arXiv subjects

Chao Zu

Publications and source records attributed to Chao Zu.

18 recordsLinked to original sources

arXiv2026

Helson Forms and Integral Operators on Bergman Spaces of Dirichlet Series

We study a family of operators $T_\alpha$, $\alpha>1$, on Hilbertian Bergman spaces of Dirichlet series. In the natural orthonormal basis, these operators are represented by weighted multiplicative Hilbert matrices involving the generalized divisor coefficients $d_\alpha$. We determine their spectral type. The essential and absolutely continuous spectra are $[0,\Lambda_\alpha]$, with absolutely continuous multiplicity one; the singular continuous spectrum is empty, and there are no embedded eigenvalues. There are only finitely many eigenvalues above $\Lambda_\alpha$, and all of them are simple. We also prove that there is a unique $\alpha_*\in(1,2)$ such that no eigenvalues occur above $\Lambda_\alpha$ for $1<\alpha<\alpha_*$, whereas such eigenvalues exist for every $\alpha>\alpha_*$. As part of the proof, we establish a spectral theorem for a general class of weighted integral Hankel operators with kernels $w(x)b(x+y)\overline{w(y)}$. Finally, we study the corresponding weighted Helson forms, give sufficient conditions for boundedness and compactness, and characterize boundedness for forms induced by finite positive measures.

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arXiv2026

Verblunsky coefficients, CMV matrices and numerical invariants of homogeneous bidisc submodules

Let $[p]$ be the principal submodule generated by a polynomial in $H^2(\mathbb D^2)$. For homogeneous $p$, the homogeneous slices of $[p]$ admit a weighted OPUC model in which the two wandering vectors are an orthonormal polynomial and its reversal. We show that the associated Verblunsky coefficients determine the singular values of the wandering-projection product and the restricted cross-commutator, as well as the non-zero spectrum of the core operator. Toeplitz-determinant and Mahler-measure identities yield exact Fredholm determinants and Schatten estimates, while $[(z-w)^N]$ rules out a uniform Hilbert--Schmidt bound. The same model gives explicit singular values of $[S_z^*,S_w]$ on the homogeneous quotient $H^2(\mathbb D^2)\ominus[p]$; for $p=(z-w)^N$, its squared Hilbert--Schmidt norm is asymptotic to $N$. For arbitrary polynomial generators, we construct a weighted bivariate model with a doubly Toeplitz, block-banded moment matrix and prove $C_p^2|_{\mathscr E_z}=\Gamma_p^*\Gamma_p$, relating the core spectrum to the cross-Gram operator between the two edge spaces. We also discuss cyclic-factor obstructions, represent higher numerical invariants by alternating CMV products, and give a quadratic counterexample to their proposed monotonicity.

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arXiv2026

Strict Monotonicity of Numerical Invariants for the Submodules $[(z-w)^k]$ in $H^2(\mathbb D^2)$

For $k\geq1$, let $M_k=[(z-w)^k]\subset H^2(\mathbb D^2)$. We first determine the banded Toeplitz matrices associated with the homogeneous components of $M_k$, together with explicit formulas for their determinants and the relevant algebraic cofactors. These formulas lead to a complete description of the spectrum of the core operator: \[ \sigma(C_{M_k}) = \{0,1\} \cup \left\{ \pm\frac{k}{n+k}:n\geq1 \right\}. \] In particular, the spectral data determine the parameter $k$. The determinant and cofactor formulas further yield a unified finite-sum representation for $\alpha_{n,j}^{(k)} =\langle w^j\phi_n,z^j\psi_n\rangle$, and hence for Yang's higher numerical invariants. We derive an adjacent relation connecting $\alpha_{n,j}^{(k)}$ and $\alpha_{n,j+1}^{(k)}$ by means of an explicit telescoping certificate, and show that the corresponding finite-section transformations are strict contractions. Combining these finite-dimensional estimates with the asymptotic behavior of $\alpha_{n,j}^{(k)}$, we prove the strict monotonicity \[ \Sigma_0(M_k)> \Sigma_1(M_k)> \Sigma_2(M_k)> \cdots . \] The cases $k\geq3$ constitute the new part of the analysis, while the previously known cases $k=1,2$ are recovered within the same framework. Consequently, Yang's monotonicity conjecture holds in strict form for the entire family $\{[(z-w)^k]:k\geq1\}$.

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arXiv2026

Block Repetition of Numerical Invariants for the Submodules $[z^k-w^k]$ in $H^2(\mathbb D^2)$

For $k\ge 2$, let $M_k=[z^k-w^k]$ be the principal homogeneous submodule of the Hardy space over the bidisk. We determine Yang's complete sequence of numerical invariants and prove $$ \Sigma_0(M_k)=\frac{\pi^2}{6},\qquad \Sigma_j(M_k)=\Sigma_{\lceil j/k\rceil}([z-w]),\quad j\ge1. $$ The proof exploits a residue-class decomposition of the Toeplitz matrices associated with the graded wandering spaces. Consequently, Yang's monotonicity conjecture holds for the family $\{M_k:k\ge2\}$. We also show that the nonzero spectral data of the core operator are independent of $k$, whereas the numerical invariant sequence recovers $k$ from the length of its constant blocks. Thus the higher numerical invariants detect module-theoretic information invisible to the core spectrum.

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arXiv2026

Corona theorem for the quaternionic Hardy space

The finite-generator corona theorem for bounded slice regular functions on the quaternionic unit ball was recently established by Colombo, Pozzi, Sabadini, and Wick. In the present paper, we extend the quaternionic $H^\infty$-corona theorem to countably many generators and obtain quantitative estimates that are independent of the cardinality of the generating family. We also establish the corresponding $H^p$-corona theorem for the full range $1\leq p<\infty$, with quantitative norm estimates for both finite and countable families of generators. In the Hilbert-space setting, we prove a quaternionic Leech factorization theorem for Hardy-space multipliers and derive, as a consequence, a Toeplitz corona characterization. Our approach is based on a fixed-slice $2\times2$ complex matrix realization of the slice regular product, together with operator-valued corona and factorization techniques.

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arXiv2026

Radial Convergence along Decreasing Coordinate Radii in \texorpdfstring{$H^\infty(\T^\infty)$}{H-infinity(T-infinity)}

Aleman, Olsen, and Saksman asked whether radial convergence may fail for bounded analytic functions on the infinite-dimensional polydisc when every radial point has non-increasing coordinates, and whether the approach may be chosen independently of the boundary point. We construct a point-dependent counterexample in which every fixed coordinate nevertheless increases to $1$. A finite discretization yields a boundary-point-independent coordinatewise decreasing approach for which convergence to the boundary function fails almost everywhere. In contrast, every boundary-point-independent approach whose fixed coordinates increase monotonically to $1$ satisfies an $L^p$ maximal inequality and the almost-everywhere Fatou theorem for $1<p<\infty$.

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arXiv2026

A Fourier Criterion for the Toeplitzness of Operators on Fock Spaces

We give a Fourier criterion for the Toeplitzness of bounded operators on Fock spaces, where Toeplitzness means representability as a Toeplitz operator with a bounded measurable symbol. For a Toeplitz operator, the anti-diagonal restriction of its canonical kernel is the Fourier transform of the Gaussian-weighted symbol. Consequently, Fourier inversion of this anti-diagonal restriction recovers the unique bounded symbol whenever such a representation exists. As applications, we characterize the Toeplitzness of weighted composition operators and generalized Volterra-type operators.

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arXiv2026

Slice Regular Composition Operators on Quaternionic Fock Spaces via Matrix Realization

We characterize the boundedness and compactness of slice regular composition operators between quaternionic Fock spaces for the full range $0<p,q<\infty$, without assuming that the composition symbol preserves a fixed complex slice. As applications of the same method, we also obtain corresponding criteria for weighted composition operators and for products of Volterra-type integral operators with slice regular composition operators. The main tool is a fixed-slice matrix realization of the regular product, which represents slice regular composition on a fixed complex slice through a holomorphic $2\times 2$ matrix functional calculus. This representation reveals a genuinely quaternionic rigidity phenomenon: boundedness imposes affine restrictions on the eigenvalue functions of the associated matrix symbol rather than on the original symbol itself. In particular, the original symbol need not be affine, and affine eigenvalue functions alone do not characterize boundedness.

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arXiv2026

Compactness of products and commutators of inner projections

In this paper, we study the compactness of the product and the commutator of two inner projections on the Hardy spaces over the unit disk and the polydisc. For the single-variable case, we provide a complete characterization of the compactness of the commutator of two inner projections by means of Douglas algebra. In the multivariable setting, we discover a rigidity phenomenon: on the bidisc, the product of two inner projections is compact if and only if it has finite rank, whereas on the polydisc of dimension strictly greater than two, any such compact product must be trivial.

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arXiv2026

On numerical invariants for submodules $[(z-w)^2]$ in $H^2(\mathbb{D}^2)$

In this paper, we study numerical invariants associated with a homogeneous submodule of the Hardy module over the bidisk. We focus on the submodule generated by the polynomial $(z-w)^2$ and obtain explicit formulas for the corresponding invariants. As an application, we verify the monotonicity property in this concrete setting. Our results provide a detailed example illustrating the behavior of these invariants beyond the linear case.

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arXiv2026

Toeplitz Operators on Quaternionic Fock Spaces

We characterize boundedness and compactness of Toeplitz operators on quaternionic Fock spaces with positive measure symbols and slice-function symbols in $\mathrm{BMO}^1$. For positive measure symbols, we derive criteria using normalized reproducing kernels and symmetric box averages, while for slice $\mathrm{BMO}^1$ symbols, the characterizations rely on the Berezin transform. We further introduce a global quaternionic Fock space $F_\alpha^p$ to define Toeplitz operators with real-valued measure symbols; this space is built by integrating slice regular functions over all complex slices of $\mathbb{H}$ and is norm-equivalent to the standard slice-based quaternionic Fock space. In the Hilbert space case $p=2$, a slice-independent orthogonal projection exists, which allows us to define Toeplitz operators with real-valued measure symbols and slice-function symbols in a unified way.

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arXiv2026

Nuclear Toeplitz operators between Fock spaces

We characterize the nuclearity of Toeplitz operators $T_\mu: F_\alpha^p \to F_\alpha^q$ with Borel measure symbols for $1\leq p,q\leq \infty$. For positive measures $\mu$ and $q\leq p$, we provide necessary and sufficient conditions in terms of the Berezin transform and establish a rigidity property for nuclearity across this range. In the case $p<q$, we obtain separate necessary and sufficient conditions, indicating that the Berezin transform alone is insufficient for a complete characterization. Our results extend to Fock spaces on $\mathbb{C}^n$.

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arXiv2026

Spectral dynamics for the infinite dihedral group and the lamplighter group

For a tuple $A=(A_0,A_1,\cdots,A_n)$ of elements in a Banach algebra $\mathfrak{B}$, its projective (joint) spectrum $p(A)$ is the collection of $z\in \mathbb{P}^n$ such that $A(z)=z_0A_0+z_1A_1+\cdots+z_nA_n$ is not invertible. If $\mathfrak{B}$ is the group $C^*$-algebra for a discrete group $G$ generated by $A_0, A_1,\dots, A_n$ with a representation $\rho$, then $p(A)$ is an invariant of (weak) equivalence for $\rho$. In \cite{BY}, B. Goldberg and R. Yang proved that the Julia set $\mathcal{J}(F)$ of the induced rational map $F$ for the infinite dihedral group $D_\infty$ is the union of the projective spectrum with the extended indeterminacy set. But the extended indeterminacy set $E_F$ is complicated. To obtain a better relationship between the projective spectrum and the Julia set, by replacing $A_\pi(z)=z_0+z_1\pi(a)+z_2\pi(t)$ with the extended pencil $A_\pi(z)=z_0+z_1\pi(a)+z_2\pi(t)+z_3\pi(at)$, where $\pi$ is the Koopman representation, and using the method of operator recursions, we show that $p(A_\pi)=\mathcal{J}(F).$ Further, we study the spectral dynamics for the Lamplighter group $\mathcal{L}$, and prove that $\mathcal{J}(Q)=E_Q$, where $Q$ is the rational map associated with $\mathcal{L}$.

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arXiv2026

A geometric approach to the compressed shift operator on the Hardy space over the bidisk

This paper studies the compressed shift operator $S_z$ on the Hardy space over the bidisk via the geometric approach. We calculate the spectrum and essential spectrum of $S_z$ on the Beurling type quotient modules induced by rational inner functions, and give a complete characterization for $S_z^*$ to be a Cowen-Douglas operator. Then we extend the concept of Cowen-Douglas operator to be the generalized Cowen-Douglas operator, and show that $S_z^*$ is a generalized Cowen-Douglas operator. Moreover, we establish the connection between the reducibility of the Hermitian holomorphic vector bundle induced by kernel spaces and the reducibility of the generalized Cowen-Douglas operator. By using the geometric approach, we study the reducing subspaces of $S_z$ on certain polynomial quotient modules.

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arXiv2025

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 Poincar\'e 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_\varphi$ with $\varphi (z) =\overline{z}^{m} + \alpha z^m + \beta$ are characterized, such as $\sigma(T_\varphi)= \overline{\varphi (\mathbb {D})}$, Fredholm index of $T_\varphi$ can only be one of $m,-m$ and $0$, $T_\varphi$ satisfies Coburn's theorem. These findings offer an illuminating example for the essential projective spectra of non-commuting operators.

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arXiv2025

Hilbert-Schmidtness of the $M_{\theta,\varphi}$-type submodules

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

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arXiv2025

Notes on a special order on $\mathbb{Z}^\infty$

In 1958, Helson and Lowdenslager extended the theory of analytic functions to a general class of groups with ordered duals. In this context, analytic functions on such a group $G$ are defined as the integrable functions whose Fourier coefficients lie in the positive semigroup of the dual of $G$. In this paper, we found some applications of their theory to infinite-dimensional complex analysis. Specifically, we considered a special order on $\mathbb{Z}^\infty$ and corresponding analytic continuous functions on $\mathbb{T}^\omega$, which serves as the counterpart of the disk algebra in infinitely many variables setting. By characterizing its maximal ideals, we have generalized the following theorem to the infinite-dimensional case: For a positive function $w$ that is integrable and log-integrable on $\mathbb{T}^d$, there exists an outer function $g$ such that $w=|g|^2$ if and only if the support of $\hat{\log w}$ is a subset of $\mathbb{N}^d\cap (-\mathbb{N})^d$. Furthermore, we have found the counterpart of the above function algebra in the closed right half-plane, and the representing measures of each point in the right half-plane for this algebra. As an application of the order, we provided a new proof of the infinite-dimensional Szeg\"{o}'s theorem.

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arXiv2023

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 $\Psi,\Psi^{\ast}$ are matrix-valued functions of bounded type and the inner part of $\Psi$ of Douglas-Shapiro-Shields factorization is a scalar inner function, then every hyponormal Toeplitz operator $T_{\Psi}$ whose square is also hyponormal must be either normal or analytic.

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