Communications of the
Korean Mathematical Society CKMS

We show that if $f \in L^{\infty}(D^{n})$ satisfies $Sf=rf$ for some $r$ in the unit circle , where $S$ is any convex combinatiom of the iterations of Berezin operator , then $f$ is $n-$ harmonic. And we give some remarks and a conjecture on the space $$M_2 = \{ f \in L^{2}(D^{2}, m \times m) | Bf=f \}.$$

Keywords: Berezin transform, joint eigenspaces, Hilbert space, n-harmonic

MSC numbers: 47A15, 46C15