Let $u$ and $\unicode[STIX]{x1D711}$ be two analytic functions on the unit disc $D$ such that $\unicode[STIX]{x1D711}(D)\subset D$ . A weighted composition operator $uC_{\unicode[STIX]{x1D711}}$ induced by $u$ and $\unicode[STIX]{x1D711}$ is defined by $uC_{\unicode[STIX]{x1D711}}f:=u\cdot f\circ \unicode[STIX]{x1D711}$ for every $f$ in $H^{p}$ , the Hardy space of $D$ . We investigate compactness of $uC_{\unicode[STIX]{x1D711}}$ on $H^{p}$ in terms of function-theoretic properties of $u$ and $\unicode[STIX]{x1D711}$ .

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We investigate the boundedness, compactness, invertibility and Fredholmness of weighted composition operators between Lorentz spaces. It is also shown that the classes of Fredholm and invertible weighted composition maps between Lorentz spaces coincide when the underlying measure space is nonatomic.

We show that Fredholm weighted composition operators on L p-spaces with nonatomic measures are precisely the invertible ones. We also characterize the classes of Fredholm and invertible weighted composition operators on l p. Furthermore, the closedness of ranges and Fredholmness of these operators on H p-spaces of the unit disk are investigated.