FACTORIZATIONS OF NONNEGATIVE SYMMETRIC OPERATORS

We prove that each closed densely defined and nonnegative symmetric operator A having disjoint nonnegative self-adjoint extensions admits infinitely many factorizations of the form A = LLo, where L-o is a closed nonnegative symmetric operator and f its nonnegative self-adjoint extension. The same factorizations are also established for a non-densely defined nonnegative closed symmetric operator with infinite deficiency indices while for operator with finite deficiency indices we prove impossibility of such a kind factorization. A construction of pairs L-o subset of L (L-o is closed and densely defined, L = L* >= 0) having the property dom (LLo) = {0} (and, in particular, dom (L-0(2))) = {0}) is given.