Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 1711.11090

Rational dilation problems associated with constrained algebras

Michael A. Dritschel·Batzorig Undrakh
View original source ↗Read source document ↗Export metadata ↓

Abstract

It is shown that rational dilation fails on broad collection of distinguished varieties associated to constrained subalgebras of the disk algebra of the form C + B A(D), where B is a finite Blaschke product with two or more zeros. This is accomplished in part by finding a minimal set of test functions. In addition, an Agler-Pick interpolation theorem is given and it is proved that there exist Kaijser-Varopoulos style examples of non-contractive unital representations where the generators are contractions.

Explore related subjects

math.FAmath.CV

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Michael A. Dritschel, Batzorig Undrakh. 2018-06-28. Rational dilation problems associated with constrained algebras. https://arxiv.org/abs/1711.11090

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

KEEP EXPLORING

Related papers

arXiv2026

Metric Poincaré type inequalities and lower bounds on the infimum of the spectrum for graphs

We study metric Poincaré type inequalities on general graphs. We characterize graphs satisfying such inequalities and then turn to the best constants in these inequalities. Invoking suitable metrics we can interpret these constants geometrically as diameters and inradii. Moreover, we can relate them to spectral theory of Laplacians once a probability measure on the graph is chosen. More specifically, we obtain a variational characterization of these constants as infimum over spectral gaps of all Laplacians on the graphs associated to probability measures

math.FA↗
arXiv2026

Natural methods of unsupervised topological alignment

In this paper, we consider methods for the diagonal multi-omics integration of heterogeneous datasets. Several approaches to the nature of biological heterogeneity are analyzed and developed to comprehend more clearly the generated differences. Specifically, the extremal trace problems for the coupled Laplacian on sets homeomorphic to the Stiefel manifold embedded in the complex Euclidean space are investigated. The gradient ascent method for the maximization problem is elaborated in the classical terms of functional analysis, which is of significant interest in itself. On this basis, we introduce a novel characteristic of dataset heterogeneity by employing the norm of the difference between the maximum and minimum points.

math.FA↗
arXiv2026

On Toeplitz operators on compact Abelian groups and discrete Wiener--Hopf operators

This paper introduces the concept of a rotation number for a continuous, non-degenerate two-dimensional vector field (a zero-free complex-valued function) on a compact connected Abelian group. This concept generalizes the notion of a finite rotation number for such groups, previously introduced by the author. Using this concept, a Gohberg-Krein index formula is derived for semi-Fredholm Toeplitz operators with continuous symbols defined on such groups. Criteria for these operators to be semi-Fredholm are established, and their essential spectra are described. As a by-product for the continuous symbol case, conditions for Fredholmness and semi-Fredholmness are established, and the Fredholm index of Wiener-Hopf operators over a linearly ordered discrete Abelian group is calculated in terms of their symbols. Spectral properties-including the spectra and essential spectra-of the Wiener-Hopf operators under consideration are also described.

math.FA↗