We provide complete characterisations for the compactness of weighted composition operators between two distinct $L^{p}$-spaces, where $1\leq p\leq \infty$. As a corollary, when the underlying measure space is nonatomic, the only compact weighted composition map between $L^{p}$-spaces is the zero operator.

中文翻译：

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空间之间的紧凑加权组合算子

我们为两个不同的加权合成算子的紧凑性提供了完整的特征$L^{p}$-空格，其中$1\leq p\leq \infty$. 作为推论，当底层测量空间是非原子的时，之间唯一的紧凑加权组合图$L^{p}$-spaces 是零运算符。