scholarly journals Index theory for skew-adjoint fredholm operators

1969 ◽  
Vol 37 (1) ◽  
pp. 5-26 ◽  
Author(s):  
M. F. Atiyah ◽  
I. M. Singer
Keyword(s):  
Index Theory ◽  
Fredholm Operators

scholarly journals On index theory for non-Fredholm operators: A (1 + 1)-dimensional example

2015 ◽  
Vol 289 (5-6) ◽  
pp. 575-609 ◽  
Author(s):  
Alan Carey ◽  
Fritz Gesztesy ◽  
Galina Levitina ◽  
Denis Potapov ◽  
Fedor Sukochev ◽  
...  
Keyword(s):  
Index Theory ◽  
Fredholm Operators ◽  
Non Fredholm Operators

Doubling perturbation sizes and preservation of operator indices in normed linear spaces

1969 ◽  
Vol 66 (2) ◽  
pp. 281-294 ◽  
Author(s):  
Karl Gustafson
Keyword(s):  
Banach Spaces ◽  
Index Theory ◽  
Fredholm Operators ◽  
Linear Spaces ◽  
Sesquilinear Forms ◽  
Normed Linear Spaces

AbstractWe extend the index theory of Fredholm operators in Banach spaces to general operators in normed linear spaces. In so doing we make use of a doubling technique, which we also apply to obtain improved results for the perturbation of semigroups and sesquilinear forms.

Higher Index Theory

2020 ◽  
Author(s):  
Rufus Willett ◽  
Guoliang Yu
Keyword(s):  
Index Theory

Hypoelliptic Laplacian and Orbital Integrals (AM-177)

2011 ◽  
Author(s):  
Jean-Michel Bismut
Keyword(s):  
Fixed Point ◽  
Symmetric Space ◽  
Heat Kernel ◽  
Geodesic Flow ◽  
Casimir Operator ◽  
Index Theory ◽  
Weighted Sum ◽  
Orbital Integrals ◽  
Hypoelliptic Heat Kernel ◽  
Wave Kernel

This book uses the hypoelliptic Laplacian to evaluate semisimple orbital integrals in a formalism that unifies index theory and the trace formula. The hypoelliptic Laplacian is a family of operators that is supposed to interpolate between the ordinary Laplacian and the geodesic flow. It is essentially the weighted sum of a harmonic oscillator along the fiber of the tangent bundle, and of the generator of the geodesic flow. In this book, semisimple orbital integrals associated with the heat kernel of the Casimir operator are shown to be invariant under a suitable hypoelliptic deformation, which is constructed using the Dirac operator of Kostant. Their explicit evaluation is obtained by localization on geodesics in the symmetric space, in a formula closely related to the Atiyah-Bott fixed point formulas. Orbital integrals associated with the wave kernel are also computed. Estimates on the hypoelliptic heat kernel play a key role in the proofs, and are obtained by combining analytic, geometric, and probabilistic techniques. Analytic techniques emphasize the wavelike aspects of the hypoelliptic heat kernel, while geometrical considerations are needed to obtain proper control of the hypoelliptic heat kernel, especially in the localization process near the geodesics. Probabilistic techniques are especially relevant, because underlying the hypoelliptic deformation is a deformation of dynamical systems on the symmetric space, which interpolates between Brownian motion and the geodesic flow. The Malliavin calculus is used at critical stages of the proof.

Global Perturbation of Nonlinear Eigenvalues

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Julián López-Gómez ◽  
Juan Carlos Sampedro
Keyword(s):  
Perturbation Theory ◽  
Spectral Parameter ◽  
Classical Theory ◽  
General Setting ◽  
Perturbation Parameter ◽  
Linear Operators ◽  
Fredholm Operators ◽  
Index Zero ◽  
Finite Dimensional ◽  
Nonlinear Eigenvalues

Abstract This paper generalizes the classical theory of perturbation of eigenvalues up to cover the most general setting where the operator surface 𝔏 : [ a , b ] × [ c , d ] → Φ 0 ⁢ ( U , V ) {\mathfrak{L}:[a,b]\times[c,d]\to\Phi_{0}(U,V)} , ( λ , μ ) ↦ 𝔏 ⁢ ( λ , μ ) {(\lambda,\mu)\mapsto\mathfrak{L}(\lambda,\mu)} , depends continuously on the perturbation parameter, μ, and holomorphically, as well as nonlinearly, on the spectral parameter, λ, where Φ 0 ⁢ ( U , V ) {\Phi_{0}(U,V)} stands for the set of Fredholm operators of index zero between U and V. The main result is a substantial extension of a classical finite-dimensional theorem of T. Kato (see [T. Kato, Perturbation Theory for Linear Operators, 2nd ed., Class. Math., Springer, Berlin, 1995, Chapter 2, Section 5]).

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