs/0707.3426

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

KEEP EXPLORING

Related papers

arXiv2026

Matrix-valued regular functions of a quaternionic variable - a decomposition and some approximation results

A quaternionic matrix-valued regular function is a map $F: Ω\rightarrow M_n(\mathbb{H})$ whose entries are (left) regular functions of a quaternion variable, where $Ω$ is a domain in $\mathbb{H}$. Given any $n \times n$ matrix-valued regular function $F(q)$ over $\mathbb{H}$ having norm at most $1$, we derive a decomposition of $F(q)$ in the form $U \star \begin{bmatrix} ||F(q_0)|| & 0\\0 & G(q)\end{bmatrix} V$ for some constant unitary matrices $U, V$, some vector $q_0$ and an $(n-1) \times (n-1)$ matrix-valued regular function $G(q)$, where $\star$ denotes the star product over $\mathbb{H}$. We use this to deduce that any matrix-valued regular function $F(q)$ having norm at most $1$ can be uniformly approximated by a convex combination of rational unitary functions.

math.FA↗
arXiv2026

Classification of bipartite observables I: stable and uniformly stable observables

When two quantum systems are prepared independently and each is driven to an idealized limit, the expectation of a bipartite observable may depend on which system is idealized first. We call an observable stable when it does not. We show that stability is equivalent to two operational properties. First, every idealized preparation of one system can be simulated to arbitrary accuracy, uniformly over all probes, by finite mixtures of actual preparations, provided that the probing system is idealized first. Second, any two idealized preparations have a unique, symmetric independent product. Under a uniform form of stability, we prove, the number of preparations that $n$ adaptive probes can tell apart at a fixed margin is bounded by a polynomial in $n$, and responses can be learned with a bounded number of mistakes; otherwise, at some margin, protocols with $n$ probes tell apart $2^n$ preparations. The swap test, Hong--Ou--Mandel coincidences and discrete equality tests turn out to be uniformly stable, with bounds independent of the dimension, while order comparisons of discrete spectra unbounded above are unstable. The proofs rest on the classical fact that a single double limit condition appears, under different names, in topology (Grothendieck), functional analysis (Krivine--Maurey), algebra (Arens), probability (Simons) and combinatorics (Pták), and in model theory as Shelah's stability, where the independent product corresponds to Harrington's lemma. Model theory thus serves as a Rosetta stone, pairing each physical question with its mathematical formulation.

math.FA↗