Rafael Dahmen
We give a sufficient criterion for complex analyticity of nonlinear maps defined on direct limits of normed spaces. This tool is then used to construct new classes of (real and complex) infinite dimensional Lie groups: (a) groups of germs of analytic diffeomorphisms around a compact set in a Banach space and (b) unions of ascending sequences of Banach Lie groups.
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Let $M$ be a compact smooth manifold of dimension $m$ (without boundary) and $G$ be a finite-dimensional Lie group, with Lie algebra $g$. Let $H^{>m/2}(M,G)$ be the group of all mappings $γ\colon M\to G$ which are $H^s$ for some $s>m/2$. We show that $H^{>m/2}(M,G)$ can be made a regular Lie group in Milnor's sense, modelled on the Silva space $H^{>m/2}(M,g)$ which is the locally convex direct limit of the Hilbert spaces $H^s(M,g)$ for $s>m/2$, such that $H^{>m/2}(M,G)$ is the direct limit of the Hilbert-Lie groups $H^s(M,G)$ for $s>m/2$ as a smooth Lie group. We also explain how the (known) L…
We generalize the classical construction principles of infinite-dimensional real (and complex) Lie groups to the case of Lie groups over non-discrete topological fields. In particular, we discuss linear Lie groups, mapping groups, test function groups, diffeomorphism groups, and weak direct products of Lie groups. The specific tools of differential calculus required for the Lie group constructions are developed. Notably, we establish differentiability properties of composition and evaluation, as well as exponential laws for function spaces. We also present techniques to deal with the subtle di…
In this work, we construct and study certain classes of infinite dimensional Lie groups that are modelled on weighted function spaces. In particular, we construct a Lie group of weighted diffeomorphisms on a Banach space. Further, we also construct certain types of weighted mapping groups. Both the weighted diffeomorphism groups and the weighted mapping groups are shown to be regular Lie groups in Milnor's sense. We also discuss semidirect products of the former groups. Moreover, we study the integrability of Lie algebras of certain vector fields.
The condition onto pair ($F,G$) of function Banach spaces under which there exists a integral operator $T:F\to G$ with analytic kernel such that the inverse mapping $T^{-1}:$im$T\to F$ does not belong to arbitrary a priori given Borel (or Baire) class is found.
We present here "the" cartesian closed theory for real analytic mappings. It is based on the concept of real analytic curves in locally convex vector spaces. A mapping is real analytic, if it maps smooth curves to smooth curves and real analytic curves to real analytic curves. Under mild completeness conditions the second requirement can be replaced by: real analytic along affine lines. Enclosed and necessary is a careful study of locally convex topologies on spaces of real analytic mappings. As an application we also present the theory of manifolds of real analytic mappings: the group of real…
Characterization of Lie centralizable mappings on B(X)
本文研究了复 Banach 空间上有界线性算子代数 B(X) 上的 Lie 中心化映射，证明了在特定算子 W 处 Lie 中心化的加性映射 f 必可表示为 f(A)=kA+h(A) 的形式。
In this work, various versions of the so-called Omega-Lemma are provided, which ensure differentiability properties of pushforwrds between spaces of C^r-sections (or compactly supported C^r-sections) in vector bundles over finite-dimensional base manifolds whose fibres are (possibly infinite-dimensional) locally convex spaces. Applications are given, including the proof of continuity for some natural module multiplications on spaces of sections and the construction of certain infinite-dimensional Lie groups of Lie group-valued maps.
The purpose of this paper is to give an updated survey on various algebraic and analytic properties of semigroups related to fixed point properties of semigroup actions on a non-empty closed convex subset of a Banach space or, more generally, a locally convex topological vector space.
We characterize boundedness and compactness of the classical Volterra operator $T_g \colon H_{v_α}^{\infty} \to H^{\infty}$ induced by a univalent function $g$ for standard weights $v_α$ with $0 \leq α< 1$, partly answering an open problem posed by A. Anderson, M. Jovovic and W. Smith. We also study boundedness, compactness and weak compactness of the generalized Volterra operator $T_g^φ$ mapping between Banach spaces of analytic functions on the unit disc satisfying certain general conditions.
It is shown that every abelian regular Lie group is a quotient of its Lie algebra via the exponential mapping.
Let $A$ be a Banach ternary algebra over a scalar field $\Bbb R$ or $\Bbb C$ and $X$ be a ternary Banach $A-$module. Let $σ,τ$ and $ξ$ be linear mappings on $A$, a linear mapping $D:(A,[]_A)\to (X,[]_X)$ is called a Lie ternary $(σ,τ,ξ)-$derivation, if $$D([abc]_A)=[[D(a)bc]_X]_{(σ,τ,ξ)}+[[D(b)ac]_X]_{(σ,τ,ξ)}+[[D(c)ba]_X]_{(σ,τ,ξ)},$$ for all $a,b,c\in A$, where $[abc]_{(σ,τ,ξ)}=aτ(b)ξ(c)-σ(c)τ(b)a.$ In this paper, we investigate the generalized Hyers--Ulam--Rassias stability of Lie ternary $(σ,τ,ξ)-$derivations on Banach ternary algebras.
Motivated by the interesting and yet scattered developments in representation theory of Banach-Lie groups, we discuss several functional analytic issues which should underlie the notion of infinite-dimensional reductive Lie group: norm ideals, triangular integrals, operator factorizations, and amenability.
We prove that a linear mapping on the algebra \(\mathfrak{sl}_n\) of all trace zero complex matrices is a local automorphism if and only if it is an automorphism or an anti-automorphism. We also show that a linear mapping on a simple Leibniz algebra of the form \(\mathfrak{sl}_n\dot +\mathcal{I}\) is a local automorphism if and only if it is an automorphism. We give examples of finite-dimensional nilpotent Lie algebras \(\mathcal{L}\) with \(\dim \mathcal{L} \geq 3\) which admit local automorphisms which are not automorphisms.
Lie groups of real analytic diffeomorphisms are $L^1$-regular
本文证明紧致实解析流形的实解析微分同胚李群具有 $L^1$-正则性，并发展无限维李群及局部凸直极限非线性映射的相关工具结果。
Geometry of infinite dimensional Cartan Developments
本文将Cartan发展推广到无限维正则李群情形，将其视为演化映射的推广，并证明Cartan发展的切映射仍是Cartan发展。
Analytic subalgebras of Beurling-Fourier algebras and complexification of Lie groups
本文通过引入解析子代数，将加权Fourier代数的Gelfand谱元素解释为李群复化中特定点上的取值，并分别给出一般连通李群的局部解、连通单连通幂零李群的全局解，以及$ax+b$-群的全局解实例。
We address the problem of studying the boundedness, compactness and weak compactness of the integral operators $T_g(f)(z)=\int_0^z f(ζ)g'(ζ)\,dζ$ acting from a Banach space $X$ into $H^\infty$. We obtain a collection of general results which are appropriately applied and mixed with specific techniques in order to solve the posed questions to particular choices of $X$.
Uniqueness results for quasi-analytic functions on compact Lie groups and homogeneous spaces
本文利用Casimir算子迭代定义拟解析函数类，证明其强唯一延拓性，从而将经典Logvinenko-Sereda型定理推广到紧连通李群及其齐性空间。
Localised analytic torsion and relative analytic torsion for non compact Lie groups of type I
本文针对I型非紧李群定义表示局部化解析挠率，并据此定义群与离散余紧子群配对的相对解析挠率，证明其等于覆盖空间的Lott $L^2$解析挠率，且以Abel群和Heisenberg群为例说明。
In this note we announce Lp multiplier theorems for invariant and non-invariant operators on compact Lie groups in the spirit of the well-known Hormander-Mikhlin theorem on Rn and its variants on tori Tn. Applications are given to the mapping properties of pseudo-differential operators on Lp-spaces and to a-priori estimates for non-hypoelliptic operators.
In this paper we study the Cauchy problem for diffusion equations associated to a class of strongly hypoelliptic pseudo-differential operators on graded Lie groups. To do so, we develop a global complex functional calculus on graded Lie groups in order to analyse the corresponding energy estimates. One of the main aspects of this complex functional calculus is that for the $(ρ,δ)$-Euclidean Hörmander classes we recover the standard functional calculus developed by Seeley [38]. In consequence the Gårding inequality that we prove for arbitrary graded Lie groups absorbs the historical 1953's ineq…
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