Non-commutative rational functions in the full Fock space
Non-commutative rational functions in the full Fock space
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DOI：
10.1090/tran/8418
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发表时间：
2020-10
影响因子：
1.3
通讯作者：
M. Jury;R. Martin;E. Shamovich
中科院分区：
文献类型：
--
作者：
M. Jury;R. Martin;E. Shamovich
当且仅当有理函数有界于复数单位圆盘内时，它属于平方可和幂级数的 Hardy 空间 $H^2$。任何这样的有理函数都必须在半径大于 1 的圆盘中进行解析。有理函数 $\mathfrak{r} \in H^2$ 的内部因式分解特别简单：$\mathfrak{r}$ 的内部因式是（有限）Blaschke 乘积，并且（因此）内部和外部因式又都是有理数。我们将 $H^2$ 中的有理函数的这些和其他基本事实扩展到 $\mathbb{C}^d$ 上的完整 Fock 空间，定义为多个 NC 变量中平方可和幂级数的 \emph{非交换 (NC) Hardy 空间}。特别是，我们描述了 NC 有理函数属于 Fock 空间的情况，证明了 NC 有理函数和 NC 多项式的内外因式分解的经典结果的类似物，并获得了 NC 有理乘子的谱结果。
A rational function belongs to the Hardy space, $H^2$, of square-summable power series if and only if it is bounded in the complex unit disk. Any such rational function is necessarily analytic in a disk of radius greater than one. The inner-outer factorization of a rational function, $\mathfrak{r} \in H^2$ is particularly simple: The inner factor of $\mathfrak{r}$ is a (finite) Blaschke product and (hence) both the inner and outer factors are again rational. We extend these and other basic facts on rational functions in $H^2$ to the full Fock space over $\mathbb{C}^d$, identified as the \emph{non-commutative (NC) Hardy space} of square-summable power series in several NC variables. In particular, we characterize when an NC rational function belongs to the Fock space, we prove analogues of classical results for inner-outer factorizations of NC rational functions and NC polynomials, and we obtain spectral results for NC rational multipliers.