能量空间上的乘法算子
Multiplication operators on the energy space
摘要：
本文从与网络上的函数$f$相关的乘法运算符$M_f$的角度研究了无限网络（加权连通图）上的“能量空间”$\mathcal{H}_{\mathcal{E}}$（有限能量函数的希尔伯特空间，又名Dirichlet有限函数）。我们证明了乘法算子$M_f$不是Hermitian的，除非$f$是常数，并计算伴随$M_f^\star$的再生核$\mathcal{H}_{\mathcal{E}}$。利用半正定函数给出了有界乘法算子的一个特征，并给出了$f$上$M_f$有界的一些条件.实例表明，$f$有界或能量有限是不够的. $M_f$的有界性条件也表示在网络上的简单随机游动的行为。我们还考虑了$\mathcal{H}_{\mathcal{E}}$的有界元素和对应于它们的（可能无界的）乘法算子。在以前的论文中，作者使用泛函积分来构造无限网络的一类边界。边界在这里描述的某些子代数的这些乘法运算符，并嵌入到该子代数的Gel'fand空间。在有限能量的唯一调和函数为常数的情况下，我们证明了Gel'fand空间是底层网络的1-点紧化。
英文摘要：
This paper studies the "energy space" $\mathcal{H}_{\mathcal{E}}$ (the Hilbert space of functions of finite energy, aka the Dirichlet-finite functions) on an infinite network (weighted connected graph), from the point of view of the multiplication operators $M_f$ associated to functions $f$ on the network. We show that the multiplication operators $M_f$ are not Hermitian unless $f$ is constant, and compute the adjoint $M_f^\star$ in terms of a reproducing kernel for $\mathcal{H}_{\mathcal{E}}$. A characterization of the bounded multiplication operators is given in terms of positive semidefinite functions, and we give some conditions on $f$ which ensure $M_f$ is bounded. Examples show that it is not sufficient that $f$ be bounded or have finite energy. Conditions for the boundedness of $M_f$ are also expressed in terms of the behavior of the simple random walk on the network. We also consider the bounded elements of $\mathcal{H}_{\mathcal{E}}$ and the (possibly unbounded) multiplication operators corresponding to them. In a previous paper, the authors used functional integration to construct a type of boundary for infinite networks. The boundary is described here in terms of a certain subalgebra of these multiplication operators, and is shown to embed into the Gel'fand space of that subalgebra. In the case when the only harmonic functions of finite energy are constant, we show that the Gel'fand space is the 1-point compactification of the underlying network.
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