An alternative approach to the parallel sum

AbstractThe parallel sum of two positive operators on a Hilbert space H is defined by the formula: A:B = limε ↓ 0 a(A + B + εI)−1 B. If the operator (A + B) is invertible then this reduces to A:B = A(A + B)−1 B. The obvious generalization to non-invertible A + B is A(A + B)†B, where (A + B)† denotes the (possibly unbounded) Moore-Penrose pseudoinverse of (A + B). This, however, does not work. In fact, from an abstract point of view, the operation (A, B) ⇒ A:B is not purely algebraic. Nevertheless, we show that the parallel sum of A and B can be written as A:B¦ker(A + B)⊥ = A~ (A + B)~ −1 B~¦ker(A + B)⊥, where A~ denotes an extension of A to the Hilbert space constructed by completing the inner product space Hker(A + B) with the inner product (x, y)A + B = ((A + B)x, y). Both sides of the above equation vanish on ker(A + B). The above results depend on the infinite network interpretations of the shorted operator (or generalized Schur complement) due to C. A. Butler and the author