All Questions
8 questions
5
votes
2
answers
149
views
Showing an operator is (or not) closed on $L^2(\mathbb{R})$
I am linearizing nonlinear waves and get operators of the form below. Everything is considered in $L^2(\mathbb{R})$.
Consider the operator $L_1=\frac{d}{dx}$. The domain is $H^1(\mathbb{R})$ and it is ...
4
votes
1
answer
228
views
Diagonalizing selfadjoint operator on core domain
Let $A$ be a densely-defined, positive, self-adjoint operator with compact resolvent on a Hilbert space $H$. Then, $\text{Range}(1+A)=H$ and there is a basis for $H$ consisting of eigenvectors of $1+A$...
- 143
1
vote
0
answers
153
views
Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 185
2
votes
1
answer
134
views
Decomposition of the spectrum of an unbounded opeator [closed]
The Wikipedia article on spectral decomposition, see here
https://en.wikipedia.org/wiki/Self-adjoint_operator
says the following:
A self-adjoint operator A on $H$ has pure point spectrum if and ...
1
vote
1
answer
2k
views
Operator theory of the Hessian
How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second ...
- 5,327
1
vote
1
answer
394
views
Pullback via flow as operator group
Let $X$ be a vector field on a manifold $M$ that induces a complete flow $\Theta_t$. Then the operator family $\Theta_t^*$,
$$\Theta_t^*u(x) = u(\Theta_t(x))$$
is a strongly continuous semigroup of ...
- 9,853
1
vote
1
answer
275
views
Laplacian on space of measures
Let $X$ be a compact Riemannian manifold and let $\mathcal{M}(X)$ be the space of regular finite Borel measures with the total variation as norm.
The Laplace-Belrami-Operator $\Delta$ on $X$ with ...
- 9,853
2
votes
0
answers
137
views
Invariant linear manifolds for multiplication by the independent variable in L^2 (R)
In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold (non-...
- 31