Normal approximation of $U$-statistics in Hilbert space

Let $\{U_n\}$, $n=1,2\ldots$ be Hilbert space $H$-valued $U$-statistics with kernel $\Phi(\cdotp,\cdot)$, corresponding to a sequence of observations (random variables) $X_1,X_2,\ldots\ $. The rate of convergence on balls in the central limit theorem for $\{U_n\}$ is investigated. The obtained estimate is of order $n^{-1/2}$ and depends explicitly on $\mathbb E\|\Phi(X_1,X_2)\|^3$ and on the trace and the first nine eigenvalues of the covariance operator of $\mathbb E(\Phi(X_1,X_2)|X_1)$.