arXiv subjects

A. Aleman

Publications and source records attributed to A. Aleman.

3 recordsLinked to original sources

arXiv2017

The class $B_\infty$

We explore properties of the class of Békollé-Bonami weights $B_\infty$ introduced by the authors in a previous work. Although Békollé-Bonami weights are known to be ill-behaved because they do not satisfy a reverse Hölder property, we prove than when restricting to a class of weights that are "nearly constant on top halves", one recovers some of the classical properties of Muckenhoupt weights. We also provide an application of this result to the study of the spectra of certain integral operators.

math.CA↗

arXiv2013

Trace ideal criteria for embeddings and composition operators on model spaces

Let $K_θ$ be a model space generated by an inner function $θ$. We study the Schatten class membership of embeddings $I : K_θ\to L^2(μ)$, $μ$ a positive measure, and of composition operators $C_ϕ:K_θ\to H^2(\mathbb D)$ with a holomprphic function $ϕ:\mathbb D\rightarrow \mathbb D$. In the case of one-component inner functions $θ$ we show that the problem can be reduced to the study of natural extensions of $I$ and $C_ϕ$ to the Hardy-Smirnov space $E^2(D)$ in some domain $D\supset \mathbb D$. In particular, we obtain a characterization of Schatten membership of $C_ϕ$ in terms of Nevanlinna counting function. By example this characterization does not hold true for general $ϕ$.

math.FA↗

arXiv2012

On a theorem of Livsic

The theory of symmetric, non-selfadjoint operators has several deep applications to the complex function theory of certain reproducing kernel Hilbert spaces of analytic functions, as well as to the study of ordinary differential operators such as Schrodinger operators in mathematical physics. Examples of simple symmetric operators include multiplication operators on various spaces of analytic functions such as model subspaces of Hardy spaces, deBranges-Rovnyak spaces and Herglotz spaces, ordinary differential operators (including Schrodinger operators from quantum mechanics), Toeplitz operators, and infinite Jacobi matrices. In this paper we develop a general representation theory of simple symmetric operators with equal deficiency indices, and obtain a collection of results which refine and extend classical works of Krein and Livsic. In particular we provide an alternative proof of a theorem of Livsic which characterizes when two simple symmetric operators with equal deficiency indices are unitarily equivalent, and we provide a new, more easily computable formula for the Livsic characteristic function of a simple symmetric operator with equal deficiency indices.

math.FA↗