arXiv subjects

Hui Dan

Publications and source records attributed to Hui Dan.

9 recordsLinked to original sources

arXiv2021

Invariant subspaces of weighted Bergman spaces in infinitely many variables

This paper is concerned with polynomially generated multiplier invariant subspaces of the weighted Bergman space $A_{\boldsymbol{\beta}}^2$ in infinitely many variables. We completely classify these invariant subspaces under the unitary equivalence. Our results not only cover cases of both the Hardy space $H^{2}(\mathbb{D}_{2}^{\infty})$ and the Bergman space $A^{2}(\mathbb{D}_{2}^{\infty})$ in infinitely many variables, but also apply in finite-variable setting.

math.FA↗

arXiv2020

Projections in Toeplitz algebra

Motivated by Barr{\'\i}a-Halmos's \cite[Question 19]{barria1982asymptotic} and Halmos's \cite[Problem 237]{Halmos1978A}, we explore projections in Toeplitz algebra on the Hardy space. We show that the product of two Toeplitz (Hankel) operators is a projection if and only if it is the projection onto one of the invariant subspaces of the shift (backward shift) operator. As a consequence one obtains new proofs of criterion for Toeplitz operators and Hankel operators to be partial isometries. Furthermore, we completely characterize when the self-commutator of a Toeplitz operator is a projection. This provides a class of nontrivial projections in Toeplitz algebra.

math.FA↗

arXiv2020

Power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces

In this paper, we concentrate on power dilation systems $\{f(z^k)\}_{k\in\mathbb{N}}$ in Dirichlet-type spaces $\mathcal{D}_t\ (t\in\mathbb{R})$. When $t\neq0$, we prove that $\{f(z^k)\}_{k\in\mathbb{N}}$ is orthogonal in $\mathcal{D}_t$ only if $f=cz^N$ for some constant $c$ and some positive integer $N$. We also give complete characterizations of unconditional bases and frames formed by power dilation systems for Drichlet-type spaces.

math.FA↗

arXiv2020

The Kozlov completeness problem

This paper concerns a long-standing problem raised by Kozlov on completeness of the dilation systems $\{\mathbf{1}_{(\alpha,\beta)}(kx):k=1,2,\cdots\}$ generated by odd periodic extensions on $\mathbb{R}$ of characteristic functions $\mathbf{1}_{(\alpha,\beta)}$, where $0\leq\alpha<\beta\leq1$. Up to now there has only some fragmentary results under the assumption $\alpha=0$. Focusing on the dilation completeness problem for characteristic functions $\mathbf{1}_V$ of open subsets $V\subset(0,1)$ that are finite unions of intervals with rational endpoints, we exhibit the exact forms of such $V$ in almost all interesting situations by using substantially techniques from analytic number theory. As a consequence, it yields a complete solution for the rational version of the Kozlov completeness problem. Moreover, our results also illustrate the fascinating connection among the Completeness Problem, the Twin Prime Conjecture and the Sophie Germain Prime Conjecture.

math.CA↗

arXiv2019

The Periodic Dilation Completeness Problem: Cyclic vectors in the Hardy space over the infinite-dimensional polydisk

The classical completeness problem raised by Beurling and independently by Wintner asks for which $\psi\in L^2(0,1)$, the dilation system $\{\psi(kx):k=1,2,\cdots\}$ is complete in $L^2(0,1)$, where $\psi$ is identified with its extension to an odd $2$-periodic function on $\mathbb{R}$. This difficult problem is nowadays commonly called as the Periodic Dilation Completeness Problem (PDCP). By Beurling's idea and an application of the Bohr transform, the PDCP is translated as an equivalent problem of characterizing cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$ over the infinite-dimensional polydisk for coordinate multiplication operators. In this paper, we obtain lots of new results on cyclic vectors in the Hardy space $\mathbf{H}_\infty^2$. In almost all interesting cases, we obtain sufficient and necessary criterions for characterizing cyclic vectors, and hence in these cases we completely solve the PDCP. Our results cover almost all previous known results on this subject.

math.FA↗

arXiv2019

A Sharp Inequality of Hardy-Littlewood Type Via Derivatives

In this paper we consider a generalized version of Carleman's inequality. An equivalent version of it states that $\|f\|_{A_\alpha^{2\alpha}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $\alpha>1$. If the norms $\|f\|_{A_\alpha^{2\alpha}}$ are decreasing in $\alpha$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\|f\|_{A_\alpha^{2\alpha}}$ in $\alpha$ and give sufficient conditions for this derivative to be non-positive. As an application, we prove the inequality for linear combinations of two reproducing kernels. Some numerical evidences are also provided.

math.FA↗

arXiv2019

Dilation theory and analytic model theory for doubly commuting sequences of $C_{.0}$-contractions

Sz.-Nagy and Foias proved that each $C_{\cdot0}$-contraction has a dilation to a Hardy shift and thus established an elegant analytic functional model for contractions of class $C_{\cdot0}$. This has motivated lots of further works on model theory and generalizations to commuting tuples of $C_{\cdot0}$-contractions. In this paper, we focus on doubly commuting sequences of $C_{\cdot0}$-contractions, and establish the dilation theory and the analytic model theory for these sequences of operators. These results are applied to generalize the Beurling-Lax theorem and Jordan blocks in the multivariable operator theory to the operator theory in countably infinitely many variables.

math.FA↗

arXiv2016

Totally Abelian Toeplitz operators and geometric invariants associated with their symbol curves

This paper mainly studies totally Abelian operators in the context of analytic Toeplitz operators on both the Hardy and Bergman space. When the symbol is a meromorphic function on $\mathbb{C}$, we establish the connection between totally Abelian property of these operators and and geometric properties of their symbol curves. It is found that winding numbers and multiplicities of self-intersection of symbol curves play an important role in this topic. Techniques of group theory, complex analysis, geometry and operator theory are intrinsic in this paper. As a byproduct, under a mild condition we provides an affirmative answer to a question raised in \cite{BDU,T1}, and also construct some examples to show that the answer is negative if the associated conditions are weakened.

math.CV↗

arXiv2014

Multiplication operators defined by a class of polynomials on L_a^2(D^2)

In this paper, we consider those multiplication operators M_p on the Bergman space L_a^2(D^2) over the bidisk, defined by a class of polynomials p. Also, this paper consider the reducing subspaces of M_p, the von Neumann algebra W^*(p) generated by M_p, and its commutant V^*(p)=W^*(p)'. The structure of V^*(p) is completely determined, along with those reducing subspaces of M_p.

math.OA↗