Factorization of nonnegative symmetric operators. (English) Zbl 1289.47044

Arlinskiĭ, Yury; Kovalev, Yury

Methods Funct. Anal. Topol. 19, No. 3, 211-226 (2013).

The authors prove that any closed densely defined nonnegative symmetric operator \(A\) having disjoint nonnegative selfadjoint extensions (that is, the intersection of their domains equals \(D(A)\)) admits infinitely many factorizations \(A=LL_0\), where \(L_0\) is a closed nonnegative symmetric operator, and \(L\) is its nonnegative selfadjoint extension. Such factorizations are established also for nondensely defined operators with infinite deficiency index; if the latter is finite, such a factorization is impossible. The authors also give a construction of such pairs \(L_0\subset L\), where \(L_0\) is closed and densely defined, \(L=L^*\geq 0\), that \(LL_0\) and \(L_0^2\) are defined only on the zero element.

Reviewer: Anatoly N. Kochubei (Kyïv)

MSC:

47B25	Linear symmetric and selfadjoint operators (unbounded)
47A68	Factorization theory (including Wiener-Hopf and spectral factorizations) of linear operators
47A64	Operator means involving linear operators, shorted linear operators, etc.

Keywords:

nonnegative symmetric operator; nonnegative selfadjoint extensions; factorization of operators