Skip to main content

All Questions

Filter by
Sorted by
Tagged with
13 votes
1 answer
465 views

One question about the $\eta$ invariant

This question is from the paper, The Analysis of Elliptic Families II. Dirac Operators, Eta Invariants, and the Holonomy Theorem, Commun. Math. Phys. 107, 103-163 (1986) --- Proposition 2.8. Suppose ...
DLIN's user avatar
  • 1,915
7 votes
0 answers
237 views

Understanding the odd-dimensional index

Given a Dirac operator $D$ on a closed odd-dimensional manifold $M$, I've sometimes heard it said that the Fredholm index of $D$ vanishes because it is an ungraded self-adjoint operator, so that $\dim\...
geometricK's user avatar
  • 1,903
5 votes
1 answer
361 views

Is this a pseudodifferential operator?

Let $M$ be a non-compact manifold and $D$ a first-order self-adjoint elliptic differential operator on $M$. Then is the bounded operator $$A:=\sqrt{(D^2+1)^{-1}}:L^2(M)\rightarrow H^1(M)$$ a ...
geometricK's user avatar
  • 1,903
5 votes
0 answers
191 views

Index of the Fredholm operator

I have two vector bundles $E_1$, $E_2$ over $M$ and an embedding of the smooth sections $\lambda : \Gamma(M, E_1) \rightarrow \Gamma(M, E_1 \oplus E_2)$. I consider a Fredholm differential operator $...
Aleksandr Alekseev's user avatar
5 votes
0 answers
211 views

Infinitesimal Generator of Billiard Flow

The Billiard flow $S_t$ on a Riemannian manifold with boundary (with corners) is the group of operators defined on continuous functions on the Co-sphere bundle as follows: To determine $S_t u(\xi)$, ...
Matthias Ludewig's user avatar
4 votes
1 answer
532 views

Diffusion semigroup generated by Laplacian

Let $M$ be a complete Riemannian manifold and $\Delta$ denote the Laplacian on it. Also assume that the spectrum of $-\Delta$ lies inside $[a, \infty)$. Let $P_t, t > 0$ denote the diffusion ...
anonymos's user avatar
3 votes
0 answers
108 views

A question on essentially self-adjoint differential operators of the type $\Delta=D^{\ast}D$

Let $(M,g)$ be a smooth (connected, complete, oriented) Riemannian manifold and let $D:C^{\infty}(M)\to C^{\infty}(M)$ be a linear partial differential operator, which I view as an operator in $L^{2}(...
B.Hueber's user avatar
  • 1,171
3 votes
0 answers
289 views

Are smooth functions with compact support a core for the Laplacian on compact manifolds with boundary?

If $M$ is a complete Riemannian manifold and $L$ is the Friedrichs extension of the Laplacian $-\Delta$, then it is known (first proven by Gaffney in the '50) that $C_0 ^\infty (M)$ is a core for $L$. ...
Alex M.'s user avatar
  • 5,407
3 votes
0 answers
91 views

Pseudodifferential operator associated to a self-adjoint extension of a symmetric operator on an incomplete manifold

Let $D$ be the Dirac operator acting on a spinor bundle $S$ over a complete Riemannian manifold $M$. Then $D$ is an essentially self-adjoint operator on $L^2(S)$. Suppose there is a compact subset $K\...
geometricK's user avatar
  • 1,903
3 votes
0 answers
406 views

Smooth perturbation of a positive self-adjoint operator with compact resolvent

Consider a one-parameter family $A_t$ of unbounded positive self-adjoint operators with discrete spectrum (for example, one can consider a one-parameter family of Laplacians on a compact Riemannian ...
user82891's user avatar
2 votes
0 answers
53 views

A question about the choice of a special harmonc spinor

Let $X$ be a complete Riemannian manifold and $H$ be the kernel of generalized Dirac operator $D$ on $L(S)$, where $S$ is the Dirac bundle. Let $K$ be a compact subset of $X$ and $K\subset \Omega$ be ...
Radeha Longa's user avatar
1 vote
1 answer
728 views

Elliptic regularity of Laplace-Beltrami operator on a manifold

I am currently trying to prove an elliptic regularity type result for the Laplace Beltrami operator $\Delta_g$ on a Riemannian manifold $(M^n,g)$. As a matter of convention, I will assume $\Delta_g$ ...
Dominic Wynter's user avatar
1 vote
0 answers
179 views

Positive square roots of inverse operators on different Sobolev spaces

Let $D$ be a self-adjoint (in the $H^0$-inner product) first-order differential operator on a manifold $M$, where $H^i$ stands for the $i$-th Sobolev space on $M$. Then $D$ extends to a bounded ...
geometricK's user avatar
  • 1,903
0 votes
0 answers
141 views

The tensor product of two Fredholm operators

What can be said about the tensor product $T\otimes S$ of two Fredholm operators $T:X_1\to Y_1$ and $S:X_2 \to Y_2$ where $X_1,X_2,Y_1, Y_2$ are Banach spaces and tensor product of ...
Ali Taghavi's user avatar
0 votes
0 answers
126 views

A question about associated operator on continuous functions space equiped with L2 norm

For M a connected compact manifold, $T$ is in $C^{1+\nu}(M,M)$ with $\nu\in(0,1)$, i.e., $DT$ is some Hölder continuous function with Hölder exponent $\nu$. Denote by $m$ the Lebesgue measure on $M$ ...
WaoaoaoTTTT's user avatar