In functional data analysis (FDA), the covariance function is fundamental not only as a critical quantity for understanding elementary aspects of functional data but also as an indispensable ingredient for many advanced FDA methods. A new class of nonparametric covariance function estimators in terms of various spectral regularizations of an operator associated with a reproducing kernel Hilbert space is developed. Despite their nonparametric nature, the covariance estimators are automatically positive semi-definite, which is an essential property of covariance functions, via a one-step procedure. An unconventional representer theorem is established to provide a finite dimensional representation for this class of covariance estimators based on data, although the solutions are searched over infinite dimensional functional spaces. To further achieve a low-rank representation, another desirable property, e.g., for dimension reduction and easy interpretation, the trace-norm regularization is particularly studied, under which an efficient algorithm is developed based on the accelerated proximal gradient method. The outstanding practical performance of the trace-norm-regularized covariance estimator is demonstrated by a simulation study and the analysis of a traffic dataset. Under both fixed and random designs, an excellent rate of convergence is established for a broad class of operator-regularized covariance function estimators, which generalizes both the trace-norm-regularized covariance estimator and other popular alternatives.