Page 55 - Textos de Matemática Vol. 44

PARALLEL SUMS OF OPERATORS
M. C. GOUVEIA
Dedicated to Nat´alia Bebiano, a colleague in profession and a friend in life.
Abstract. A solution to a problem in Halmos’ Hilbert Space Problem Book is presented using the formula for the orthogonal projection on the intersec- tion of two subspaces in terms of their respective projections. This formula is obtained by the extension to singular matrices of the definition of parallel sum of matrices. It is observed that such solution is given as an example of the applications of generalized inverse theory.
1. Introduction
“What can you do with generalized inverses that you could not do without?” This is a common question to, and between, those who are interested in the profits of generalized inverse theory. Among several answers we remark that one referred by Ben-Israel and Greville in [1, p. 282], not only because it is an example of a result that has never been obtained outside generalized inverse theory, but also because it deals with matrix theory.
In the following we use common notation Cm×n for the vector-space of
m × n complex matrices, capital letters for matrices or their corresponding lin-
ear transformations and ∗ for the involution defined by A∗ = (a )∗ = (a¯ )T. ij ij
With R(A) and N(A) we mean, respectively, the range and the null space of the operator A. The orthogonal projection onto R(A) will be denoted by PR(A) or by PA. If S is a n-dimensional complex space, we use S⊥ to denote the complementary subspace of S. An hermitian operator A is said to be PSD if it is positive semidefinite.
An operator on a complex Hilbert space H is understood to be a bounded linear transformation from H into itself. The algebra of all bounded operators on H will be denoted by B(H). If T ∈ B(H), then R(T), N(T) and T∗ denote, respectively, the range, the null space and the adjoint of T.
With respect to generalized inverses we recall some definitions and we refer [1] for the general theory.
2010 Mathematics Subject Classification. 15A09
Key words and phrases. Parallel sums, generalized inverses.
The work was supported by Instituto de Telecomunica¸co˜es, Portugal.
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