On the delbar-equation in a Banach space

Banach代数值 polydisk 上 H^∞ 函数的 Bézout 方程可解性

Solvability of the Bézout Equation for Banach Algebra-Valued $H^\infty$ Functions on the Polydisk

研究Banach代数值polydisk上H^∞函数的Bézout方程可解性，结合维度归纳与拓扑结构分析，推导出更广泛子代数类的可解性。

$L^\infty$ Estimates for the Banach-valued $\bar\partial$ problem in a Disk

We study the differential equation $\frac{\partial G}{\partial\bar z}=g$ with an unbounded Banach-valued Bochner measurable function $g$ on the open unit disk $\mathbb D\subset\mathbb C$. We prove that under some conditions on the growth and essential support of $g$ such equation has a bounded solution given by a continuous linear operator. The obtained results are applicable to the Banach-valued corona problem for the algebra of bounded holomorphic functions on $\mathbb D$ with values in a complex commutative unital Banach algebra.

The Sylvester equation in Banach algebras

Let $\mathcal{A}$ be a unital complex semisimple Banach algebra, and $M_{\mathcal{A}}$ denote its maximal ideal space. For a matrix $M\in {\mathcal{A}}^{n\times n}$, $\widehat{M}$ denotes the matrix obtained by taking entry-wise Gelfand transforms. For a matrix $M\in {\mathbb{C}}^{n\times n}$, $σ(M)\subset \mathbb{C}$ denotes the set of eigenvalues of $M$. It is shown that if $A\in {\mathcal{A}}^{n\times n}$ and $B\in {\mathcal{A}}^{m\times m}$ are such that for all $φ\in M_{\mathcal{A}}$, $σ(\widehat{A}(φ))\cap σ(\widehat{B}(φ))=\emptyset$, then for all $C\in {\mathcal{A}}^{n\times m}$, the S…