  This article generalises the concept of realised covariation to
Hilbert-space-valued stochastic processes. More precisely, based on
high-frequency functional data, we construct an estimator of the trace-class
operator-valued integrated volatility process arising in general mild solutions
of Hilbert space-valued stochastic evolution equations in the sense of Da Prato
and Zabczyk (2014). We prove a weak law of large numbers for this estimator,
where the convergence is uniform on compacts in probability with respect to the
Hilbert-Schmidt norm. In addition, we show that the conditions on the
volatility process are valid for most common stochastic volatility models in
Hilbert spaces.
