Conditionally heteroskedastic processes are commonly described by GARCH models. Such models have been intensively analyzed in the univariate and multivariate settings, and more recently in the settings of high-dimensional and function-valued data. So far in the functional setting, GARCH models have only been extended to ``pointwise" models akin to the multivariate diagonal GARCH. In this talk we consider functional GARCH models that describe the evolution of the entire conditional covariance operator of the data, which we term ``operator-level fGARCH" models. We derive sufficient conditions for the existence of unique, strictly stationary solutions of operator-level fGARCH recursions, moment properties, and consistency properties of regularized Yule-Walker estimators for the infinite dimensional model parameters. We demonstrate in several simulation experiments and data analyses to high-frequency asset returns data the usefulness of this new class of models.