Extensions of Symmetric Operators - Positive Symmetric Operators

Positive Symmetric Operators

A symmetric operator A is called positive if <Ax, x> ≥ 0 for all x in Dom(A). It is known that for every such A, one has dim(K₊) = dim(K_-). Therefore every positive symmetric operator has self-adjoint extensions. The more interesting question in this direction is whether A has positive self-adjoint extensions.

For two positive operators A and B, we put A ≤ B if

in the sense of bounded operators.