scholarly journals An Interpolation Theorem for Quasimartingales in Noncommutative Symmetric Spaces

2021 ◽  
Vol 2021 ◽  
pp. 1-5
Author(s):  
Congbian MA ◽  
Guoxi Zhao
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Interpolation Theorem ◽  
Noncommutative Space ◽  
Boyd Indices ◽  
Interpolation Result

Let E be a separable symmetric space on 0 , ∞ and E M the corresponding noncommutative space. In this paper, we introduce a kind of quasimartingale spaces which is like but bigger than E M and obtain the following interpolation result: let E ^ M be the space of all bounded E M -quasimartingales and 1 < p < p E < q E < q < ∞ . Then, there exists a symmetric space F on 0 , ∞ with nontrivial Boyd indices such that E ^ M = L ^ p M , L ^ q M F , K .

scholarly journals THE SMOOTHNESS OF ORBITAL MEASURES ON NONCOMPACT SYMMETRIC SPACES

2021 ◽  
pp. 1-20
Author(s):  
SANJIV KUMAR GUPTA ◽  
KATHRYN E. HARE
Keyword(s):  
Symmetric Space ◽  
Symmetric Spaces ◽  
Continuous Measure ◽  
Convolution Product ◽  
Absolutely Continuous ◽  
Regular Elements ◽  
Connected Subgroup ◽  
Decay Properties ◽  
Absolutely Continuous Measure ◽  
Special Case

Abstract Let $G/K$ be an irreducible symmetric space, where G is a noncompact, connected Lie group and K is a compact, connected subgroup. We use decay properties of the spherical functions to show that the convolution product of any $r=r(G/K)$ continuous orbital measures has its density function in $L^{2}(G)$ and hence is an absolutely continuous measure with respect to the Haar measure. The number r is approximately the rank of $G/K$ . For the special case of the orbital measures, $\nu _{a_{i}}$ , supported on the double cosets $Ka_{i}K$ , where $a_{i}$ belongs to the dense set of regular elements, we prove the sharp result that $\nu _{a_{1}}\ast \nu _{a_{2}}\in L^{2},$ except for the symmetric space of Cartan class $AI$ when the convolution of three orbital measures is needed (even though $\nu _{a_{1}}\ast \nu _{a_{2}}$ is absolutely continuous).

Corwin–Greenleaf multiplicity functions for complex semisimple symmetric spaces

2015 ◽  
Vol 26 (05) ◽  
pp. 1550039
Author(s):  
Salma Nasrin
Keyword(s):  
Symmetric Space ◽  
Lie Algebras ◽  
Lie Group ◽  
Symmetric Spaces ◽  
Coadjoint Orbit ◽  
Natural Projection ◽  
Real Form ◽  
Single Orbit ◽  
Complex Semisimple ◽  
Complex Symmetric

Let Gℂ be a complex simple Lie group, GU a compact real form, and [Formula: see text] the natural projection between the dual of the Lie algebras. We prove that, for any coadjoint orbit [Formula: see text] of GU, the intersection of [Formula: see text] with a coadjoint orbit [Formula: see text] of Gℂ is either an empty set or a single orbit of GU if [Formula: see text] is isomorphic to a complex symmetric space.

DECAY RATES FOR THE SHIFTED WAVE EQUATION ON A SYMMETRIC SPACE OF NONCOMPACT TYPE

2013 ◽  
Vol 10 (04) ◽  
pp. 677-701
Author(s):  
CARLOS ALMADA
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Symmetric Space ◽  
Decay Rate ◽  
Wave Operator ◽  
Symmetric Spaces ◽  
Decay Rates ◽  
Noncompact Type ◽  
Killing Fields ◽  
The Cauchy Problem

We derive L∞–L1 decay rate estimates for solutions of the shifted wave equation on certain symmetric spaces (M, g). The Cauchy problem for the shifted wave operator on these spaces was studied by Helgason, who obtained a closed form for its solution. Our results extend to this new context the classical estimates for the wave equation in ℝn. Then, following an idea from Klainerman, we introduce a new norm based on Lie derivatives with respect to Killing fields on M and we derive an estimate for the case that n = dim M is odd.

scholarly journals On projective-symmetric spaces

1964 ◽  
Vol 4 (1) ◽  
pp. 113-121 ◽  
Author(s):  
Bandana Gupta
Keyword(s):  
Symmetric Space ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Covariant Differentiation ◽  
Metric Tensor ◽  
Projective Curvature ◽  
Projective Curvature Tensor ◽  
Image Position ◽  
Riemannian Spaces

This paper deals with a type of Remannian space Vn (n ≧ 2) for which the first covariant dervative of Weyl's projective curvature tensor is everywhere zero, that is where comma denotes covariant differentiation with respect to the metric tensor gij of Vn. Such a space has been called a projective-symmetric space by Gy. Soós [1]. We shall denote such an n-space by ψn. It will be proved in this paper that decomposable Projective-Symmetric spaces are symmetric in the sense of Cartan. In sections 3, 4 and 5 non-decomposable spaces of this kind will be considered in relation to other well-known classes of Riemannian spaces defined by curvature restrictions. In the last section the question of the existence of fields of concurrent directions in a ψ will be discussed.

An Interpolation Theorem

2000 ◽  
Vol 6 (4) ◽  
pp. 447-462 ◽  
Author(s):  
Martin Otto
Keyword(s):  
Interpolation Theorem ◽  
Order Logic ◽  
First Order Logic ◽  
Elementary Model ◽  
Relation Symbol ◽  
First Order ◽  
Theoretic Proof ◽  
Universal Quantification ◽  
The Given ◽  
Interpolation Result

AbstractLyndon's Interpolation Theorem asserts that for any valid implication between two purely relational sentences of first-order logic, there is an interpolant in which each relation symbol appears positively (negatively) only if it appears positively (negatively) in both the antecedent and the succedent of the given implication. We prove a similar, more general interpolation result with the additional requirement that, for some fixed tuple of unary predicates U, all formulae under consideration have all quantifiers explicitly relativised to one of the U. Under this stipulation, existential (universal) quantification over U contributes a positive (negative) occurrence of U.It is shown how this single new interpolation theorem, obtained by a canonical and rather elementary model theoretic proof, unifies a number of related results: the classical characterisation theorems concerning extensions (substructures) with those concerning monotonicity, as well as a many-sorted interpolation theorem focusing on positive vs. negative occurrences of predicates and on existentially vs. universally quantified sorts.

CLASSES MINIMALES DE RÉSEAUX ET RÉTRACTIONS GÉOMÉTRIQUES ÉQUIVARIANTES DANS LES ESPACES SYMÉTRIQUES

2001 ◽  
Vol 64 (2) ◽  
pp. 275-286 ◽  
Author(s):  
CHRISTOPHE BAVARD
Keyword(s):  
Finite Group ◽  
Euclidean Space ◽  
Symmetric Space ◽  
Group Action ◽  
Symmetric Spaces ◽  
Classification Problem ◽  
Finite Group Action ◽  
Natural Geometry ◽  
A Finite Group

Equivariant and cocompact retractions of certain symmetric spaces are constructed. These retractions are defined using the natural geometry of symmetric spaces and in relation to the theory of lattices of euclidean space. The following cases are considered: the symmetric space corresponding to lattices endowed with a finite group action, from which is obtained some information relating to the classification problem of these lattices, and the Siegel space Sp2g(R)/Ug, for which a natural Sp2g(Z)-equivariant cocompact retract of codimension 1 is obtained.

scholarly journals Vassiliev invariants from symmetric spaces

2016 ◽  
Vol 25 (10) ◽  
pp. 1650055 ◽  
Author(s):  
Indranil Biswas ◽  
Niels Leth Gammelgaard
Keyword(s):  
Symmetric Space ◽  
Lie Algebra ◽  
Curvature Tensor ◽  
Symmetric Spaces ◽  
Weight System ◽  
Riemannian Symmetric Space ◽  
Riemann Curvature ◽  
Riemann Curvature Tensor ◽  
Vassiliev Invariants ◽  
Chord Diagrams

We construct a natural framed weight system on chord diagrams from the curvature tensor of any pseudo-Riemannian symmetric space. These weight systems are of Lie algebra type and realized by the action of the holonomy Lie algebra on a tangent space. Among the Lie algebra weight systems, they are exactly characterized by having the symmetries of the Riemann curvature tensor.

INTEGRABILITY AND SYMMETRIC SPACES II: THE COSET SPACES

1989 ◽  
Vol 04 (03) ◽  
pp. 675-699 ◽  
Author(s):  
L. A. FERREIRA
Keyword(s):  
Symmetric Space ◽  
Poisson Bracket ◽  
Algebraic Structure ◽  
Symmetric Spaces ◽  
Sufficient Condition ◽  
Conserved Quantities ◽  
Coset Space ◽  
Canonical Variables ◽  
Coset Spaces

It is shown that a sufficient condition for a model describing the motion of a particle on a coset space to possess a Fundamental Poisson bracket Relation, and consequently charges in involution, is that it must be a symmetric space. The conditions, a Hamiltonian, or any functions of the canonical variables, has to satisfy in order to commute with these charges, are studied. It is shown that, for the case of the noncompact symmetric spaces, these conditions lead to an algebraic structure which plays an important role in the construction of conserved quantities.

scholarly journals Traces of generalized potentials

2017 ◽  
Vol 25 ◽  
pp. 84
Author(s):  
B.I. Peleshenko ◽  
T.N. Semirenko
Keyword(s):  
Symmetric Space ◽  
Integral Operators ◽  
Interpolation Theorem

The theorem on finding traces of integral operators defined on a symmetric space with a certain kernel is obtained using the Krein-Semenov interpolation theorem.

Some Hypersurfaces of Symmetric Spaces

1983 ◽  
Vol 26 (3) ◽  
pp. 303-311 ◽  
Author(s):  
Yoshio Matsuyama
Keyword(s):  
Symmetric Space ◽  
Projective Space ◽  
Symmetric Spaces ◽  
Real Projective Space ◽  
Principal Curvatures

AbstractIn this paper we consider how much we can say about an irreducible symmetric space M which admits a hypersurface N with at most two distinct principal curvatures. Then we will obtain that (1) if N is locally symmetric, then M must be a sphere, a real projective space and their noncompact duals (2) if N is Einstein, then M must be rank 1.

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