Non positively curved metric in the space of positive definite infinite matrices

We introduce a Riemannian metric with non positive curvature in the (infinite dimensional) manifold Σ∞ of positive invertible operators of a Hilbert space H, which are scalar perturbations of Hilbert-Schmidt operators. The (minimal) geodesics and the geodesic distance are computed. It is shown that this metric, which is complete, generalizes the well known non positive metric for positive definite complex matrices. Moreover, these spaces of finite matrices are naturally imbedded in Σ∞.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; ArgentinaFil: Varela, Alejandro. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentin

Metric geodesics of isometries in a Hilbert space and the extension problem

We consider the problem of finding short smooth curves of isometries in a Hilbert space H. The length of a smooth curve γ(t), t ∈ [0, 1], is measured by means of ∫^1-0 γ^. (t)ǀǀ dt, where ǀǀ ǀǀ denotes the usual norm of operators. The initial value problem is solved: for any isometry Vo and each tangent vector at V0 (which is an operator of the form iXV0 with X* = X) with norm less than or equal to π, there exist curves of the form e^itZ V0, with initial velocity iZV0 = iXV0, which are short along their path. These curves, which we call metric geodesics, need not be unique, and correspond to the so called extension problem considered by M.G. Krein and others: in our context, given asymmetric operator X0|R(V0) : R(V0)→H, find all possible Z* = Z extending X0|R(V0) to all H, with ǀǀZǀǀ= ǀǀX0ǀǀ. We also consider the problem of finding metric geodesics joining two given isometries V0 and V1. It is well known that if there exists a continuous path joining V0 and V1, then both ranges have the same codimension. We show that if this number is finite, then there exist metric geodesics joining V0 and V1.Fil: Andruchow, Esteban. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Recht, Lázaro. Universidad Simón Bolívar; Venezuela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; ArgentinaFil: Varela, Alejandro. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento. Instituto de Ciencias; Argentin

Unitary operators with decomposable corners

We study pairs (U, L) , where U is a unitary operator in H and L⊂ H is a closed subspace, such that PL0U|L0:L0→L0has a singular value decomposition. Abstract characterizations of this condition are given, as well as relations to the geometry of projections and pairs of projections. Several concrete examples are examined.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderón; Argentina. Universidad Nacional de General Sarmiento; Argentin

A note on a question by T. Ando

In 1979 T. Ando posed the following question: suppose E and F are two projectionvalued measures defined on an algebra Σ of subsets of Ω, which verifykE(∆) − F(∆)k ≤ 1 − δ, ∆ ∈ Σ,for some δ > 0. Does there exist a unitary operator u such that u∗E(∆)u = F(∆) for all∆ ∈ Σ? He knew that the answer was affirmative if both measures were strongly σ-additiveand maximal (i.e. E and F have ciclic vectors). In this note, we show that the answer isalso affirmative if both measures take values in a common finite von Neumann algebra.Fil: Andruchow, Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina. Universidad Nacional de General Sarmiento; Argentin