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2 votes
0 answers
120 views

Closure of Laplacian

Let $(M,g)$ be a complete Riemannian manifold and $\Delta$ the (positive) Laplace-Beltrami operator. Now, consider this operator as an operator $$\Delta:\mathcal{D}(\Delta)\to L^{2}(M)$$ There are two ...
11 votes
2 answers
478 views

$x f'$ bounded by $x^2f $ and $f''$?

Consider the Hilbert space of functions $f \in L^2(\mathbb R)$ such that $x^2f \in L^2(\mathbb R) $ and $ f'' \in L^2(\mathbb R).$ I am wondering whether it is true that $xf'\in L^2(\mathbb R)$ as ...
7 votes
1 answer
311 views

Almost orthonormal projection and orthonormal projection in Hilbert space

Let $(e_i)_i$ be a family of vectors in a Hilbert space being almost orthonormal but not quite, i.e. $$\langle e_i, e_j \rangle \approx \delta_{i,j} + \alpha e^{-\vert i-j \vert} $$ and $\alpha$ is ...
1 vote
0 answers
922 views

A Question on certain Hilbert space of continuous functions, and a characteristic of convergence in it

Define $T^k(\Omega)$, $\Omega$ an open subset of $\mathbb{R}^m$ (with a smooth boundary), as a space of function equivalance classes, with the norm defined as $$ \|f\|_{T^k(\Omega)}^2 = \|f\|_{L^2(...
4 votes
0 answers
174 views

Constant in trace theorem for balls

Consider the standard open ball $B_r:=\left\{x ; \left\lvert x \right\rvert \le R \right\}.$ The trace theorem tells us any function in $W^{k,p}(B_r)$ can be restricted to a function $W^{k-1,p}(\...
7 votes
1 answer
337 views

Flows in Hilbert spaces

Let $\varphi: [0,T] \rightarrow H$ be a Hilbert space valued $C^1$-function. Let $H = X \oplus X^{\perp}$ such that $\varphi(0) \in X$ and the implication $\varphi(t) \in X \Rightarrow \varphi'(t) \in ...