Holomorphy and Convexity in Lie Theory

Neeb, Karl-Hermann

doi:10.1515/9783110808148

Cited by 152 publications

(167 citation statements)

References 0 publications

Supporting

Mentioning

167

Contrasting

Order By: Relevance

“…The necessary background material concerning convex cones and Laplace transforms (of positive measures) needed here can be found, e.g., in [10] or [8].…”

Section: Results On General Dirichlet Series and Laplace Transformsmentioning

confidence: 99%

General Dirichlet series, arithmetic convolution equations and Laplace transforms

2009

View full text Add to dashboard Cite

Abstract. In the earlier paper [Proc. Amer. Math. Soc. 135 (2007)], we studied solutions g : N → C to convolution equations of the formwhere a0, . . . , a d : N → C are given arithmetic functions associated with Dirichlet series which converge on some right half plane, and also g is required to be such a function. In this article, we extend our previous results to multidimensional general Dirichlet series of the form P x∈X f (x)e −sx (s ∈ C k ), where X ⊆ [0, ∞) k is an additive subsemigroup. If X is discrete and a certain solvability criterion is satisfied, we determine solutions by an elementary recursive approach, adapting an idea of Fečkan [Proc. Amer. Math. Soc. 136 (2008)]. The solution of the general case leads us to a more comprehensive question: Let X be an additive subsemigroup of a pointed, closed convex cone C ⊆ R k . Can we find a complex Radon measure on X whose Laplace transform satisfies a given polynomial equation whose coefficients are Laplace transforms of such measures?

show abstract

“…The necessary background material concerning convex cones and Laplace transforms (of positive measures) needed here can be found, e.g., in [10] or [8].…”

Section: Results On General Dirichlet Series and Laplace Transformsmentioning

confidence: 99%

General Dirichlet series, arithmetic convolution equations and Laplace transforms

2009

View full text Add to dashboard Cite

show abstract

“…to generalize the theory of analytic extension of unitary group representations ( [27], [23,24], [37,38], [33]) to the context of Banach-Lie groups.…”

Section: Annales De L'institut Fouriermentioning

confidence: 99%

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Neeb¹

2011

Ann. inst. Fourier

View full text Add to dashboard Cite

Let G be a connected and simply connected Banach-Lie group or, more generally, a BCH-Lie group. On the complex enveloping algebra U C (g) of its Lie algebra g we define the concept of an analytic functional and show that every positive analytic functional λ is integrable in the sense that it is of the form λ(D) = dπ(D)v, v for an analytic vector v of a unitary representation of G. On the way to this result we derive criteria for the integrability of * -representations of infinite dimensional Lie algebras of unbounded operators to unitary group representations.For the matrix coefficient π v,v (g) = π(g)v, v of a vector v in a unitary representation of an analytic Fréchet-Lie group G we show that v is an analytic vector if and only if π v,v is analytic in an identity neighborhood. Combining this insight with the results on positive analytic functionals, we derive that every local positive definite analytic function on a 1-connected Fréchet-BCH-Lie group G extends to a global analytic function.

show abstract

“…The general approach to positive definite functions on semigroups and related objects is developed in [Ne99]. For the convenience of the reader we refer to that book for details also on background material like reproducing kernel spaces which could be found elsewhere (e.g.…”

Section: K Q(s) Q(t) = ϕ Q(st * ) (*)mentioning

confidence: 99%

“…Let X be a right S-space. According to [Ne99,Prop. II.4.3], a positive definite kernel K on X is invariant if and only if the action of S on C X given by (s.f )(x) := f(x.s) leaves the pre-Hilbert space H 0 K invariant and defines on this space an hermitian representation (π K , H 0 K ).…”

Section: Positive Definite Spherical Functions On Positive Domainsmentioning

confidence: 99%

Positive definite spherical functions on Olshanskii domains

Hilgert

Neeb

1999

Trans. Amer. Math. Soc.

Self Cite

View full text Add to dashboard Cite

Abstract. Let G be a simply connected complex Lie group with Lie algebra g, h a real form of g, and H the analytic subgroup of G corresponding to h. The symmetric space M = H\G together with a G-invariant partial order ≤ is referred to as an Ol shanskiȋ space. In a previous paper we constructed a family of integral spherical functions φµ on the positive domain M + := {Hx : Hx ≥ H} of M. In this paper we determine all of those spherical functions on M + which are positive definite in a certain sense.

show abstract

Holomorphy and Convexity in Lie Theory

Cited by 152 publications

References 0 publications

General Dirichlet series, arithmetic convolution equations and Laplace transforms

General Dirichlet series, arithmetic convolution equations and Laplace transforms

On Analytic Vectors for Unitary Representations of Infinite Dimensional Lie Groups

Positive definite spherical functions on Olshanskii domains

Contact Info

Product

Resources

About