The Topological Structure of Weighted Composition Operators from Lipschitz Spaces to $$H^{\infty }$$ H ∞

We give some simple norm estimates of the difference of weighted composition operators acting from Lipschitz space $$\mathcal {L}_\alpha $$ to the space $$H^\infty $$ of all analytic functions on the unit disk, and showing that any two bounded weighted composition operators are pathwise connected. We also obtain some necessary or sufficient conditions for the difference to be compact. Unlike the equivalence of the boundedness and compactness for weighted composition operators from $$\mathcal {L}_\alpha $$ to $$H^\infty ,$$ we show that there exists a difference which is bounded, but not compact.