An Introduction to Hilbert Module Approach to Multivariable Operator Theory

摘要: Let $\{T_1, \ldots, T_n\}$ be a set of $n$ commuting bounded linear operators on Hilbert space $\mathcal{H}$. Then the $n$-tuple $(T_1, T_n)$ turns $\mathcal{H}$ into module over $\mathbb{C}[z_1, z_n]$ in following sense: \[\mathbb{C}[z_1, z_n] \times \mathcal{H} \raro \clh, \quad (p, h) \mapsto p(T_1, T_n)h,\]where $p \in \mathbb{C}[z_1, and $h \mathcal{H}$. The above is usually called z_n]$. modules (or natural function algebras) were first introduced by R. G. Douglas C. Foias 1976. two main driving forces algebraic complex geometric views to multivariable operator theory. This article gives an introduction algebras surveys some recent developments. Here theory presented as combination commutative algebra, geometry spaces its applications $n$-tuples ($n \geq 1$) operators. topics which are studied include: model from point view, holomorphic functions, tensor products, localizations, dilations, submodules quotient modules, free resolutions, curvature Fredholm modules. More developments study approach can found companion paper, "Applications Module Approach Multivariable Operator Theory".