Geometry of ℑ-Stiefel manifolds

Let ℑ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space ℋ and U(ℋ) ℑ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in ℑ. In this paper we study the geometry of the unitary orbits {UV : U ε U(ℋ) ℑ} and {UVW * : U,W ε U(ℋ) ℑ}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + ℑ, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V *V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + ℑ and homogeneous reductive spaces of U(ℋ) ℑ and U(ℋ) ℑ ×U(ℋ) ℑ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(ℋ) ℑ (or U(ℋ) ℑ × U(ℋ)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.Facultad de Ciencias Exacta

Examples of homogeneous manifolds with uniformly bounded metric projection

Let M be a finite von Neumann algebra with a faithful normal trace τ. Denote by Lp(M)sh the skew-Hermitian part of the non-commutative Lp space associated with (M, τ). Let 1 < p < ∞, z ∈ Lp(M)sh and S be a real closed subspace of Lp(M)sh. The metric projection Q : Lp(M)sh −→ S is defined for every z ∈ Lp(M)sh as the unique operator Q(z) ∈ S such that kz − Q(z)kp = miny∈ S kz − ykp. We show the relation between metric projection and metric geometry of homogeneous spaces of the unitary group UM of M, endowed with a Finsler quotient metric induced by the p-norms of τ, kxkp = τ(|x| p) 1/p, p an even integer. The problem of finding minimal curves in such homogeneous spaces leads to the notion of uniformly bounded metric projection. Then we show examples of metric projections of this type.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

Geometry of ℑ-Stiefel manifolds

Let ℑ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space ℋ and U(ℋ) ℑ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in ℑ. In this paper we study the geometry of the unitary orbits {UV : U ε U(ℋ) ℑ} and {UVW * : U,W ε U(ℋ) ℑ}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + ℑ, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V *V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + ℑ and homogeneous reductive spaces of U(ℋ) ℑ and U(ℋ) ℑ ×U(ℋ) ℑ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(ℋ) ℑ (or U(ℋ) ℑ × U(ℋ)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.Facultad de Ciencias Exacta

On normal operator logarithms

Let X,Y be normal bounded operators on a Hilbert space such that e X=eY. If the spectra of X and Y are contained in the strip S of the complex plane defined by |I(z)|≤π, we show that |X|=|Y|. If Y is only assumed to be bounded, then |X|Y=Y|X|. We give a formula for X-Y in terms of spectral projections of X and Y provided that X,Y are normal and e X=eY. If X is an unbounded self-adjoint operator, which does not have (2k+1)π,k∈ℤ, as eigenvalues, and Y is normal with spectrum in S satisfying eiX=eY, then Y∈{ e iX}″. We give alternative proofs and generalizations of results on normal operator exponentials proved by Schmoeger.Facultad de Ciencias Exacta

Examples of homogeneous manifolds with uniformly bounded metric projection

Let M be a finite von Neumann algebra with a faithful normal trace τ. Denote by Lp(M)sh the skew-Hermitian part of the non-commutative Lp space associated with (M, τ). Let 1 < p < ∞, z ∈ Lp(M)sh and S be a real closed subspace of Lp(M)sh. The metric projection Q : Lp(M)sh −→ S is defined for every z ∈ Lp(M)sh as the unique operator Q(z) ∈ S such that kz − Q(z)kp = miny∈ S kz − ykp. We show the relation between metric projection and metric geometry of homogeneous spaces of the unitary group UM of M, endowed with a Finsler quotient metric induced by the p-norms of τ, kxkp = τ(|x| p) 1/p, p an even integer. The problem of finding minimal curves in such homogeneous spaces leads to the notion of uniformly bounded metric projection. Then we show examples of metric projections of this type.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

Metric geometry in infinite dimensional Stiefel manifolds

Let J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredholm index of a pair of projections. Let UI(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. Then StI(v) coincides with the orbit of v under the action of UI(H)×UI(H) on I given by (u,w)·v0=uv0w*, u,w∈UI(H) and v0∈StI(v). We endow StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v).The other results concern with minimal curves in I-Stiefel manifolds when the ideal I is fixed as the compact operators in H. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.Facultad de Ciencias Exacta

Geometry of ℑ-Stiefel manifolds

Let ℑ be a separable Banach ideal in the space of bounded operators acting in a Hilbert space ℋ and U(ℋ) ℑ the Banach-Lie group of unitary operators which are perturbations of the identity by elements in ℑ. In this paper we study the geometry of the unitary orbits {UV : U ε U(ℋ) ℑ} and {UVW * : U,W ε U(ℋ) ℑ}, where V is a partial isometry. We give a spatial characterization of these orbits. It turns out that both are included in V + ℑ, and while the first one consists of partial isometries with the same kernel of V , the second is given by partial isometries such that their initial projections and V *V have null index as a pair of projections. We prove that they are smooth submanifolds of the affine Banach space V + ℑ and homogeneous reductive spaces of U(ℋ) ℑ and U(ℋ) ℑ ×U(ℋ) ℑ respectively. Then we endow these orbits with two equivalent Finsler metrics, one provided by the ambient norm of the ideal and the other given by the Banach quotient norm of the Lie algebra of U(ℋ) ℑ (or U(ℋ) ℑ × U(ℋ)I) by the Lie algebra of the isotropy group of the natural actions. We show that they are complete metric spaces with the geodesic distance of these metrics.Facultad de Ciencias Exacta

On normal operator logarithms

Let X,Y be normal bounded operators on a Hilbert space such that e X=eY. If the spectra of X and Y are contained in the strip S of the complex plane defined by |I(z)|≤π, we show that |X|=|Y|. If Y is only assumed to be bounded, then |X|Y=Y|X|. We give a formula for X-Y in terms of spectral projections of X and Y provided that X,Y are normal and e X=eY. If X is an unbounded self-adjoint operator, which does not have (2k+1)π,k∈ℤ, as eigenvalues, and Y is normal with spectrum in S satisfying eiX=eY, then Y∈{ e iX}″. We give alternative proofs and generalizations of results on normal operator exponentials proved by Schmoeger.Facultad de Ciencias Exacta

Metric geometry in infinite dimensional Stiefel manifolds

Let J be a separable Banach ideal in the space of bounded operators acting in a Hilbert space H and I the set of partial isometries in H. Fix v∈I. In this paper we study metric properties of the I-Stiefel manifold associated to v, namely. StI(v)={v0∈: v-v0∈I,j(v0*v0,v*v)=0}, where j(,) is the Fredholm index of a pair of projections. Let UI(H) be the Banach-Lie group of unitary operators which are perturbations of the identity by elements in I. Then StI(v) coincides with the orbit of v under the action of UI(H)×UI(H) on I given by (u,w)·v0=uv0w*, u,w∈UI(H) and v0∈StI(v). We endow StI(v) with a quotient Finsler metric by means of the Banach quotient norm of the Lie algebra of UI(H)×UI(H) by the Lie algebra of the isotropy group. We give a characterization of the rectifiable distance induced by this metric. In fact, we show that the rectifiable distance coincides with the quotient distance of UI(H)×UI(H) by the isotropy group. Hence this metric defines the quotient topology in StI(v).The other results concern with minimal curves in I-Stiefel manifolds when the ideal I is fixed as the compact operators in H. The initial value problem is solved when the partial isometry v has finite rank. In addition, we use a length-reducing map into the Grassmannian to find some special partial isometries that can be joined with a curve of minimal length.Facultad de Ciencias Exacta

Geometría en espacios homogéneos de grupos unitarios

En este trabajo estamos interesados en propiedades geométricas de espacios homogéneos de grupos unitarios. Estos espacios homogéneos, construidos a partir de elementos de la teoría de operadores, son variedades de dimensión infinita donde definimos una métrica de Finsler natural. Principalmente, estudiaremos aspectos concernientes a la geometría métrica de estos espacios homogéneos, como la existencia y unicidad de curvas minimales o propiedades de la distancia rectificable, y en menor medida, estudiaremos aspectos diferenciales, como la presencia de una estructura reductiva. En variedades Riemannianas o de Finsler de dimensión finita, o más aún, en espacios métricos de longitud localmente compactos, el Teorema de Hopf-Rinow relaciona la existencia de curvas de longitud minimal con la completitud en la distancia rectificable. Dado que este teorema no es válido en variedades de dimensión infinita, resulta necesario desarrollar técnicas ad-hoc en cada ejemplo para hallar curvas minimales o analizar la completitud. En particular, en los espacios homogéneos tratados aquí, este tipo de cuestiones métricas generan diversos problemas, interesantes por sí mismos en la teoría de operadores y que permiten observar como funciona o falla la teoría finito dimensional de variedades. Estudiaremos dos tipos de espacios homogéneos. El primero, dentro de un marco general, donde la métrica de Finsler está inducida por la traza finita de un álgebra, y el segundo es un ejemplo concreto acerca de isometrías parciales, donde la métrica proviene de la norma de ideales de Banach.Facultad de Ciencias Exacta