Diagonalization and Representation Results for Nonpositive Sesquilinear Form Measures

We study decompositions of operator measures and more general sesquilinear form measures E into linear combinations of positive parts, and their diagonal vector expansions. The underlying philosophy is to represent E as a trace class valued measure of bounded variation on a new Hilbert space related to E. The choice of the auxiliary Hilbert space fixes a unique decomposition with certain properties, but this choice itself is not canonical. We present relations to Naimark type dilations and direct integrals. Mathematics Subject Classification. 47A70 (Primary); 47A07 (Secondary). Date: February 1, 2008. Corresponding author. Telephone: +358-2-333 5737, telefax: +358-2-333 5070. 1 2 HYTÖNEN, PELLONPÄÄ, AND YLINEN