All Questions
33 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 ...
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 ...
- 356
0
votes
0
answers
29
views
On constructing the canonical boundary operator for a given differential operator
Given an $n\times n$ matrix $$X=\begin{pmatrix}
x_{11} & x_{12} & \cdots & x_{1n} \\
x_{21} & x_{22} & \cdots & x_{2n} \\
\vdots & \vdots & \ddots & \vdots \\
x_{n1}...
- 765
2
votes
1
answer
644
views
Reference request: inverse of differential operators
I have asked a similar question on MSE but I did not receive any replies, so I am reposting here in case it is more appropriate (though I have slightly generalized the question).
As an example ...
- 721
3
votes
2
answers
147
views
Lumer-Phillips-type theorem for non-autonomous evolutions
The classical Lumer-Phillips theorem characterizes the generators of contraction semigroups. I am looking for a similar characterization or at least a sufficient condition for a family of unbounded, ...
- 300
4
votes
1
answer
228
views
Diagonalizing selfadjoint operator on core domain
Let $A$ be a densely-defined, positive, self-adjoint operator with compact resolvent on a Hilbert space $H$. Then, $\text{Range}(1+A)=H$ and there is a basis for $H$ consisting of eigenvectors of $1+A$...
- 143
4
votes
1
answer
156
views
approximation of a Feller semi-group with the infinitesimal generator
Let $T_t$ a Feller semigroup (see this) and let $(A,D(A))$ its infinitesimal generator.
If A is a bounded operator it is easy to show that the Feller semi-group is $e^{tA}$.
Is this formula always ...
- 293
1
vote
0
answers
153
views
Maximal domain of an unbounded linear operator in a weighted Hilbert-space
Let's consider the following (unbounded) linear operator. (So called transport operator in some context.)
$$ \mathrm{T}: \mathcal{H} \supset \mathcal{D}(\mathrm{T}) \to \mathcal{H} , f \mapsto \mathrm{...
- 185
2
votes
1
answer
149
views
On a core for Neumann Laplacian on $C(\overline{D})$
Let $D \subset \mathbb{R}^d$ be a bounded $C^1$ domain. We consider a reflected Brownian motion $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in \overline{D}})$ on $\overline{D}$. Let $\{p_t\}_{t>0}$ denote ...
- 721
6
votes
1
answer
576
views
Spectrum of the complex harmonic oscilllator
Let
$$
H_\lambda=-\frac{d^2}{dx^2}+\lambda^2 x^2,\quad\lambda>0.
$$ It is known that the spectrum of $H_\lambda$ is the set $\{(2n-1)\lambda,n\in \Bbb N^*\}$. Now put
$$
(U_\mu \phi)(x)= e^{\mu\...
- 521
2
votes
1
answer
206
views
On which subspace $W\subset C^{\infty}[0,1]$ is $(Df)(x)=xf'(x)$ a bounded operator provided all functions in $W$ are flat functions?
I have already asked this question on MSE; now I repeat it on MO.
https://math.stackexchange.com/questions/4132346/on-which-subspace-w-subset-c-infty0-1-is-df-xfx-a-bounded-operator
First we ...
- 356
3
votes
1
answer
140
views
Infinite-dimensional analogue of "positive-negative splitting implies non-degeneracy"
(This question is related to Splitting a space into positive and negative parts but different.)
Given a finite-dimensional vector space $V$ over $\mathbb{R}$, what I call a "positive-negative ...
- 1,804
1
vote
0
answers
92
views
Closure of $f\mapsto\sigma f''$ on $\mathcal{C}^2(\,[0,1]\,)$
Let $\sigma\in\mathcal{C}^0(]0,1])$ a positive function such that $\lim\limits_{t\rightarrow 0}\sigma(t)=0$, and $f\in\mathcal{C}^0(\,[0,1]\,)\cap\mathcal{C}^2(\,]0,1]\,)$ such that $\lim\limits_{t\...
- 449
3
votes
0
answers
774
views
The exponential derivative operator
Thank you very much for the interesting responses in my previous question The Quotient exponential operator.
I have another complicated formula related to the previous one in the following
$$
\exp\...
- 159
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 $...
0
votes
0
answers
113
views
References for a proof or interpretation of deficiency indices theorem (von Neumann)
I am looking for a proof or some interpretation around why the domain of the new extension $D(A_U)$ in the Theorem below is given by its specific formula.
I have already searched in papers and here ...
- 41
2
votes
0
answers
99
views
1D Schrödinger Equation with Measure-Valued Coefficients
I've been looking at one of the simplest systems I can think of: a one-dimensional infinite square well on $[0,1]$ with Hamiltonian given by the following:
$$\hat{H}=-\frac{\hbar^2}{2m}\frac{d^2}{dx^2}...
- 21
4
votes
0
answers
169
views
Inverse Laplacian and convolution in Albeverio's “Solvable Models in quantum mechanics”
I asked this question on math.stackexchange.com two weeks ago but got no answers so far and I got no clues from literature, so maybe someone here knows a reference. I hope it is ok to ask this ...
- 141
2
votes
0
answers
306
views
Interesting examples of spectral decompositions of BOUNDED operators with both continuous and discrete spectrum
I would like to have a few basic examples of bounded self-adjoint operators $T$ (more generally bounded normal would be fine) on a Hilbert space $(H,\langle,\rangle)$ for which the following criteria ...
- 7,609
4
votes
0
answers
174
views
Superposition operator from Sobolev space to Lebesgue space
Let $\Omega$ be a bounded, connected set in $\mathbb{R}^2$ with smooth boundary. I am wondering under what conditions on the real function $f(x):\mathbb{R}\to \mathbb{R}$ the superposition operator $F(...
- 395
4
votes
0
answers
154
views
What is the generator of the heat semigroup on non-complete manifolds?
If $M$ is a complete Riemannian manifold, it possesses a unique self-adjoint positive operator $-\Delta$ on $L^2(M)$. If $M$ is not complete, though, it is known that the Laplace-Beltrami operator $-\...
- 5,407
2
votes
2
answers
460
views
A possible dynamical approach to the "Invariant Subspace Problem"
In the literature, is there any paper or research investigating the invariant subspace problem with consideration of differential operators acting on an appropriate Sobolev space?In particular is ...
- 356
3
votes
0
answers
182
views
Prove a certain function maps to upper half plane
Suppose $M$ is a bounded self-adjoint operator on space of complex valued functions on the real line $S_1=L^2(\mathbb{R},a(x)dx)$, where $a(x)$ is a nice real positive analytical function ( I have in ...
- 31
2
votes
1
answer
238
views
Why is index unchanged after applying functional calculus?
Suppose $D$ is the Dirac operator on a closed spin manifold $M$, with spinors $S$. One can take the functional calculus of $D$ with respect to the continuous function $f:\mathbb{R}\rightarrow\mathbb{R}...
- 1,903
1
vote
1
answer
187
views
Spectrum on an unbounded operator
Consider the operator $T_{c}=-\frac{d}{dx^{2}}+ c x^{2}$ with $c\in C^{*}$, $Re(c)>0$ defined on its domain $D_{c}=\{u\in L^{2}; T_{c}(u)\in L^{2}\}$.
Put $c=a+ib$ avec $a>0$ et $b\in R$.
...
- 135
2
votes
1
answer
291
views
analytic continuation argument
In "Pseudo-spectra, the harmonic oscillator and
complex resonances" (login required), the author says
Sections $2$ and $3$ of this paper concern the operator $Hf(x)=(-\frac{d^{2}}{dx^{2}}+...
- 135
4
votes
1
answer
254
views
Strongly continuous semigroups and symbols of pseudo differential operators
I am considering the Cauchy IVP for the evolution equation
$$u_t + \Psi u =0$$
where $\Psi$ is a linear pseudo differential operator with symbol $\widehat{\Psi}\left(\underline{\xi}\right)$.
The ...
- 155
1
vote
1
answer
2k
views
Operator theory of the Hessian
How can I learn more about the operator theory of the Hessian? The Hessian of a function $u : \Omega \rightarrow \mathbb R$ over a domain $\Omega \subseteq \mathbb R^n$ is the matrix of second ...
- 5,327
3
votes
1
answer
277
views
adjoint of this closed (?) operator
I am currently dealing with an unbounded operator
$T:\{f \in L^2(-2\pi,2\pi); f \in AC((-2\pi,2\pi)), T(f) \in L^2, \lim_{x \rightarrow \pm 2 \pi} f(x)g(x)=0\} \subset L^2(-2\pi,2\pi)\rightarrow L^2(...
2
votes
1
answer
136
views
Proper domain for operators
in this paper on arxiv in equation 27, two operators
$$A_m^* = (1-x^2)^{\frac{1}{2}} \frac{d}{dx} + \frac{mx}{\sqrt{1-x^2}}$$
and $$A_m = - \frac{d}{dx}(1-x^2)^{\frac{1}{2}} + \frac{mx}{\sqrt{1-x^2}...
1
vote
1
answer
210
views
Functional Calculus and Fredholm index
Let $-\Delta: W^{2,2} \subset L^2(\mathbb{S}^2) \rightarrow L^2(\mathbb{S}^2)$. Then it is "easy" to show that $-\Delta $ is self-adjoint. Now, I am looking for closed operators $T$ and $T^*$ of order ...
- 11
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
0
votes
1
answer
795
views
Can we construct a Hilbert space where the operator following differencial operator is symmetric?
I'd like to know if one can define a pertinent Hilbert space where the operator
$$A_p v := -\frac{1}{2} v" + (vF + v\int_\mathbb{R} Sp + p\int_\mathbb{R} Sv )'$$ is symmetric. Here, $p$ satisfies ...