Abstract. We present a von Neumann--Wold decomposition of a two-isometric operator on a general Hilbert space. A pure two-isometry is shown to be unitarily equivalent to a shift operator (multiplication by the independent variable) on a Dirichlet space $D(\mu )$ corresponding to a positive operator measure $\mu $ on the unit circle. Our result contains a previous result by S. Richter [Richter2] as well as the classical von Neumann--Wold decomposition of an isometry.

AMS Subject Classification (1991): 47A15, 31C25

Keyword(s): Two-isometry, von Neumann--Wold decomposition, Dirichlet space

Received February 26, 2004. (Registered under 5841/2009.)