arXiv subjects

Zeyou Zhu

Publications and source records attributed to Zeyou Zhu.

5 recordsLinked to original sources

arXiv2026

Essential normality of quotient modules vs. Hilbert-Schmidtness of submodules in $H^2(\mathbb D^2)$

In the present paper, we prove that all the quotient modules in $H^2(\mathbb D^2)$, associated to the finitely generated submodules containing a distinguished homogenous polynomial, are essentially normal, which is the first result on the essential normality of non-algebraic quotient modules in $H^2(\mathbb D^2)$. Moreover, we obtain the equivalence of the essential normality of a quotient module and the Hilbert-Schmidtness of its associated submodule in $H^2(\mathbb D^2)$, in the case that the submodule contains a distinguished homogenous polynomial. As an application, we prove that each finitely generated submodule containing a polynomial is Hilbert-Schmidt, which partially gives an affirmative answer to the conjecture of Yang \cite{Ya3}.

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arXiv2026

Arveson's version of the Gauss-Bonnet-Chern formula for Hilbert modules over the polynomial rings

In this paper, we refine the framework of Arveson's version of the Gauss-Bonnet-Chern formula by proving that a submodule in the Drury-Arveson module being locally algebraic is equivalent to Arveson's version of the Gauss-Bonnet-Chern formula holding true for the associated quotient module. Moreover, we establish the asymptotic Arveson's curvature invariant and the asymptotic Euler characteristic for contractive Hilbert modules over polynomial rings in infinitely many variables, and obtain the infinitely-many-variables analogue of Arveson's version of Gauss-Bonnet-Chern formula. Finally, we solve the finite defect problem for submodules of the Drury-Arveson module $H^2$ in infinitely many variables by proving that $H^2$ has no nontrivial submodules of finite rank.

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arXiv2026

Arveson's Gauss-Bonnet-Chern Formula for Hilbert Modules in the Multiplier-Algebra Framework

The present paper continues Arveson's program on curvature, Euler characteristic, and Gauss-Bonnet-Chern type formulae for Hilbert modules. For regular unitarily invariant complete Nevanlinna--Pick spaces on the unit ball in $\mathbb{C}^d$, we prove that Arveson's Gauss-Bonnet-Chern formula holds unconditionally over the full multiplier algebra. Together with the polynomial-ring case, this shows that the validity of the formula depends essentially on the coefficient algebra. We then solve Arveson's finite defect problem in the same framework by giving a sharp classification of finite defect submodules. This problem is natural in curvature theory, since both curvature and Euler characteristic are governed by the defect structure. Finally, using commutative-algebra methods, we apply this classification to unitary-equivalence rigidity for finite-codimensional submodules of vector-valued reproducing kernel Hilbert modules.

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arXiv2026

The norms for symmetric and antisymmetric tensor products of the weighted shift operators

In the present paper, we study the norms for symmetric and antisymmetric tensor products of weighted shift operators. By proving that for $n\geq 2$, $$\|S_α^{l_1}\odot\cdots \odot S_α^{l_k}\odot S_α^{*l_{k+1}}\odot\cdots \odot S_α^{*l_{n}}\| =\mathop{\prod}_{i=1}^n\left \| S_α^{l_{i}}\right\|, \text{ for any} \ (l_1,l_2\cdots l_n)\in\mathbb N^n$$ if and only if the weight satisfies the regularity condition, we partially solve \cite[Problem 6 and Problem 7]{GA}. It will be seen that most weighted shift operators on function spaces, including weighted Bergman shift, Hardy shift, Dirichlet shift, etc, satisfy the regularity condition. Moreover, at the end of the paper, we solve \cite[Problem 1 and Problem 2]{GA}.

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arXiv2024

Fredholm index of Toeplitz pairs with $H^{\infty}$ symbols

In the present paper, we characterize the Fredholmness of Toeplitz pairs on Hardy space over the bidisk with the bounded holomorphic symbols, and hence we obtain the index formula for such Toeplitz pairs. The key to obtain the Fredholmness of such Toeplitz pairs is the $L^p$ solution of Corona Problem over $\mathbb{D}^2$.

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