arXiv subjects

Aleksei B. Aleksandrov

Publications and source records attributed to Aleksei B. Aleksandrov.

4 recordsLinked to original sources

arXiv2024

Haagerup tensor products and Schur multipliers

In this paper we compare various classes of Schur multipliers: classical matrix Schur multipliers, discrete Schur multipliers, Schur multipliers with respect to measures and Schur multipliers with respect to spectral measures. The main result says that in the case of Schur multipliers with respect to measures and spectral measures such Schur multipliers coincide isometrically with the Haagerup tensor products of the corresponding $L^\infty$ spaces. We deduce this result from a well known analogue of it for discrete Schur multipliers.

math.FA↗

arXiv2023

Dominant sets for model spaces in several variables

Let $I$ be an inner function in $\mathcal{D} = B_{n_1}\times B_{n_2}\cdots \times B_{n_k}$, where $B_n$ denotes the open unit ball of $\mathbb{C}^n$, $n\ge 1$. We construct dominant sets for the space $H^2 \ominus I H^2$, where $H^2 = H^2(\mathcal{D})$ denotes the standard Hardy space.

math.CV↗

arXiv2021

Clark measures and de Branges-Rovnyak spaces in several variables

Let $B_n$ denote the unit ball of $\mathbb{C}^n$, $n\ge 1$, and let $\mathcal{D}$ denote a finite product of $B_{n_j}$, $j\ge 1$. Given a non-constant holomorphic function $b: \mathcal{D} \to B_1$, we study the corresponding family $σ_α[b]$, $α\in\partial B_1$, of Clark measures on the distinguished boundary $\partial\mathcal{D}$. We construct a natural unitary operator from the de Branges-Rovnyak space $\mathcal{H}(b)$ onto the Hardy space $H^2(σ_α)$. As an application, for $\mathcal{D}= B_n$ and an inner function $I: B_n \to B_1$, we show that the property $σ_1[I]\llσ_1[b]$ is directly related to the membership of an appropriate explicit function in $\mathcal{H}(b)$.

math.CV↗

arXiv2019

Clark measures on the complex sphere

Let $B_d$ denote the unit ball of $\mathbb{C}^d$, $d\ge 1$. Given a holomorphic function $φ: B_d \to B_1$, we study the corresponding family $σ_α[φ]$, $α\in\partial B_1$, of Clark measures on the unit sphere $\partial B_d$. If $φ$ is an inner function, then we introduce and investigate related unitary operators $U_α$ mapping analogs of model spaces onto $L^2(σ_α)$, $α\in\partial B_1$. In particular, we explicitly characterize the set of $U_α^* f$ such that $fσ_α$ is a pluriharmonic measure. Also, for an arbitrary holomorphic $φ: B_d \to B_1$, we use the family $σ_α[φ]$ to compute the essential norm of the composition operator $C_φ: H^2(B_1)\to H^2(B_d)$.

math.CV↗