arXiv subjects

Puyu Cui

Publications and source records attributed to Puyu Cui.

5 recordsLinked to original sources

arXiv2026

Finite-rank Toeplitz commutators on the harmonic Bergman space

We completely characterize finite-rank commutators of Toeplitz operators with bounded harmonic symbols on the harmonic Bergman space of the unit disk. Apart from affine dependence, finite rank occurs precisely when both symbols are analytic with rational primitives, or both are coanalytic and their complex conjugates have rational primitives. Our approach combines a block-matrix decomposition with Luecking's theorem, which rules out nonzero finite-rank Toeplitz operators with bounded symbols on the Bergman space. In particular, every such finite-rank commutator has square zero.

math.FA↗

arXiv2026

Norms of the Hilbert Matrix Operator on Some Analytic Function Spaces

For $\mathcal H:\mathcal B^α\to\mathcal B^α$ with $1<α<2$, we obtain an exact norm representation. In particular, the norm is given by an explicit Gamma-function formula for $1<α\leq3/2$, whereas for $3/2<α<2$ it strictly exceeds the corresponding Gamma-function expression. For $\mathcal H:\mathcal B^α\to\mathcal B^α_{\log}$, we determine the exact norm: it is equal to $3/2$ for $1<α\leq4/3$ and is given by an explicit Gamma-function formula for $4/3<α<2$. We also determine the exact norm of $\mathcal H:H^1\to H^1_{\log}$ and show that it is equal to $π$. Finally, for $0<α<1$, we determine the exact norm of $\mathcal H:H^\infty_α\to\mathcal B^{α+1}$, which is given by an explicit Gamma-function formula.

math.FA↗

arXiv2026

Commuting slant Toeplitz operators for polyanalytic and harmonic symbols

We study commutativity and zero products of slant Toeplitz operators on the Bergman space of the unit disk. For bounded polyanalytic symbols and for bounded harmonic symbols with boundary values in $W^{1,2}(\mathbb T)$, we prove that two such operators commute if and only if their symbols are linearly dependent over $\mathbb C$, with both symbols belonging to the same class. In the polyanalytic case, no boundedness assumption is imposed on the individual analytic components. We also show that every finite-rank commutator in this class is zero and that every finite product associated with nonzero polyanalytic symbols has infinite rank. Finally, for $s\ge2$, we prove that a product of $s$ slant Toeplitz operators with bounded harmonic symbols whose boundary values belong to $W^{s-1,2}(\mathbb T)$ is zero if and only if at least one symbol vanishes identically.

math.FA↗

arXiv2026

Spectral properties of Toeplitz operators with harmonic function symbols on the Bergman space

This paper investigates the spectral properties of Toeplitz operators on the Bergman space of unit disk. We present an integral representation of $ T^*_{z^m}$, which establishes a connection between the Bergman functions and the solutions of PDE theory. In fact, by leveraging the Poincaré theorem in difference equations and the solution forms of differential equations, this paper describes the kernels of certain Toeplitz operators with harmonic polynomial symbols, and further gives the sufficient conditions for the connectedness of the spectra of these Toeplitz operators. The spectral properties of $ T_φ$ with $φ(z) =\overline{z}^{m} + αz^m + β$ are characterized, such as $σ(T_φ)= \overline{φ(\mathbb {D})}$, Fredholm index of $T_φ$ can only be one of $m,-m$ and $0$, $T_φ$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators.

math.FA↗

arXiv2009

Chaos for Cowen-Douglas operators

In this article, we provide a sufficient condition which gives Devaney chaos and distributional chaos for Cowen-Douglas operators. In fact, we obtain a distributionally chaotic criterion for bounded linear operators on Banach spaces.

math.FA↗