On the nonidentity of weak and strong extensions of differential operators

Since the celebrated theorem of Friedrichs [2], various authors have studied the question of the identity of weak and strong extension of partial differential operators (e.g., see [1], [3], [4], [5], [8] and [6]). In the last work, Phillips and Sarason have shown that their versions of weak and strong extensions need not agree. Recently, Ralston [7] has modified this example to show that the same is true for the usual weak and strong extensions. Here we would like to indicate another example whose basic feature is that the boundary matrix has constant nonzero eigenvalues. The operator is the Cauchy-Riemann operator