We study generalization under label shift in domain adaptation where the learner has access to labeled samples from the source domain but unlabeled samples from the target domain. Prior works deploy label classifiers and introduce various methods to estimate the importance weights from source to target domains. They use these estimates in importance weighted empirical risk minimization to learn classifiers. In this work, we theoretically compare the prior approaches, relax their strong assumptions, and generalize them from requiring label classifiers to general functions. This latter generalization improves the conditioning on the inverse operator of the induced inverse problems by allowing for broader exploitation of the spectrum of the forward operator. The prior works in the study of label shifts are limited to categorical label spaces. In this work, we propose a series of methods to estimate the importance weight functions for arbitrary normed label spaces. We introduce a new operator learning approach between Hilbert spaces defined on labels (rather than covariates) and show that it induces a perturbed inverse problem of compact operators. We propose a novel approach to solve the inverse problem in the presence of perturbation. This analysis has its own independent interest since such problems commonly arise in partial differential equations and reinforcement learning. For both categorical and general normed spaces, we provide concentration bounds for the proposed estimators. Using the existing generalization analysis based on Rademacher complexity, R\'enyi divergence, and MDFR lemma in Azizzadenesheli et al. , we show the generalization property of the importance weighted empirical risk minimization on the unseen target domain.