Page 62 - Textos de Matemática Vol. 44

52 M. C. GOUVEIA
Proof. Since R((E : F)1/2) is closed, by Theorem 2.5 we have R(E : F) = R(E) ∩ R(F) = R(E ∧ F).
By definition, CC∗ +DD∗ is the projection of H onto N((E +F)1/2)⊥. Then, E1/2C∗ + F1/2D∗ = (E + F)1/2(CC∗ + DD∗) = (E + F)1/2.
Let z = Ex = Fy ∈ R(E) ∩ R(F). Then
(E + F)1/2CE1/2x = z = (E + F)1/2DF1/2y.
It follows that CE1/2x = DF1/2y. Therefore,
z=E1/2E1/2x =
= (E1/2C∗ + F 1/2D∗)CE1/2x
(E+F)1/2CE1/2x
= E1/2C∗DF 1/2y + F 1/2D∗CE1/2x
= (E:F)y+(F:E)x
= 2(E:F)z.
Hence, 2(E : F ) and E ∧ F agree on their common range.
Moreover, since by definition and Theorem 2.5 both operators are self-
adjoint, then 2(E : F ) = E ∧ F .
References
[1] Adi Ben-Israel, T. N. E. Greville, Generalized Inverses, Theory and Applications, Springer Verlag New York Inc., 2003.
[2] W. N. Anderson, and R. J. Duffin, Series and parallel addition of matrices, J. Math. Anal. Appl. 26 (1969), 576-594.
[3] P. A. Filmore and J. P. Williams, On operator ranges, Adv. in Math. 7 (1971), 254-281.
[4] P. R. Halmos, A Hilbert Space Problem Book, D. Van Nostrand, Princeton, NJ, 1967.
[5] C. R. Rao and S. K. Mitra, Generalized Inverse of Matrices and its Applications, Wiley,
New York, 1971.
(M. C. Gouveia) Department of Mathematics, University of Coimbra, Apartado 3008, EC Santa Cruz, 3001-501 Coimbra, Portugal
E-mail address: mcag@mat.uc.pt