The continuity properties of discrete-spectrum families of Fredholm operators

The Index Bundle for Selfadjoint Fredholm Operators and Multiparameter Bifurcation for Hamiltonian Systems

The index of a selfadjoint Fredholm operator is zero by the well-known fact that the kernel of a selfadjoint operator is perpendicular to its range. The Fredholm index was generalised to families by Atiyah and Jänich in the sixties, and it is readily seen that on complex Hilbert spaces this so called index bundle vanishes for families of selfadjoint Fredholm operators as in the case of a single operator. The first aim of this note is to point out that for every real Hilbert space and every compact topological space $X$ there is a family of selfadjoint Fredholm operators parametrised by $X\time…

On the spectral $ν$-continuity

In this paper we study the $ν$-continuity of the spectrum and some of its parts. We show that the approximate point spectrum $σ_{ap}$ is upper semi-$ν$-continuous at every Fredholm operator, then we give sufficient conditions to guarantee the $ν$-continuity of $σ_{ap}$. Also we show that the restriction of the Weyl spectrum on the class of essentially $G_1$ operators is $ν$-continuous. Finally, we investigate the $ν$-continuity of the spectrum on the class of $p$-hyponormal operators.

强连续和局部等连续算子族及其与双连续性的关系

Strongly continuous and locally equicontinuous families of operators and their relation to bi-continuity

本文研究了在序数完备Hausdorff局部凸空间上的强连续和局部等连续算子族，探讨了Saks空间中这些概念与双连续性及等紧性之间的关系，推广了已知结果至特定算子族。

On the continuity of spectra for families of magnetic pseudodifferential operators

For families of magnetic pseudodifferential operators defined by symbols and magnetic fields depending continuously on a real parameter $ε$, we show that the corresponding family of spectra also varies continuously with $ε$.

Index of continuous families of bounded linear operators in Banach spaces and application

In this paper, we define an analytical index for continuous families of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Banach space $X.$ We also consider the Weyl spectrum for continuous families of bounded linear operators and we study its continuity.

A new approach in index theory

In this paper, we define an analytical index for a continuous family of Fredholm operators parameterized by a topological space $\mathbb{X}$ into a Hilbert space $H,$ as a sequence of integers, extending naturally the usual definition of the index and we prove the homotopy invariance of the index. We give also an extension of the Weyl theorem for normal continuous families and we prove that if $H$ is separable, then the space of B-Fredholm operators on $H$ is path connected.

Minimal Families of Limit Operators

We study two abstract scenarios, where an operator family has a certain minimality property. In both scenarios, it is shown that norm, spectrum and resolvent are the same for all family members. Both abstract settings are illustrated by practically relevant examples, including discrete Schrödinger operators with periodic, quasiperiodic, almost-periodic, Sturmian and pseudo-ergodic potential. The main tool is the method of limit operators, known from studies of Fredholm operators and convergence of projection methods. We close by connecting this tool to the study of subwords of the operator pot…

Fredholm conditions on non-compact manifolds: theory and examples

We give explicit Fredholm conditions for classes of pseudodifferential operators on suitable singular and non-compact spaces. In particular, we include a "user's guide" to Fredholm conditions on particular classes of manifolds including asymptotically hyperbolic manifolds, asymptotically Euclidean (or conic) manifolds, and manifolds with poly-cylindrical ends. The reader interested in applications should be able read right away the results related to those examples, beginning with Section 5. Our general, theoretical results are that an operator adapted to the geometry is Fredholm if, and only …

Stable and Norm-stable Invariant Subspaces

We prove that if T is an operator on an infinite-dimensional Hilbert space whose spectrum and essential spectrum are both connected and whose Fredholm index is only 0 or 1, then the only nontrivial norm-stable invariant subspaces of T are the finite-dimensional ones. We also characterize norm-stable invariant subspaces of any weighted unilateral shift operator. We show that quasianalytic shift operators are points of norm continuity of the lattice of the invariant subspaces. We also provide a necessary condition for strongly stable invariant subspaces for certain operators.

Exhaustive families of representations of $C^*$-algebras associated to $N$-body Hamiltonians with asymptotically homogeneous interactions

We continue the analysis of algebras introduced by Georgescu, Nistor and their coauthors, in order to study $N$-body type Hamiltonians with interactions. More precisely, let $Y$ be a linear subspace of a finite dimensional Euclidean space $X$, and $v_Y$ be a continuous function on $X/Y$ that has uniform homogeneous radial limits at infinity. We consider, in this paper, Hamiltonians of the form $H = - Δ+ \sum_{Y \in S} v_Y$, where the subspaces $Y$ belong to some given family S of subspaces. We prove results on the spectral theory of the Hamiltonian when $S$ is any family of subspaces and exten…

Lipschitz Continuity of Spectra of Pseudodifferential Operators in a Weighted Sjöstrand Class and Gabor Frame Bounds

We study one-parameter families of pseudodifferential operators whose Weyl symbols are obtained by dilation and a smooth deformation of a symbol in a weighted Sjöstrand class. We show that their spectral edges are Lipschitz continuous functions of the dilation or deformation parameter. Suitably local estimates hold also for the edges of every spectral gap. These statements extend Bellissard's seminal results on the Lipschitz continuity of spectral edges for families of operators with periodic symbols to a large class of symbols with only mild regularity assumptions. The abstract results are us…

On the norm-continuity for evolution family arising from non-autonomous forms

We consider evolution equations of the form \begin{equation*}\label{Abstract equation} \dot u(t)+ A(t)u(t)=0,\ \ t\in[0,T],\ \ u(0)=u_0, \end{equation*} where $A(t),\ t\in [0,T],$ are associated with a non-autonomous sesquilinear form $\mathfrak a(t,\cdot,\cdot)$ on a Hilbert space $H$ with constant domain $V\subset H.$ In this note we continue the study of fundamental operator theoretical properties of the solutions. We give a sufficient condition for norm-continuity of evolution families on each spaces $V, H$ and on the dual space $V'$ of $V.$ The abstract results are applied to a class of e…

伪微分算子的极端谱值、Gabor框架界与饱和的Hölder连续性

Hölder-Continuity of Extreme Spectral Values of Pseudodifferential Operators, Gabor Frame Bounds, and Saturation

本文基于加权Sjöstrand类伪微分算子极端谱值的研究，证明了更大符号类具有参数Hölder连续性，并据此推导了Gabor系统族框架界在特定条件下的连续性与Hölder连续性。

On continuity of measurable group representations and homomorphisms

Let G be a locally compact group, and let U be its unitary representation on a Hilbert space H. Endow the space L(H) of linear bounded operators on H with weak operator topology. We prove that if U is a measurable map from G to L(H) then it is continuous. This result was known before for separable H. To prove this, we generalize a known theorem on nonmeasuralbe unions of point finite families of null sets. We prove also that the following statement is consistent with ZFC: every measurable homomorphism from a locally compact group into any topological group is continuous. This relies, in turn, …

正算子半群的序有界性与序连续性

Order boundedness and order continuity properties of positive operator semigroups

本文在巴拿赫格框架下对比相对一致连续（ruc）半群与$C_0$-半群，证明ruc半群即短时间轨道序有界的正$C_0$-半群，并关联谱界与增长界相等性等三个相关研究主题。

Regularity of stochastic Volterra equations by functional calculus methods

We establish pathwise continuity properties of solutions to a stochastic Volterra equation with an additive noise term given by a local martingale. The deterministic part is governed by an operator with an $H^\infty$-calculus and a scalar kernel. The proof relies on the dilation theorem for positive definite operator families on a Hilbert space.