A mean value theorem on bounded symmetric domains

Let $\Omega$ be a Cartan domain of rank $r$ and genus $p$ and $B_\nu$, $\nu >p-1$, the Berezin transform on $\Omega$; the number $B_{\nu }f(z)$ can be interpreted as a certain invariant-mean-value of a function $f$ around $z$. We show that a Lebesgue integrable function satisfying $f=B_\nu f=B_{\nu +1}f=\dots =B_{\nu +r}f$, $\nu \ge p$, must be $\mathcal{M}$-harmonic. In a sense, this result is reminiscent of Delsarte's two-radius mean-value theorem for ordinary harmonic functions on the complex $n$-space $\mathbf{C}^{n}$, but with the role of radius $r$ played by the quantity $1/\nu$.