Abstract. Let $S = (S_1,\ldots, S_n)$ be a tuple of symmetric operators $S_j\colon D(S_j)\subset H\rightarrow H$ on a Hilbert space $H$. We show that $S$ is jointly essentially self-adjoint in the sense of Fuglede if and only if a suitable associated matrix operator is essentially self-adjoint as a single operator. As an application we obtain an elementary proof of Nelson's famous commutativity criterion for essentially self-adjoint operators.

AMS Subject Classification (1991): 47B25, 47B15

Received December 2, 1999. (Registered under 2789/2009.)