Metrics in the set of partial isometries with finite rank

Esteban Andruchow; Gustavo Corach

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Finite-rank perturbations of positive operators and isometries

Man-Duen Choi, Pei Yuan Wu (2006)

Studia Mathematica

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We completely characterize the ranks of A - B and $A^{1 / 2} - B^{1 / 2}$ for operators A and B on a Hilbert space satisfying A ≥ B ≥ 0. Namely, let l and m be nonnegative integers or infinity. Then l = rank(A - B) and $m = r a n k (A^{1 / 2} - B^{1 / 2})$ for some operators A and B with A ≥ B ≥ 0 on a Hilbert space of dimension n (1 ≤ n ≤ ∞) if and only if l = m = 0 or 0 < l ≤ m ≤ n. In particular, this answers in the negative the question posed by C. Benhida whether for positive operators A and B the finiteness of rank(A - B) implies that...

Metrics with homogeneous geodesics on flag manifolds

Dimitri V. Alekseevsky, Andreas Arvanitoyeorgos (2002)

Commentationes Mathematicae Universitatis Carolinae

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A geodesic of a homogeneous Riemannian manifold $(M = G / K, g)$ is called homogeneous if it is an orbit of an one-parameter subgroup of $G$ . In the case when $M = G / H$ is a naturally reductive space, that is the $G$ -invariant metric $g$ is defined by some non degenerate biinvariant symmetric bilinear form $B$ , all geodesics of $M$ are homogeneous. We consider the case when $M = G / K$ is a flag manifold, i.eȧn adjoint orbit of a compact semisimple Lie group $G$ , and we give a simple necessary condition that $M$ admits a non-naturally...

A classification of projectors

Gustavo Corach, Alejandra Maestripieri, Demetrio Stojanoff (2005)

Banach Center Publications

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A positive operator A and a closed subspace of a Hilbert space ℋ are called compatible if there exists a projector Q onto such that AQ = Q*A. Compatibility is shown to depend on the existence of certain decompositions of ℋ and the ranges of A and $A^{1 / 2}$ . It also depends on a certain angle between A() and the orthogonal of .

A Weitzenbôck formula for the second fundamental form of a Riemannian foliation

Paolo Piccinni (1984)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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Si considera la seconda forma fondamentale $\alpha$ di foliazioni su varietà riemanniane e si ottiene una formula per il laplaciano $\nabla^{2}\alpha$ - Se ne deducono alcune implicazioni per foliazioni su varietà a curvatura costante.

Filling Radius and Short Closed Geodesics of the $2$ -Sphere

Stéphane Sabourau (2004)

Bulletin de la Société Mathématique de France

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We show that the length of the shortest nontrivial curve among the simple closed geodesics of index zero or one and the figure-eight geodesics of null index provides a lower bound on the area and the diameter of the Riemannian $2$ -spheres.

Boundary behaviour of invariant distances and complex geodesics

Marco Abate (1986)

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni

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In questa Nota viene studiato il comportamento al bordo delle distanze di Carathéodory e Kobayashi in domini fortemente pseudoconvessi di classe $\mathbf{C}^{2}$ . Come applicazione si dimostra che ogni geodetica complessa in tali domini è estendibile al bordo di classe $\mathbf{C}^{0,\frac{1}{2}}$ .

The $C^{*}$ -algebra of a Hilbert bimodule

Sergio Doplicher, Claudia Pinzari, Rita Zuccante (1998)

Bollettino dell'Unione Matematica Italiana

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Un $C^{*}$ -modulo hilbertiano destro $X$ su una $C^{*}$ -algebra $A$ dotato di uno $^{*}$ -omomorfismo isometrico $ϕ : A \to L_{A} (X)$ viene qui considerato come un oggetto $X_{A}$ della $C^{*}$ -categoria degli $A$ -moduli Hilbertiani destri. Come in [11], associamo ad esso una $C^{*}$ -algebra $O_{X_{A}}$ contenente $X$ come un « $A$ -bimodulo hilbertiano in $O_{X_{A}}$ ». Se $X$ è pieno e proiettivo finito $O_{X_{A}}$ è la $C^{*}$ -algebra $C^{*} (X)$ , la generalizzazione delle algebre di Cuntz-Krieger introdotta da Pimsner [27] (e in un caso particolare da Katayama [31]). Più in generale, $C^{*} (X)$ è canonicamente...

Naturally reductive homogeneous $(α, β)$ -metric spaces

M. Parhizkar, H.R. Salimi Moghaddam (2021)

Archivum Mathematicum

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In the present paper we study naturally reductive homogeneous $(α, β)$ -metric spaces. We show that for homogeneous $(α, β)$ -metric spaces, under a mild condition, the two definitions of naturally reductive homogeneous Finsler space, given in the literature, are equivalent. Then, we compute the flag curvature of naturally reductive homogeneous $(α, β)$ -metric spaces.

Metrics in the sphere of a C*-module

Esteban Andruchow, Alejandro Varela (2007)

Open Mathematics

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Given a unital C*-algebra $𝒜$ and a right C*-module $𝒳$ over $𝒜$ , we consider the problem of finding short smooth curves in the sphere $𝒮_{𝒳}$ = x ∈ $𝒳$ : 〈x, x〉 = 1. Curves in $𝒮_{𝒳}$ are measured considering the Finsler metric which consists of the norm of $𝒳$ at each tangent space of $𝒮_{𝒳}$ . The initial value problem is solved, for the case when $𝒜$ is a von Neumann algebra and $𝒳$ is selfdual: for any element x 0 ∈ $𝒮_{𝒳}$ and any tangent vector ν at x 0, there exists a curve γ(t) = e tZ(x 0), Z ∈ $ℒ_{𝒜} (𝒳)$ , Z* = −Z and...