A Krein space H and bounded linear operators B, C on H are given. Then, some min and max problems about the operators (BX − C) # (BX − C), where X runs over the space of all bounded linear operators on H, are discussed. In each case, a complete answer to the problem, including solvability conditions and characterization of the solutions, is presented. Also, an adequate decomposition of B is considered and the min-max problem is addressed. As a by-product the Moore-Penrose inverse of B is characterized as the only solution of a variational problem. Other generalized inverses are described in a similar fashion as well.Mathematics Subject Classification (2010). 47A58, 47B50, 41A65.Keywords. Operator approximation, Krein spaces, Moore-Penrose inverse. 1 L(H) stands for the space of all the bounded linear operators from H to H. Vol. 99 (9999) Operator LSP and Moore-Penrose inverses 3 characterized as the indefinite inverses of B. The results about this indefinite minimization problem and their counterparts for the symmetric maximization problem are applied in Section 5 where B is factorized as B = B + + B − and the min-max problem min X∈L(H) max Y ∈L(H)is addressed. Section 6 contains the main results about the Moore-Penrose inverse and the generalized inverses of a given Krein space operator.
PreliminariesIn the following all Hilbert spaces are complex and separable. If H and K are Hilbert spaces, L(H, K) stands for the space of all the bounded linear operators from H to K and CR(H, K) for the subset of L(H, K) comprising all the operators with closed ranges. When H = K we write, for short, L(H) and CR(H). The range and null space of any given A ∈ L(H, K) are denoted by R(A) and N (A), respectively. The direct sum of two closed subspaces M and N of H is represented by M+N . If H is decomposed as H = M+N , the projection onto M with null space N is denoted P M//N and abbreviated P M when N = M ⊥ . In general, Q is used to indicate the subset of all the oblique projections in L(H), namely, Q := {Q ∈ L(H) : Q 2 = Q}.The following is a well-known result about range inclusion and factorizations of operators. We will refer to it along the paper. Theorem 2.1 (Douglas' Theorem [9]). Let Y, Z ∈ L(H). Then R(Z) ⊆ R(Y ) if and only if there exists D ∈ L(H) such that Z = Y D.
Krein SpacesAlthough familiarity with operator theory on Krein spaces is presumed, we hereafter include some basic notions. Standard references on Krein spaces and operators on them are [1,4,5]. We also refer to [10,11] as authoritative accounts of the subject.Consider a linear space H with an indefinite metric, i.e., a sesquilinear Hermitian form [ , ]. A vector x ∈ H is said to be positive if [ x, x ] > 0. A subspace S of H is positive if every x ∈ S, x = 0, is a positive vector. Negative, nonnegative, nonpositive and neutral vectors and subspaces are defined likewise.We say that two closed subspaces M and N are orthogonal, and we write M [⊥] N , if [ m, n ] = 0 for every m ∈ M and n ∈ N . We denote the orthogonal direct sum of two closed subspaces M and ...
