arXiv subjects

Kunyu Guo

Publications and source records attributed to Kunyu Guo.

25 records · Page 2Linked to original sources

arXiv2020

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 $ψ\in L^2(0,1)$, the dilation system $\{ψ(kx):k=1,2,\cdots\}$ is complete in $L^2(0,1)$, where $ψ$ 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 $\|f\|_{A_α^{2α}}\leq\|f\|_{H^2}$, where $f$ is a holomorphic function and $α>1$. If the norms $\|f\|_{A_α^{2α}}$ are decreasing in $α$, then the inequality holds for $f$. For a dense set of functions, we calculate the derivative of the norms $\|f\|_{A_α^{2α}}$ in $α$ 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↗

arXiv2018

Composition Operators with Quasiconformal Symbols

This paper seeks to extend the theory of composition operators on analytic functional Hilbert spaces from analytic symbols to quasiconformal ones. The focus is the boundedness but operator-theoretic questions are discussed as well. In particular, we present a thorough analysis of $L^p$-estimates of a class of singular integral operators $P_φ$ associated with a quasiconformal mapping $φ$.

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

Cowen's class and Thomson's class

In studying commutants of analytic Toeplitz operators, Thomson proved a remarkable theorem which states that under a mild condition, the commutant of an analytic Toeplitz operator is equal to that of Toeplitz operator defined by a finite Blaschke product. Cowen gave an significant improvement of Thosom's result. In this paper, we will present examples in Cowen's class which does not lie in Thomson's class.

math.CV↗

arXiv2013

Operator analogues of Mahler's measure

Motivated by a geometric meaning of Mahler's measure, we introduce two operator analogues of Mahler's measure. This leads to some interesting equalities and inequalities between the two operator-theoretic Mahler measures and the classical Mahler measure. In order to apply these results to the operator version of Lehmer's problem, we introduce and study an important class of operators, the so-called subharmonic operators. It is shown that the operator version of Lehmer's problem fails under some mild condition.

math.FA↗

arXiv2013

Geometric constructions of thin Blaschke products and reducing subspace problem

In this paper, we mainly study geometric constructions of thin Blaschke products $B$ and reducing subspace problem of multiplication operators induced by such symbols $B$ on the Bergman space. Considering such multiplication operators $M_B$, we present a representation of those operators commuting with both $M_B$ and $M_B^*$. It is shown that for "most" thin Blaschke products $B$, $M_B$ is irreducible, i.e. $M_B$ has no nontrivial reducing subspace; and such a thin Blaschke product $B$ is constructed. As an application of the methods, it is proved that for "most" finite Blaschke products $ϕ$, $M_ϕ$ has exactly two minimal reducing subspaces. Furthermore, under a mild condition, we get a geometric characterization for when $M_B$ defined by a thin Blaschke product $B$ has a nontrivial reducing subspace.

math.FA↗