We prove that the group of weighted composition operators induced by continuous automorphism groups of the upper half plane $$\mathbb{U}$$ is strongly continuous on the weighted Dirichlet space of $$\mathbb{U}$$ , $${{\cal D}_\alpha }(\mathbb{U})$$ . Further, we investigate when they are isometries on $${{\cal D}_\alpha }(\mathbb{U})$$ . In each case, we determine the semigroup properties while in the case that the induced composition group is an isometry, we apply similarity theory to determine the spectral properties of the group.

中文翻译：

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上半平面加权狄利克雷空间上的组成群谱

我们证明了由上半平面$$\mathbb{U}$$ 的连续自同构群引出的加权复合算子群在$$\mathbb{U}$$ , $${ 的加权狄利克雷空间上是强连续的{\cal D}_\alpha }(\mathbb{U})$$ 。此外，我们调查了它们何时是 $${{\cal D}_\alpha }(\mathbb{U})$$ 上的等距。在每种情况下，我们确定半群属性，而在诱导组合群是等距的情况下，我们应用相似理论来确定该群的谱属性。