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Armin Rainer

Publications and source records attributed to Armin Rainer.

66 records · Page 4Linked to original sources
arXiv2009

Perturbation of complex polynomials and normal operators

We study the regularity of the roots of complex monic polynomials $P(t)$ of fixed degree depending smoothly on a real parameter $t$. We prove that each continuous parameterization of the roots of a generic $C^\infty$ curve $P(t)$ (which always exists) is locally absolutely continuous. Generic means that no two of the continuously chosen roots meet of infinite order of flatness. Simple examples show that one cannot expect a better regularity than absolute continuity. This result will follow from the proposition that for any $t_0$ there exists a positive integer $N$ such that $t \mapsto P(t_0\pm (t-t_0)^N)$ admits smooth parameterizations of its roots near $t_0$. We show that $C^n$ curves $P(t)$ (where $n = °P$) admit differentiable roots if and only if the order of contact of the roots is $\ge 1$. We give applications to the perturbation theory of normal matrices and unbounded normal operators with compact resolvents and common domain of definition: The eigenvalues and eigenvectors of a generic $C^\infty$ curve of such operators can be arranged locally in an absolutely continuous way.

math.CA↗
arXiv2009

Many parameter Hoelder perturbation of unbounded operators

If $u\mapsto A(u)$ is a $C^{0,α}$-mapping, for $0< α\le 1$, having as values unbounded self-adjoint operators with compact resolvents and common domain of definition, parametrized by $u$ in an (even infinite dimensional) space, then any continuous (in $u$) arrangement of the eigenvalues of $A(u)$ is indeed $C^{0,α}$ in $u$.

math.FA↗
arXiv2009

The Convenient Setting for non-Quasianalytic Denjoy--Carleman Differentiable Mappings

For Denjoy--Carleman differential function classes $C^M$ where the weight sequence $M=(M_k)$ is logarithmically convex, stable under derivations, and non-quasianalytic of moderate growth, we prove the following: A mapping is $C^M$ if it maps $C^M$-curves to $C^M$-curves. The category of $C^M$-mappings is cartesian closed in the sense that $C^M(E,C^M(F,G))\cong C^M(E\x F, G)$ for convenient vector spaces. Applications to manifolds of mappings are given: The group of $C^M$-diffeomorphisms is a $C^M$-Lie group but not better.

math.FA↗
arXiv2008

Invariant functions in Denjoy-Carleman classes

Let $V$ be a real finite dimensional representation of a compact Lie group $G$. It is well-known that the algebra $\mathbb R[V]^G$ of $G$-invariant polynomials on $V$ is finitely generated, say by $σ_1,...,σ_p$. Schwarz proved that each $G$-invariant $C^\infty$-function $f$ on $V$ has the form $f=F(σ_1,...,σ_p)$ for a $C^\infty$-function $F$ on $\mathbb R^p$. We investigate this representation within the framework of Denjoy-Carleman classes. One can in general not expect that $f$ and $F$ lie in the same Denjoy-Carleman class $C_M$ (with $M=(M_k)$). For finite groups $G$ and (more generally) for polar representations $V$ we show that for each $G$-invariant $f$ of class $C_M$ there is an $F$ of class $C_N$ such that $f=F(σ_1,...,σ_p)$, if $N$ is strongly regular and satisfies $N_k \ge M_{km} \ep^{k+1}$, for all $k$, with $m$ an (explicitly known) integer depending only on the representation and $ε>0$ independent of $k$. In particular, each $G$-invariant $(1+δ)$-Gevrey function $f$ has the form $f=F(σ_1,...,σ_p)$ for a $(1+δm)$-Gevrey function $F$. Applications to equivariant functions and basic differential forms are given.

math.RT↗
arXiv2007

Orbit projections of proper Lie groupoids as fibrations

Let $\mathcal{G} \rightrightarrows M$ be a source locally trivial proper Lie groupoid such that each orbit is of finite type. The orbit projection $M \to M/\mathcal{G}$ is a fibration if and only if $\mathcal{G} \rightrightarrows M$ is regular.

math.DG↗
arXiv2006

Orbit projections as fibrations

The orbit projection $π: M \to M/G$ of a proper $G$-manifold $M$ is a fibration if and only if all points in $M$ are regular. Under additional assumptions we show that $π$ is a quasifibration if and only if all points are regular. We get a full answer in the equivariant category: $π$ is a $G$-quasifibration if and only if all points are regular.

math.DG↗
arXiv2006

Choosing roots of polynomials with symmetries smoothly

The roots of a smooth curve of hyperbolic polynomials may not in general be parameterized smoothly, even not $C^{1,α}$ for any $α> 0$. A sufficient condition for the existence of a smooth parameterization is that no two of the increasingly ordered continuous roots meet of infinite order. We give refined sufficient conditions for smooth solvability if the polynomials have certain symmetries. In general a $C^{3n}$ curve of hyperbolic polynomials of degree $n$ admits twice differentiable parameterizations of its roots. If the polynomials have certain symmetries we are able to weaken the assumptions in that statement.

math.CA↗
arXiv2005

Singular Poisson reduction of cotangent bundles

We consider the Poisson reduced space $(T^*Q)/K$ with respect to a cotangent lifted action. It is assumed that $K$ is a compact Lie group which acts by isometries on the Riemannian manifold $Q$ and that the action on $Q$ is of single isotropy type. Realizing $(T^*Q)/K$ as a Weinstein space we determine the induced Poisson structure and its symplectic leaves. We thus extend the Weinstein construction for principal fiber bundles to the case of surjective Riemannian submersions $Q\to Q/K$.

math.SG↗
arXiv2005

Lifting smooth curves over invariants for representations of compact Lie groups, III

Any sufficiently often differentiable curve in the orbit space $V/G$ of a real finite-dimensional orthogonal representation $G \to O(V)$ of a finite group $G$ admits a differentiable lift into the representation space $V$ with locally bounded derivative. As a consequence any sufficiently often differentiable curve in the orbit space $V/G$ can be lifted twice differentiably. These results can be generalized to arbitrary polar representations. Finite reflection groups and finite rotation groups in dimensions two and three are discussed in detail.

math.RT↗
arXiv2003

Lifting mappings over invariants of finite groups

We characterize those regular, holomorphic or formal maps into the orbit space $V/G$ of a complex representation of a finite group $G$ which admit a regular, holomorphic or formal lift to the representation space $V$. In particular, the case of complex reflection groups is investigated.

math.AG↗