Analytic cohomology in a Banach space

On real analytic Banach manifolds

Let $X$ be a real Banach space with an unconditional basis (e.g., $X=\ell_2$ Hilbert space), $Ω\subset X$ open, $M\subsetΩ$ a closed split real analytic Banach submanifold of $Ω$, $E\to M$ a real analytic Banach vector bundle, and ${\Cal A}^E\to M$ the sheaf of germs of real analytic sections of $E\to M$. We show that the sheaf cohomology groups $H^q(M,{\Cal A}^E)$ vanish for all $q\ge1$, and there is a real analytic retraction $r:U\to M$ from an open set $U$ with $M\subset U\subsetΩ$ such that $r(x)=x$ for all $x\in M$. Some applications are also given, e.g., we show that any infinite dimensi…

解析函数在特定 Banach 空间上的边界行为

Boundary behavior of analytic functions on certain Banach spaces

本文证明了在多项式稠密且点求值连续的圆盘解析函数 Banach 空间中，若某函数模的上极限在单位圆上几乎处处无穷大，则该性质对残余生成的函数集同样成立。

An analytic Koszul complex in a Banach space

We show that the holomorphic ideal sheaf of a linear section of a pseudoconvex open subset $Ω$ of, say, a Hilbert space $X=\ell_2$ is acyclic. We also prove an analog of Hefer's lemma, i.e., if $f:Ω\timesΩ\to\CC$ is holomorphic and $f(x,x)=0$ for $x\inΩ$, then there is a holomorphic $g:Ω\timesΩ\to X^*$ with values in the dual space $X^*$ of $X$ such that $f(x,y)=g(x,y)(x-y)$

Banach Analytic Sets and a Non-Linear Version of the Levi Extension Theorem

We prove a certain non-linear version of the Levi extension theorem for meromorphic functions. This means that the meromorphic function in question is supposed to be extendable along a sequence of complex curves, which are arbitrary, not necessarily straight lines. Moreover, these curves are not supposed to belong to any finite dimensional analytic family. The conclusion of our theorem is that nevertheless the function in question meromorphically extends along an (infinite dimensional) analytic family of complex curves and its domain of existence is a pinched domain filled in by this analytic …

Cohomology of semi 1-coronae and extension of analytic subsets

We generalize some of the results in [arXiv: math.CV/0503430], and prove a bump-lemma for closed sets in semi 1-coronae. From this we obtain some finite cohomology results and an extension theorem for analytic subsets in 1-coronae.

Volterra type operators on weighted Banach spaces of analytic functions

Smith et al. recently gave the sufficient and necessary conditions for the boundedness of Volterra type operators on Banach spaces of bounded analytic functions when the symbol functions are univalent. In this paper, we give the complete characterizations of the conditions for the boundedness and compactness of Volterra type operators $T_g$ and their companion operators $S_g$ between weighted Banach spaces of analytic functions, which essentially generalize their works.

Note on the quasi-proper direct image with value in a Banach analytic set

We give a rather simple proof of the generalization of Kuhlmann's quasi-proper direct image theorem to the case of a map with values in a Banach analytic set. The proof uses a generalization of the Remmert-Stein's theorem to this context. AMS Classification. 32 H 02-32 K 05-32 C 25.

Representing analytic cohomology groups of complex manifolds

Consider a holomorphic vector bundle $L\to X$ and an open cover ${\frak U}=\{U_a\colon a\in A\}$ of $X$, parametrized by a complex manifold $A$. We prove that the sheaf cohomology groups $H^q(X,L)$ can be computed from the complex $C^{\bullet}_{\text{hol}}$ $({\frak U},L)$ of cochains $(f_{a_0\ldots a_q})_{a_0,\ldots, a_q\in A}$ that depend holomorphically on the $a_j$, provided $S=\{(a,x)\in A\times X\colon x\in U_a\}$ is a Stein open subset of $A\times X$. The result is proved in the setting of Banach manifolds, and is applied to study representations on cohomology groups induced by a holomo…

Some applications of interpolating sequences for Banach spaces of analytic functions

M. J. Beltrán-Meneua et al. \cite{beltran1} and E. Jordá and A. Rodríguez-Arenas \cite{jorda} characterized the (uniformly) mean ergodic composition operators on $H^\infty(\mathbb{D})$ and $H^\infty_ν(\mathbb{D})$, respectively. In this paper, by using the interpolating sequences, we give other necessary and sufficient conditions for the (uniformly) mean ergodicity of composition operators on these spaces.

一个解析常数小于1的2维实Banach空间

A $2$-dimensional real Banach space with constant of analyticity less than one

本文证明了在实2维Banach空间$\ell_1^2$上存在解析函数，其某点幂级数的均匀收敛半径严格小于$1-\\|a\\|$，揭示了实解析函数中解析半径与均匀收敛半径的根本区别，并首次为解析常数提供了非平凡上界。

Compact multipliers on spaces of analytic functions

In the paper compact multiplier operators on Banach spaces of analytic functions on the unit disk with the range in Banach sequence lattices are studied. If the domain space $X$ is such that $H_\infty\hookrightarrow X\hookrightarrow H_1$, necessary and sufficient conditions for compactness are presented. Moreover, the calculation of the Hausdorff measure of noncompactness for diagonal operators between Banach sequence lattices is applied to obtaining the characterization of compact multipliers in case the domain space $X$ satisfies $H_\infty\hookrightarrow X\hookrightarrow H_2$.

解析多项式Banach空间的局部常数与Bohr现象

Local constants and Bohr's phenomenon for Banach spaces of analytic polynomials

本研究构建通用框架分析多元解析多项式Banach空间的投影常数等基本常数的渐近最优行为，推导一般Banach序列格的Bohr半径估计，并在有限维Lorentz序列空间中得到宽参数范围的最优结果。

On holomorphic domination, I

Let $X$ be a separable Banach space and $u{:} X\to\Bbb{R}$ locally upper bounded. We show that there are a Banach space $Z$ and a holomorphic function $h{:} X\to Z$ with $u(x)<\|h(x)\|$ for $x\in X$. As a consequence we find that the sheaf cohomology group $H^q(X,\Cal{O})$ vanishes if $X$ has the bounded approximation property (i.e., $X$ is a direct summand of a Banach space with a Schauder basis), $\Cal{O}$ is the sheaf of germs of holomorphic functions on $X$, and $q\ge1$. As another consequence we prove that if $f$ is a $C^1$-smooth $\overline\partial$-closed $(0,1)$-form on the space $X=L_…

Note on vector valued Hardy spaces related to analytic functions having distributional boundary values

Analytic functions defined on a tube domain $T^{C}\subset \mathbb{C}^{n}$ and taking values in a Banach space $X$ which are known to have $X$-valued distributional boundary values are shown to be in the Hardy space $H^{p}(T^{C},X)$ if the boundary value is in the vector valued Lebesgue space $L^{p}(\mathbb{R}^{n},X)$, where $1\leq p \leq \infty$ and $C$ is a regular open convex cone. Poisson integral transform representations of elements of $H^{p}(T^{C}, X)$ are also obtained for certain classes of Banach spaces, including reflexive Banach spaces.