Yu. Kovalev
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Factorizations of nonnegative symmetric operators
MFAT 19 (2013), no. 3, 211-226
211-226
We prove that each closed denselydefined and nonnegative symmetric operator ˙A having disjointnonnegative self-adjoint extensions admits infinitely manyfactorizations of the form ˙A=LL0, where L0 is aclosed nonnegative symmetric operator and L its nonnegativeself-adjoint extension. The same factorizations are also establishedfor a non-densely defined nonnegative closed symmetric operator withinfinite deficiency indices while for operator with finitedeficiency indices we prove impossibility of such a kindfactorization. A construction of pairs L0⊂L (L0 isclosed and densely defined, L=L∗≥0) having the propertydom(LL0)={0} (and, in particular, dom(L20)={0}) is given.