AI 中文总结
该研究确定了使乘法算子为弗雷德霍姆算子的解析函数巴拿赫空间性质，刻画了多种经典空间上的弗雷德霍姆乘法算子，还描述了这类空间上$M_z$不变的闭有限余维子空间并证明其循环性。
AI 中文摘要
我们确定复平面上开单位圆盘$\boldsymbol{\udbD1}$上解析函数巴拿赫空间$\boldsymbol{\udcB1}$的性质，确保乘法算子$\boldsymbol{M_\boldsymbol{\udc11}: \boldsymbol{\udcB1} \to \boldsymbol{\udcB1}}$为弗雷德霍姆算子当且仅当其符号$\boldsymbol{\udc12}$在$\boldsymbol{\udbD1}$边界附近远离0有界。这些性质为多种已深入研究的空间所共有，包括哈代空间$\boldsymbol{H^p(\boldsymbol{\udbD1})}$、加权伯格曼空间$\boldsymbol{A^p_\boldsymbol{\udc13}(\boldsymbol{\udbD1})}$、哈代-索伯列夫空间$\boldsymbol{H^2_\boldsymbol{\udb55}(\boldsymbol{\udbD1})}$、第$j$阶导数属于$\boldsymbol{H^p(\boldsymbol{\udbD1})}$的函数空间$\boldsymbol{S_j^p(\boldsymbol{\udbD1})}$以及圆盘代数$\boldsymbol{A}$。因此，作为推论，本研究刻画了这些空间上的弗雷德霍姆乘法算子。此外，我们根据$\boldsymbol{M_z: \boldsymbol{\udcB1} \to \boldsymbol{\udcB1}}$不变的$\boldsymbol{\udcB1}$的闭有限余维子空间所共有的零点，描述了这些子空间。我们讨论了这些子空间与刻画$\boldsymbol{\udcB1}$上弗雷德霍姆乘法算子问题之间的联系，并证明限制在这类子空间上的$\boldsymbol{M_z}$总是循环的，其多项式循环向量的次数等于该子空间的余维数。
英文摘要
We identify properties of a Banach space $\mathcal{B}$ of analytic functions on the open unit disk $\mathbb{D}$ in the complex plane ensuring that a multiplication operator $M_ψ: \mathcal{B} \to\mathcal{B}$ is Fredholm if and only if its symbol $ψ$ is bounded away from $0$ near $\partial \mathbb{D}$. The properties we identify are shared by a wide variety of much-studied spaces, including the Hardy spaces $H^p(\mathbb{D})$, weighted Bergman spaces $A^p_ω(\mathbb{D})$, Hardy-Sobolev spaces $H^2_β(\mathbb{D})$, the spaces $S_j^p(\mathbb{D})$ of functions having $j$-th derivative in $H^p(\mathbb{D})$, and the disk algebra $A$. Thus, as a corollary, our work characterizes Fredholm multiplication operators on these spaces. In addition, we describe the closed, finite-codimensional subspaces of $\mathcal{B}$ that are invariant under $M_z: \mathcal{B} \to \mathcal{B}$ in terms of the zeros that the functions in such subspaces have in common. We discuss connections between these subspaces and the problem of characterizing the Fredholm multiplication operators on $\mathcal{B}$, and we prove that $M_z$ restricted to such a subspace is always cyclic, with a polynomial cyclic vector having degree equal to the codimension of the subspace.
Comments27 pages, no figures
