Page 60 - Textos de Matemática Vol. 44

50 M. C. GOUVEIA
is, if A,B ∈ Cm×n the parallel sum of A and B is defined by (3.1) A : B = A(A + B)†B.
Obviously, one advantage of considering the parallel sum (3.1) is the unique- ness of A†. In such case we have the following
Lemma 3.1. Let A, B be hermitian PSD operators on the finite dimensional complex vector space V. Then
(i) A:B=B:A.
(ii) A : B is hermitian.
(iii) R(A:B)=R(A)∩R(B).
(iv) (A:B)†=A†+B†.
(v) A : B = (PR(A)∩R(B)(A† + B†)PR(A)∩R(B))†.
Proof. It is clear that (i) and (iii) are proved just like (i) and (iii) in Theo-
rem 2.3. Next we prove (ii)
(A : B)∗ = (A(A + B)†B)∗ = B∗(A + B)∗†A∗
= B∗:A∗
= B:A=A:B, inviewof(i).
(iv) Following the reasoning in the proof of (ii) in Theorem 2.3, we obtain
(A:B)(A† +B†)(A:B)=A:B.
Now we prove that A† +B† (A† + B†)(A ∓ B)(A† + B†) = =
is a (2,3,4)-inverse of A∓B. (A† + B†)A(A + B)†B(A† + B†)
A†A(A + B)†B(A† + B†) +
B†A(A + B)†B(A† + B†) (A+B)†(A+B)(A† +B†)=A† +B†.
=
Since A, B and A : B are hermitian, from known results it follows
((A:B)(A:B)†)∗ =
= (A† + B†)A(A + B)†B
((A:B)∗)†(A:B)∗ =(A:B)†(A:B) = (A+B)†(A+B)
= PR(A+B)∗ = PR(A+B)
= (A+B)(A+B)† =(A:B)(A:B)†.
That A† + B† is a (4)-inverse of (A : B) is now trivial .