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A definition of frames for Krein spaces is proposed, which extends the notion of J-orthonormal basis of Krein spaces. A J-frame for a Krein space (H, [ , ]) is in particular a frame for H in the Hilbert space sense. But it is also compatible with the indefinite inner product [ , ], meaning that it determines a pair of maximal uniformly J-definite subspaces with different positivity, an analogue to the maximal dual pair associated to a J-orthonormal basis.Also, each J-frame induces an indefinite reconstruction formula for the vectors in H, which resembles the one given by a J-orthonormal basis.

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We characterize the sets X of all products P Q, and Y of all products P QP , where P, Q run over all orthogonal projections and we solve the problems arg min{ P − Q : (P, Q) ∈ Z}, for Z = X or Y. We also determine the polar decompositions and Moore-Penrose pseudoinverses of elements of X. * Partially supported by PICT 5272 FONCYT and UBACYT I023.

A positive operator A and a closed subspace S of a Hilbert space H are called compatible if there exists a projector Q onto S such that AQ = Q * A. Compatibility is shown to depend on the existence of certain decompositions of H and the ranges of A and A 1/2 . It also depends on a certain angle between A(S) and the orthogonal of S.2000 Mathematics Subject Classification: 47A64, 47A07, 46C99. Key words and phrases: oblique projectors, least square problems, scaled projectors, positive operators.Research partially supported by CONICET (PIP 2083/02), Universidad de Buenos Aires (UBACYT X050), Universidad de La Plata 11 X350 and ANPCYT (PICT03-9521).The paper is in final form and no version of it will be published elsewhere.[145] 146G. CORACH ET AL.For each closed subspace S of H let Q S denote the set of all projectors with range S. For each (bounded linear semidefinite) positive operator A on H consider the set P(A, S) = {Q ∈ Q S : AQ = Q * A}, i.e., all Q with range S which are Hermitian with respect to the sesquilinear form ξ, η A = Aξ, η . Of course, P(A, S) can be empty (see examples below); we say that A, S are compatible if P(A, S) is not empty. This condition can be read in terms of different space decompositions, range inclusions and angles between certain closed subspaces of H. It is known [19] that, if A and S are compatible then a distinguished element P A,S of P(A, S) exists which has optimal properties. We show explicit formulas for P A,S which are computationally useful.Many results on oblique projectors can be found in the papers by Afriat [1], Davis [22], Ljance [43], Mizel and Rao [45], Halmos [33], Greville [32], Gerisch [30], Pták [49]. Projectors which are Hermitian with respect to a positive matrix have been studied by Mitra and Rao [44] and Baksalary and Kala [9]. More recently, Hassi and Nordstrom [35]studied projectors which are Hermitian with respect to a self-adjoint operator but with emphasis on the case in which P(A, S) is a singleton. In [47], Pasternak-Winiarski studied the analyticity of the map A → P A,S , where A runs over the set of positive invertible operators. The map (A, S) → P A,S is studied by Andruchow, Corach and Stojanoff [6], for positive invertible A. For general selfadjoint A, several results on P(A, S) can be found in [19] and the present paper can be seen as its continuation. Additional results by the authors are contained in [20] and [21]. The latter makes a link between oblique projectors and abstract splines in the sense of Atteia [8]. It is natural that this type of least square approximation results appears in this context, because P A,S is a kind of orthogonal projector for an appropriate inner product. In particular, oblique projectors, mainly in the finite-dimensional setting, appear frequently under the form of "scaled projectors", i.e., projectors which are Hermitian with respect to a positive diagonal matrix. The reader is referred to the papers by Stewart [53], O'Leary [46], Hanke and Neumann [34], Gonzaga and Lara [31], Wei [60], Forsgren [28], Vavasis [1...