Friedrichs extension and essential spectrum of systems of differential operators of mixed order

A general construction for the Friedrichs extension of symmetric semi-bounded block operators L-0 = ((A)(C) (B)(D)), with not necessarily bounded entries, acting in the product of Hilbert spaces has been given by Konstantinov and Mennicken via the form gamma(mu)[u]: = <(A-mu)u,u > - <((D) over bar-mu)(-1) Cu,Cu >, u is an element of D(A). There the entry A was assumed to be essentially self-adjoint. Here it will be shown that the result remains true if A is only symmetric and that the closability of gamma(mu) follows from the semiboundedness of L-0. This will be applied to a 2 x 2 system of singular mixed-order differential equations satisfying the quasi-regularity condition, thus enabling us to give a much simpler calculation for the essential spectrum than in papers by Hardt, Mennicken, Naboko and Faierman, Mennicken, Moller, respectively, for a related 3 x 3-system. (c) 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim.