arXiv subjects

Chao Zu

Publications and source records attributed to Chao Zu.

20 records · Page 2Linked to original sources

arXiv2025

Which hyponormal block Toeplitz operators are either normal or analytic?

In this paper, we continue Curto-Hwang-Lee's work to study the connection between hyponormality and subnormality for block Toeplitz operators acting on the vector-valued Hardy space of the unit circle. Curto-Hwang-Lee's work focuses primarily on hyponormality and subnormality of block Toeplitz operators with rational symbols. By studying the greatest common divisor of matrix-valued inner functions and the ``weak" commutativity of matrix-valued inner functions, we extended Curto-Hwang-Lee's result to block Toeplitz operators with symbols of bounded type. More precisely, we proved that if $Ψ,Ψ^{\ast}$ are matrix-valued functions of bounded type and the inner part of $Ψ$ of Douglas-Shapiro-Shields factorization is a scalar inner function, then every hyponormal Toeplitz operator $T_Ψ$ whose square is also hyponormal must be either normal or analytic.

math.FA↗

arXiv2025

Hilbert-Schmidtness of the $M_{θ,φ}$-type submodules

Let $θ(z),φ(w)$ be two nonconstant inner functions and $M$ be a submodule in $H^2(\mathbb{D}^2)$. Let $C_{θ,φ}$ denote the composition operator on $H^2(\mathbb{D}^2)$ defined by $C_{θ,φ}f(z,w)=f(θ(z),φ(w))$, and $M_{θ,φ}$ denote the submodule $[C_{θ,φ}M]$, that is, the smallest submodule containing $C_{θ,φ}M$. Let $K^M_{λ,μ}(z,w)$ and $K^{M_{θ,φ}}_{λ,μ}(z,w)$ be the reproducing kernel of $M$ and $M_{θ,φ}$, respectively. By making full use of the positivity of certain de Branges-Rovnyak kernels, we prove that \[K^{M_{θ,φ}}= K^M \circ B~ \cdot R,\] where $B=(θ,φ)$, $R_{λ,μ}(z,w)=\frac{1-\overline{θ(λ)}θ(z)}{1-\barλz} \frac{1-\overline{φ(μ)}φ(w)}{1-\barμw}$. This implies that $M_{θ,φ}$ is a Hilbert-Schmidt submodule if and only if $M$ is. Moreover, as an application, we prove that the Hilbert-Schmidt norms of submodules $[θ(z)-φ(w)]$ are uniformly bounded.

math.FA↗