On the geometry of normal projections in Krein spaces

Let $\mathcal{H}$ be a Krein space with fundamental symmetry $J$. Along this paper, the geometric structure of the set of $J$-normal projections $\mathcal{Q}$ is studied. The group of $J$-unitary operators $\mathcal{U}_J$ naturally acts on $\mathcal{Q}$. Each orbit of this action turns out to be an analytic homogeneous space of $\mathcal{U}_J$, and a connected component of $\mathcal{Q}$. The relationship between $\mathcal{Q}$ and the set $\mathcal{E}$ of $J$-selfadjoint projections is analized: both sets are analytic submanifolds of $L(\mathcal{H})$ and there is a natural real analytic submersion from $\mathcal{Q}$ onto $\mathcal{E}$, namely $Q\mapsto QQ^\#$. The range of a $J$-normal projection is always a pseudo-regular subspace. Then, for a fixed pseudo-regular subspace $\mathcal{S}$, it is proved that the set of $J$-normal projections onto $\mathcal{S}$ is a covering space of the subset of $J$-normal projections onto $\mathcal{S}$ with fixed regular part.Comment: 19 pages, accepted for publication in the Journal of Operator Theor

The effect of finite rank perturbations on Jordan chains of linear operators

A general result on the structure and dimension of the root subspaces of a matrix or a linear operator under finite rank perturbations is proved: The increase of dimension from the $n$-th power of the kernel of the perturbed operator to the $(n+1)$-th power differs from the increase of dimension of the corresponding powers of the kernels of the unperturbed operator by at most the rank of the perturbation and this bound is sharp

Problemas de aproximación en espacios con métrica indefinida

La teoría de operadores lineales en espacios con métrica indefinida aparece por primera vez en el artículo “Hermitian operators in spaces with an indefinite metric” de L. S. Pontryagin, publicado en el año 1944; si bien anteriormente algunos físicos teóricos se habían topado con estos espacios, este trabajo marca la aparición de una nueva rama del análisis funcional en la década de 1940. Cabe destacar que nos referimos a espacios de dimensión infinita, ya que las transformaciones lineales en espacios (de dimensión finita) con métrica indefinida han sido estudiadas desde fines del siglo XIX por G. F. Frobenius, entre otros.Facultad de Ciencias Exacta

Problemas de aproximación en espacios con métrica indefinida

La teoría de operadores lineales en espacios con métrica indefinida aparece por primera vez en el artículo “Hermitian operators in spaces with an indefinite metric” de L. S. Pontryagin, publicado en el año 1944; si bien anteriormente algunos físicos teóricos se habían topado con estos espacios, este trabajo marca la aparición de una nueva rama del análisis funcional en la década de 1940. Cabe destacar que nos referimos a espacios de dimensión infinita, ya que las transformaciones lineales en espacios (de dimensión finita) con métrica indefinida han sido estudiadas desde fines del siglo XIX por G. F. Frobenius, entre otros.Doctor en Ciencias Exactas, área MatemáticaUniversidad Nacional de La PlataFacultad de Ciencias Exacta

Normal projections in Krein spaces

Given a complex Krein space H with fundamental symmetry J, the aim of this note is to characterize the set of J-normal projections Q={Q∈L(H):Q2=QandQ#Q=QQ#}. The ranges of the projections in QQ are exactly those subspaces of HH which are pseudo-regular. For a fixed pseudo-regular subspace S , there are infinitely many J-normal projections onto it, unless SS is regular. Therefore, most of the material herein is devoted to parametrizing the set of J-normal projections onto a fixed pseudo-regular subspace S.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

Shorting selfadjoint operators in Hilbert spaces

Given a closed subspace S of a Hilbert space H and a (bounded) selfadjoint operator B acting on H, a min-max representation of the shorted operator (or Schur complement) of B to S is obtained under compatibility hypotheses. Also, an extension of Pekarev's formula is given.Facultad de Ciencias Exacta

Weak matrix majorization

Given X,Y∈Rn×m we introduce the following notion of matrix majorization, called weak matrix majorization,X≻wYifthereexistsarow- stochasticmatrixA∈Rn×nsuchthatAX=Y,and consider the relations between this concept, strong majorization (≻s) and directional majorization (≻). It is verified that ≻s ⇒ ≻ ⇒ ≻w, but none of the reciprocal implications is true. Nevertheless, we study the implications ≻w ⇒ ≻s and ≻ ⇒ ≻s under additional hypotheses. We give characterizations of strong, directional and weak matrix majorization in terms of convexity. We also introduce definitions for majorization between Abelian families of selfadjoint matrices, called joint majorizations. They are induced by the previously mentioned matrix majorizations. We obtain descriptions of these relations using convexity arguments.Facultad de Ciencias Exacta