Factorization and Reflexivity on Fock spaces

Non-commutative rational functions in the full Fock space

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{…

The $H^\infty$ algebras of higher rank graphs

We begin the study of a new class of operator algebras that arise from higher rank graphs. Every higher rank graph generates a Fock space Hilbert space and creation operators which are partial isometries acting on the space. We call the weak operator topology closed algebra generated by these operators a 'higher rank semigroupoid algebra'. A number of examples are discussed in detail, including the single vertex case and higher rank cycle graphs. In particular the cycle graph algebras are identified as matricial multivariable function algebras. We obtain reflexivity for a wide class of graphs …

Blaschke-Singular-Outer factorization of free non-commutative functions

By classical results of Herglotz and F. Riesz, any bounded analytic function in the complex unit disk has a unique inner-outer factorization. Here, a bounded analytic function is called \emph{inner} or \emph{outer} if multiplication by this function defines an isometry or has dense range, respectively, as a linear operator on the Hardy Space, $H^2$, of analytic functions in the complex unit disk with square-summable Taylor series. This factorization can be further refined; any inner function $θ$ decomposes uniquely as the product of a \emph{Blaschke inner} function and a \emph{singular inner} …

Small Hankel operators on generalized Fock spaces

We consider Fock spaces $F^{p,\ell}_α$ of entire functions on ${\mathbb C}$ associated to the weights $e^{-α|z|^{2\ell}}$, where $α>0$ and $\ell$ is a positive integer. We compute explicitly the corresponding Bergman kernel associated to $F^{2,\ell}_α$ and, using an adequate factorization of this kernel, we characterize the boundedness and the compactness of the small Hankel operator $\mathfrak{h}^{\ell}_{b,α}$ on $F^{p,\ell}_α$. Moreover, we also determine when $\mathfrak{h}^{\ell}_{b,α}$ is a Hilbert-Schmidt operator on $F^{2,\ell}_α$.

Fock空间上的里程表映射：块分解、Toeplitz型实现与伴随

Odometer maps on Fock spaces: block decompositions, Toeplitz-type realizations, and the adjoint

本文研究向量值全Fock空间上的里程表映射$W_L$，通过上三角块分解和Hardy空间中的解析Toeplitz算子$M_\Theta$实现，刻画了其等距、酉、可逆性及范数恒等式和Douglas型分解性质，并推导了$W_L^*$的显式公式，进而分析了核、多重性、Fredholm性、本质正规性及谱性质。