Alejandra Maestripieri

Generalized Schur complements and oblique projections

Let S be a closed subspace of a Hilbert space H and A a bounded linear selfadjoint operator on H. In this note, we show that the existence of A-selfadjoint projections with range S is related to some properties of shorted operators, Schur complements (in Ando’s generalization of the classical concept) and compressions.

Weighted Generalized Inverses, Oblique Projections, and Least-Squares Problems

ABSTRACT A generalization with singular weights of Moore–Penrose generalized inverses of closed range operators in Hilbert spaces is studied using the notion of compatibility of subspaces and positive operators.

Redundant decompositions, angles between subspaces and oblique projections

Let Η be a complex Hilbert space. We study the relationships between the angles between closed subspaces of H, the oblique projections associated to non direct decompositions of H and a notion of compatibility between a positive (semidefinite) operator A acting on H and a closed subspace S of H. It turns out that the compatibility is ruled by the values of the Dixmier angle between the orthogonal complement S _l_ of S and the closure of AS. We show that every redundant decomposition H = S+M_l_ (where redundant means that S ∩M_l_ is not trivial) occurs in the presence of a certain compatibility. We also show applications of these results to some signal processing problems (consistent reconstruction) and to abstract splines problems which come from approximation theory.

GEOMETRY OF POSITIVE OPERATORS AND UHLMANN'S APPROACH TO THE GEOMETRIC PHASE

Abstract In Uhlmanns description of the differential geometry of the space Ω of density operators, a relevant role is played by the parallel condition ω*ω = ω*ω, where ω is a lifting of a curve γ in Ω, i.e. ω( t )ω( t )* = γ( t ) for all t . In this paper we get a principal bundle with a natural connection over the space G + of all positive invertible elements of a C *-algebra such that the parallel transport is ruled by Uhlmanns parallel equation.

Weighted Procrustes problems

Abstract Let H be a Hilbert space, L ( H ) the algebra of bounded linear operators on H and W ∈ L ( H ) a positive operator such that W 1 / 2 is in the p-Schatten class, for some 1 ≤ p ∞ . Given A ∈ L ( H ) with closed range and B ∈ L ( H ) , we study the following weighted approximation problem: analyze the existence of m i n X ∈ L ( H ) ‖ A X − B ‖ p , W , where ‖ X ‖ p , W = ‖ W 1 / 2 X ‖ p . In this paper we prove that the existence of this minimum is equivalent to a compatibility condition between R ( B ) and R ( A ) involving the weight W , and we characterize the operators which minimize this problem as W -inverses of A in R ( B ) .

Projections in operator ranges

If H is a Hilbert space, A is a positive bounded linear operator on H and S is a closed subspace of H, the relative position between S and A -1 (S⊥) establishes a notion of compatibility. We show that the compatibility of (A, S) is equivalent to the existence of a convenient orthogonal projection in the operator range R(A 1/2 ) with its canonical Hilbertian structure.

Spectrum of J-frame operators

A \(J\)-frame is a frame \(\mathcal{F}\) for a Krein space \((\mathcal{H},[\cdot,\cdot ])\) which is compatible with the indefinite inner product \([\cdot,\cdot ]\) in the sense that it induces an indefinite reconstruction formula that resembles those produced by orthonormal bases in \(\mathcal{H}\). With every \(J\)-frame the so-called \(J\)-frame operator is associated, which is a self-adjoint operator in the Krein space \(\mathcal{H}\). The \(J\)-frame operator plays an essential role in the indefinite reconstruction formula. In this paper we characterize the class of \(J\)-frame operators in a Krein space by a \(2\times 2\) block operator representation. The \(J\)-frame bounds of \(\mathcal{F}\) are then recovered as the suprema and infima of the numerical ranges of some uniformly positive operators which are build from the entries of the \(2\times 2\) block representation. Moreover, this \(2\times 2\) block representation is utilized to obtain enclosures for the spectrum of \(J\)-frame operators, which finally leads to the construction of a square root. This square root allows a complete description of all \(J\)-frames associated with a given \(J\)-frame operator.

Differential structure of the Thompson components of selfadjoint operators

Different equivalence relations are defined in the set L(H) s of selfadjoint operators of a Hilbert space H in order to extend a very well known relation in the cone of positive operators. As in the positive case, for a ∈ L(H) s the equivalence class C a admits a differential structure, which is compatible with a complete metric defined on C a . This metric coincides with the Thompson metric when a is positive.

Shorted operators and minus order

ABSTRACT Let be a Hilbert space, the algebra of bounded linear operators on and a positive operator. Given a closed subspace of , we characterize the shorted operator of W to as the maximum and as the infimum of certain sets, for the minus order . Also, given with closed range, we study the following operator approximation problem considering the minus order: We show that, under certain conditions, the shorted operator of is the minimum of this problem and we characterize the set of solutions.

Polar decomposition under perturbations of the scalar product

Let A be a unital C* algebra with involution * represented in a Hilbert space H, G the group of invertible elements of A, U the unitary group of A, G^s the set of invertible selfadjoint elements of A, Q={e in G : e^2 = 1} the space of reflections and P = Q\cap U. For any positive a in G consider the a-unitary group U_a={g in G : a^{-1} g^* a = g^{-1}}, i.e. the elements which are unitary with respect to the scalar product _a = for \xi, \eta in H. If \pi denotes the map that assigns to each invertible element its unitary part in the polar decomposition, we show that the restriction \pi|_{U_a}: U_a \to U is a diffeomorphism, that \pi(U_a \cap Q) = P and that \pi(U_a\cap G^s) = U_a\cap G^s = {u in G: u=u^*=u^{-1} and au = ua}.