The Jordan decomposition and Kaplansky's second test problem for Hermitian holomorphic vector bundles
Bingzhe Hou, Chunlan Jiang
TL;DR
The paper reframes Kaplansky's second test problem in complex geometry by studying push-forward Hermitian holomorphic vector bundles $E_{h(S_{\beta})}$ on weighted Hardy spaces $H^2_{\beta}$. It proves a Jordan-type decomposition: for any $f\in\mathrm{Hol}(\overline{\mathbb{D}})$ there exists a unique $m$ and $h$ such that $E_{f(S_{\beta})}\sim\bigoplus_{1}^{m} E_{h(S_{\beta})}$, with $E_{h(S_{\beta})}$ indecomposable and $h$ unique up to disk automorphisms. It further provides a similarity classification: $E_{h_1(S_{\beta})}$ and $E_{h_2(S_{\beta})}$ are similar precisely when $h_1=h\circ B_1$ and $h_2=h\circ B_2$ for Blaschke products $B_1,B_2$ of the same order, reducing Kaplansky's second test problem to a geometric setting. The results also affirm a geometric version of a problem posed by Douglas (2007) and yield $K_0$-group identifications for the commutant algebras of multiplication operators on $H^2_{\beta}$, highlighting the role of polynomial growth. The paper closes with counterexamples showing the polynomial growth assumption is necessary and discusses broader implications for Cowen-Douglas operators and invariant deformation algebras.
Abstract
In 1954, I. Kaplansky proposed three test problems for deciding the strength of structural understanding of a class of mathematical objects in his treatise "Infinite abelian groups", which can be formulated for very general mathematical systems. In this paper, we focus on Kaplansky's second test problem in a context of complex geometry. Let $H^2_β$ be a weighted Hardy space. The Cowen-Douglas operator theory tells us that each $h\in\textrm{Hol}(\overline{\mathbb{D}})$ induces a Hermitian holomorphic vector bundle on $H^2_β$, denoted by $E_{h(S_β)}(Ω)$, where $Ω$ is a domain. We show that the vector bundle $E_{h(S_β)}$ is a push-forwards Hermitian holomorphic vector bundle and study the similarity deformation problems. Our main theorem is that if $H^2_β$ is a weighted Hardy space of polynomial growth, then for any $f\in \textrm{Hol}(\overline{\mathbb{D}})$, there exists a unique positive integer $m$ and an function $h\in\textrm{Hol}(\overline{\mathbb{D}})$ inducing an indecomposable vector bundle $E_{h(S_β)}$, such that $E_{f(S_β)}$ is similar to $\bigoplus_1^m E_{h(S_β)}$, where $h$ is unique in the sense of analytic automorphism group action. That could be seemed as a Jordan decomposition theorem for the push-forwards Hermitian holomorphic vector bundles. Furthermore, we give the similarity classification of those push-forwards Hermitian holomorphic vector bundles induced by analytic functions, and give an affirmative answer to Kaplansky's second test problem for those objects. We also give an affirmative answer to the geometric version and generalized version of a problem proposed by R. Douglas in 2007, and obtain the $K_0$-group of the commutant algebra of a multiplication operator on a weighted Hardy space of polynomial growth. In addition, we give an example to show the setting of polynomial growth condition is necessary.