On Semi-Groups of Selfadjoint Transformations in Hilbert Space.

In a paper dealing with the representations of Lie groups by linear transformations in the n-dimensional and Hilbert space,' I have obtained the following particular result: Let Ht (c < t < co) be afamily of bounded selfadjoint transformations in Hilbert space i,? forming a group, i.e., Ho = E (the identical transformation), HtHs = Ht + s Suppose that (for every element f in O) the numerical function (Htf, f) = IHt,2f 12 of t is Lebesgue-measurable in a suitable neighborhood of t = 0. Let us denote by m resp. M the lower resp. the upper bound of H1 (we have m > 0). Then there is a resolution of the identity, EA, continuous from the left, such that E, = O for X _ m, E, = Efor X > M and that