论文标题
关于广义Hartogs三角形的操作者理论
Operator theory on generalized Hartogs triangles
论文作者
论文摘要
我们考虑$ n $ $ n $ tuples $ p $由多项式$ p_1，\ ldots，p_n $组成的$ n $ p $的家庭$ \ mathcal p $ p $ p_n $，可满足$ \ partial_i p_j（0）=δ__=δ__{i，j}，$ i，$ i，$ i，$ i，$ i，j = 1，$ ndots， $ \ triangle^{\！n} _ {_ p} $，我们将称为广义Hartogs Triangle。我们对选择$ p_a =（p_ {1，a}，\ ldots，p_ {n，a}），$ $ $ a \ geq 0，$ p_ {j { $ p_a $由\ begin {qore} \ triangle^{\！ N-1， | z_n |^2 + a | z_1 |^2 <1 \ big \}。 \ end {equation}域$ \ triangle^{\！n} _ {_ p}，$ $ n \ geq 2 $从来都不是多项式凸。但是，$ \ triangle^{\！n} _ {_ p} $始终是holomorphorthensy convex。在\ Mathcal p $中的任何$ p \ in \ mathbb n^n中，我们将$ \ triangle^{\！n} _ p}的积极的半定义核$ \ mathscr $ \ mathscr k _ {_ {p，m}} $与$ \ triangle^{\！ space $ \ mathscr h^2_m（\ triangle^{\！n} _ {_ p}）$ holomorphic函数在$ \ triangle^{\！n} _ {_ p}上。乘法$ n $ -tuple $ \ mathscr m_z $作用于$ \ mathscr h^2_m（\ triangle^{\！n} _ {_ p}）。$，事实证明，$ \ Mathscr m_z $从来都不是合理的。尽管$ \ mathscr m^*_ z-λ$的关节内核的维度是每一个$ 1 $的$ 1 $的常量$ \ infty。$我们利用联合亚正态的概念来定义$ \ triangle^{\！n} _ {_ 0}的耐力空间。
We consider the family $\mathcal P$ of $n$-tuples $P$ consisting of polynomials $P_1, \ldots, P_n$ with nonnegative coefficients which satisfy $\partial_i P_j(0) = δ_{i, j},$ $i, j=1, \ldots, n.$ With any such $P,$ we associate a Reinhardt domain $\triangle^{\!n}_{_P}$ that we will call the generalized Hartogs triangle. We are particularly interested in the choices $P_a = (P_{1, a}, \ldots, P_{n, a}),$ $a \geq 0,$ where $P_{j, a}(z) = z_j + a \prod_{k=1}^n z_k,~ j=1, \ldots, n.$ The generalized Hartogs triangle associated with $P_a$ is given by \begin{equation} \triangle^{\!n}_a = \Big\{z \in \mathbb C \times \mathbb C^{n-1}_* : |z_j|^2 < |z_{j+1}|^2(1-a|z_1|^2), ~j=1, \ldots, n-1, |z_n|^2 + a|z_1|^2 < 1\Big\}. \end{equation} The domain $\triangle^{\!n}_{_P},$ $n \geq 2$ is never polynomially convex. However, $\triangle^{\!n}_{_P}$ is always holomorphically convex. With any $P \in \mathcal P$ and $m \in \mathbb N^n,$ we associate a positive semi-definite kernel $\mathscr K_{_{P, m}}$ on $\triangle^{\!n}_{_P}.$ This combined with the Moore's theorem yields a reproducing kernel Hilbert space $\mathscr H^2_m(\triangle^{\!n}_{_P})$ of holomorphic functions on $\triangle^{\!n}_{_P}.$ We study the space $\mathscr H^2_m(\triangle^{\!n}_{_P})$ and the multiplication $n$-tuple $\mathscr M_z$ acting on $\mathscr H^2_m(\triangle^{\!n}_{_P}).$ It turns out that $\mathscr M_z$ is never rationally cyclic. Although the dimension of the joint kernel of $\mathscr M^*_z-λ$ is constant of value $1$ for every $λ\in \triangle^{\!n}_{_P}$, it has jump discontinuity at the serious singularity $0$ of the boundary of $\triangle^{\!n}_{_P}$ with value equal to $\infty.$ We capitalize on the notion of joint subnormality to define a Hardy space on $\triangle^{\!n}_{_0}.$ This in turn gives an analog of the von Neumann's inequality for $\triangle^{\!n}_{_0}.$
