Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 0710.1880

Operator Theory and Complex Geometry

Ronald G. Douglas
View original source ↗Read source document ↗Export metadata ↓

Abstract

One approach to multivariate operator theory involves concepts and techniques from algebraic and complex geometry and is formulated in terms of Hilbert modules. In these notes we provide an introduction to this approach including many proofs. We are particularly interested in examples related to hermitian holomorphic vector bundles and we study submodules and reducing submodules in such cases. We go into some detail concerning a problem of Zhu on the reducing subspaces of powers of the Bergman shift as well as more recent work of the author and J. Sarkar on proper submodules which are unitarily equivalent to the orginal. Although the basic results are not new, there is some novelty in the details and the organization of the material.

Explore related subjects

math.FAmath.SP

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Ronald G. Douglas. 2007-11-28. Operator Theory and Complex Geometry. https://arxiv.org/abs/0710.1880

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

KEEP EXPLORING

Related papers

arXiv2026

Mixed Bourgain-Morrey spaces and their applications to boundedness of operators

We introduce the mixed Bourgain-Morrey spaces and obtain their preduals. The boundedness of Hardy-Littlewood maximal operator, iterated maximal operator, fractional integral operator, singular integral operator on these spaces is proved. In addition, we give a description of the dual of mixed Bourgain-Morrey spaces and conclude the reflexivity of these spaces.

math.FA↗
arXiv2026

The stability landscape in wave-packet scattering: geometric rigidity and sharp Sobolev thresholds

A central problem in harmonic analysis is to understand how frequency resolution constrains stability under nonlinear changes of coordinates. Motivated by Mallat's wavelet scattering theory, we establish a sharp resolution--robustness principle for nonlinear representations built from two-dimensional wave-packet systems, showing that the transverse geometry of the frequency channels determines the critical Sobolev regularity. Whenever the channels are thinner than in the wavelet regime, arbitrarily small smooth deformations can move high-frequency mass between adjacent channels, producing instability already at the level of the linear filter bank and the first scattering layer. We match this obstruction with scale-sharp commutator estimates propagated through the full scattering cascade, obtaining Lipschitz stability at the critical regularity threshold. Together, the results provide the first intrinsic deformation-stability theory for non-wavelet Euclidean scattering transforms and identify a microlocal transport mechanism governing robustness across nonlinear multiscale representations.

math.FA↗
arXiv2026

Unbounded Helson Forms with a Moment Representation

We study the positive Helson form $h_0[a,b]=\sum_{m,n\ge1}α(mn)a_n\overline{b_m}$ on the finitely supported sequences $c_{00}\subset\ell^2(\mathbb{N})$, where $α(n)=\int_{\mathbb{R}}n^{-s}dμ(s)<\infty$ for a finite positive Borel measure $μ$. Under the Carleman condition $\sum_{k\ge1}α(4^k)^{-1/(2k)}=\infty$, closability of $h_0$ is equivalent to $\sqrt{n}α(n)\to0$, and hence to $μ((-\infty,1/2])=0$. For measures concentrated on $(1/2,\infty)$, we identify the closure of the operator $N_0a=\sum_{n\ge1}a_nn^{-s}$, defined on $c_{00}$, with its maximal realization from $\ell^2(\mathbb{N})$ to $L^2(μ)$. We further prove that the sequences $a\in c_{00}$ satisfying $\sum_{n\ge1}a_n/\sqrt{n}=0$ form an operator core for the positive self-adjoint operator associated with the closure of $h_0$.

math.FA↗