On normal operator logarithms

Let $X,Y$ be normal bounded operators on a Hilbert space such that $e^X=e^Y$. If the spectra of $X$ and $Y$ are contained in the strip \s of the complex plane defined by $|\Im(z)|\leq \pi$, 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 $e^X=e^Y$. If $X$ is an unbounded self-adjoint operator, which does not have $(2k+1) \pi$, k \in \ZZ, as eigenvalues, and $Y$ is normal with spectrum in \s satisfying $e^{iX}=e^Y$, then $Y \in \{\, e^{iX} \, \}"$. We give alternative proofs and generalizations of results on normal operator exponentials proved by Ch. Schmoeger

Homogeneous manifolds from noncommutative measure spaces

Let M be a finite von Neumann algebra with a faithful trace $\tau$. In this paper we study metric geometry of homogeneous spaces O of the unitary group U of M, endowed with a Finsler quotient metric induced by the p-norms of $\tau$, $||x||_p=\tau(|x|^p)^{1/p}$, $p\ge 1$. The main results include the following. The unitary group carries on a rectifiable distance d_p induced by measuring the length of curves with the p-norm. If we identify O as a quotient of groups, then there is a natural quotient distance d'_p that metrizes the quotient topology. On the other hand, the Finsler quotient metric defined in O provides a way to measure curves, and therefore, there is an associated rectifiable distance d_{O,p}. For $p\ge 2$, we prove that the distances d'_p and d_{O,p} coincide. Based on this fact, we show that the metric space (O,d'_p) is a complete path metric space. The other problem treated in this article is the existence of metric geodesics, or curves of minimal length, in O. We give two abstract partial results in this direction. The first concerns the initial values problem and the second the fixed endpoints problem. We show how these results apply to several examples. In the process, we improve some results about the metric geometry of U with the p-norm.Comment: 30 pages. The examples in section 4 have been removed, those of section 5 have been cut dow

The group of L^2 isometries on H^1_0

Let U be an open subset of R^n. Let L^2=L^2(U,dx) and H^1_0=H^1_0(U) be the standard Lebesgue and Sobolev spaces of complex-valued functions. The aim of this paper is to study the group G of invertible operators on H^1_0 which preserve the L^2-inner product. When U is bounded and the border $\partial U$ is smooth, this group acts as the intertwiner of the H^1_0 solutions of the non-homogeneous Helmholtz equation $u-\Delta u=f$, $u|_{\partial U}=0$. We show that G is a real Banach-Lie group, whose Lie algebra is (i times) the space of symmetrizable operators. We discuss the spectrum of operators belonging to G by means of examples. In particular, we give an example of an operator in G whose spectrum is not contained in the unit circle. We also study the one parameter subgroups of G. Curves of minimal length in G are considered. We introduce the subgroups G_p:=G \cap (I - B_p(H^1_0)), where B_p(H_0^1) is a Schatten ideal of operators on H_0^1. An invariant (weak) Finsler metric is defined by the p-norm of the Schatten ideal of operators of L^2. We prove that any pair of operators g_1,g_2 in G_p can be joined by a minimal curve of the form $a(t)=g_1 e^{itX}$, where X is a symmetrizable operator in B_p(H^1_0).Comment: 22 page

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

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.Fil: Chiumiento, Eduardo Hernan. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Saavedra 15. Instituto Argentino de Matemática Alberto Calderon; Argentina. Universidad Nacional de La Plata. Facultad de Ciencias Económicas; Argentin

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