On the spectral kernel of symmetric extensions

In a Hilbert space H we consider closed and symmetric operators A and à with closed ranges such that AÃ. We prove a necessary and sufficient condition for the existence of a closed and symmetric operator B with ABà the range of which is not closed. We show that this condition can be fulfilled and, by the way, we get a counter example to the assertion that the continuous part of the spectral kernel of a symmetric operator is contained in the corresponding part of a symmetric extension, as is claimed in the books of Achieser-Glasmann 1], Neumark 2] and Smirnow 3].