All Questions
42 questions
15
votes
3
answers
1k
views
Version of Banach-Steinhaus theorem
I am wondering about the following version of the Banach-Steinhaus theorem.
Let $A$ be a closed convex subset contained in the unit ball of a Banach space $X$ and consider bounded operators $T_n \in \...
- 536
11
votes
3
answers
3k
views
Dual space of $L^2(\mathbb{R},L^1(0,1))$?
I was wondering what the dual space of $L^2(\mathbb{R},L^1(0,1))$ is? (equipped with Lebesgue measures)
Formally, one would suspect that it is just $L^2(\mathbb{R},L^{\infty}(0,1))$. But this may be a ...
- 175
11
votes
2
answers
1k
views
Concentration compactness. Can this concept be stated in a theorem?
I recently attended a talk on NLS which is rather not my main field of interest. Yet, I got interested in a concept called concentration compactness during the talk.
When I approached the speaker ...
- 501
10
votes
1
answer
3k
views
Trace of integral trace-class operator
I have seen many answers to the converse question (which seems to be difficult in general), but I would like to ask the following:
Let $T: L^2 \rightarrow L^2$ be a trace-class operator that is also ...
user69109
7
votes
2
answers
682
views
Hölder continuity for operators
Let $x,y$ be positive real numbers then
$$|\sqrt{x}-\sqrt{y}|=\dfrac{|x-y|}{\sqrt{x}+\sqrt{y}}=\sqrt{|x-y|}\cdot \dfrac{\sqrt{|x-y|}}{\sqrt{x}+\sqrt{y}}\leq 1\cdot |x-y|^{\frac{1}{2}}$$
we obtain $1/...
user127612
7
votes
2
answers
997
views
Uniform continuity of heat semigroup
I would like to illustrate my question with an example:
It is well-known that $\Delta$ is the generator of a strongly continuous semigroup $(T(t))$ on $L^2(\mathbb R^n),$ i.e. the heat-semigroup.
It ...
- 536
5
votes
2
answers
459
views
Backward heat equation and forward perturbed heat equation well posed?
I consider the following scenario. Let $I$ be a compact interval in space and $f$ a nice function in the space $C^{\infty}(I)$. In the following we consider a self-adjoint realization of our operators ...
- 536
5
votes
1
answer
571
views
Schrödinger operator with Coulomb potential
The free Laplacian $-\Delta$ has absolutely continuous spectrum $[0,\infty).$ The Coulomb Hamiltonian $H=-\Delta-\frac{1}{\vert x\vert}$ on $L^2(\mathbb R^3)$ has absolutely continuous spectrum $[0,\...
- 119
5
votes
1
answer
171
views
Invariant subspace in infinite dimensions
Let $A(t)$ be a family of skew self-adjoint operator defined on some Hilbert space $H$ with common domain $D(A).$
The dependence on $t$ is in the strongly continuous sense, i.e. for all $x \in D(A)$ ...
- 177
5
votes
1
answer
211
views
Pointwise convergence in functional calculus
Let $A_n$ be a family of (bounded) self-adjoint operator converging pointwise to some (unbounded) self-adjoint operator $A,$ i.e. for all $x$ in the domain of $A$
$$\left\lVert A_n x-Ax \right\rVert \...
- 65
5
votes
1
answer
1k
views
Trace-norm of integral operator
Let me start by saying that I do appreciate any insight on this. So also if you have a partial result, please share it as a comment or answer.
This is somewhat unrelated to what I normally do, so I ...
- 175
4
votes
1
answer
366
views
Dissipative operator on Banach spaces
An operator $A$ is called dissipative if for all $x \in D(A)$ and $\lambda >0$
$$ \left\lVert (A-\lambda)x \right\rVert \ge \lambda \left\lVert x \right\rVert.$$
On a Hilbert space this is ...
- 501
4
votes
1
answer
279
views
Schroedinger operator in 2 dimensions with singular potential
Consider the Schroedinger operator
$$H = -\Delta + \frac{c}{\vert x \vert^2}$$
in two dimensions with $c >0$
This operator has a self-adjoint realization, since it is a positive symmetric operator ...
4
votes
1
answer
201
views
Spectrum Cauchy-Euler operator
A Cauchy-Euler operator is an operator that leaves homogeneous polynomial of a certain degree invariant, named after the Cauchy-Euler differential equations
We consider the operator
$$(Lf)(x) = \...
- 536
4
votes
1
answer
213
views
Mapping properties of backward and forward heat equation
In a previous question on mathoverflow, I asked about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions.
The ...
- 536
4
votes
0
answers
135
views
Zygmund class, Schwartz class and Littlewood-Paley projection operators
I'm studying Littlewood-Paley theory in harmonic analysis, where I encountered the following problem related to the Zygmund class of functions:
Consider the Zygmund class of functions defined as ...
- 349
3
votes
1
answer
224
views
Extension of Sobolev function defined on unit cube
Im wondering about theorems concerning extending Sobolev functions defined on the $d$-dimensional unit cube to all of $\mathbb{R}^d$. More precisely, given $f:[0,1]^d \to \mathbb{R}$ with $f\in H^k([0,...
- 93
3
votes
1
answer
255
views
Closure of tensor product /tensor product semigroup
In this reference the following claim is made in Remark 2
Let $A,B$ be closable operators on Banach spaces $X,Y$, then $A \otimes 1$ and $1 \otimes B$ are closable operators on the Banach space $X \...
user127612
3
votes
1
answer
876
views
Is Quantum Mechanics (norm)-consistent?
I edited a few small comments to the question in order to make it perhaps more comprehensible.
Today I came across the following question in quantum mechanics.
In Quantum mechanics it is common to ...
- 39
3
votes
1
answer
274
views
Function square-integrable
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \left(\frac{xy}{(x^2+y^2+1)}\right)^2 \ dx$$
where $x_0$ is an ...
3
votes
0
answers
322
views
Heat equation damps backward heat equation?
In a previous question on mathoverflow, I was wondering about the following:
Let $\Delta$ be the Laplacian on some compact interval $I$ of the real line with let's say Dirichlet boundary conditions. ...
- 536
2
votes
3
answers
303
views
Uniqueness of solution depending on constant?
I am a physicist and I am aware that this forum is for professional mathematical questions, but please be not too hard on my notation.
I encountered the following integral equation for functions $f:[...
2
votes
1
answer
93
views
Lipschitz bound on semigroups
Let $T$ be a self-adjoint operator (possibly unbounded) and $S$ a bounded self-adjoint operator.
Then one can study the unitary groups $R_T(t):=e^{itT}$ and $R_S(t):=e^{itS}.$
Now if you think about ...
2
votes
1
answer
102
views
Evolution equation invariance of sets
Let $A: D(A) \subset X \rightarrow X$ be a generator of a $C_0-$semigroup and $Z$ be a bounded operator on $X$, then the evolution equation for $u \in C([0,T], \mathbb{R})$
$$\varphi'(t) = A \varphi(t)...
- 115
2
votes
1
answer
289
views
Laplacian dissipative?
is it true that the Laplacian $\Delta:=\frac{d^2}{dx^2}$ on $(0,1)$ with Neumann boundary conditions is dissipative on $C[0,1]?$
For this we have to show that there is for any $x \in D(\Delta)$a $x' \...
- 21
2
votes
0
answers
180
views
Approximating $L^p$ functions by eigenfunctions of Laplacian
I'm reading a paper https://www.sciencedirect.com/science/article/pii/S0022039608004932.
In this paper, the authors assume that $\mathcal{O}$ is a bounded domain of $\mathbb{R}^N$ with $C^m$ boundary ...
- 121
2
votes
0
answers
190
views
Absence of fixed points
Let $f$ be an arbitrary function in $L^2(0,\infty)$ and consider the function
$$(g_f)(y) = \frac{1}{y-x_0} \int_{0}^{\infty} f(x) \frac{xy}{(x^2+y^2+1)} \ dx$$
where $x_0$ is an arbitrary but fixed ...
2
votes
0
answers
78
views
Generalization of supersymmetry to dimension 3
in two dimensions there is a simple trick to study the spectrum of operators of the form
$$\textbf{A}:=\left( \begin{matrix}0 && A^* \\ A && 0 \end{matrix}\right)$$
The trick is to ...
- 167
2
votes
0
answers
86
views
I have an embedding $\iota$ between two Hilbert spaces and want to know if $\iota\iota^\ast$ is something simple like an orthogonal projection
I'm reading A Concise Course on Stochastic Partial Differential Equations. In Proposition 2.5.2 the authors define the notion of a cylindrical $Q$-Wiener process $W$. It turns out that $W$ is just a $...
user74319
1
vote
1
answer
737
views
$L^2$ function in Schwartz space?
Let $f:\mathbb R^n \rightarrow \mathbb R$ be a smooth function whose derivatives are all polynomially bounded and $f \in L^{\infty}.$
Such a function has the property that when multiplied with any ...
- 177
1
vote
1
answer
165
views
Integral function $z(x):=\int_{Y} f(x,y)d\mu(y)$ continuous?
Let $z(x):=\int_{Y} f(x,y)d\mu(y)$ for $x \in \mathbb R$ be an integral function where $\mu$ is a finite(!) Borel measure on $Y$ and $x \mapsto f(x,y)$ is continuous for every $y.$
Moreover, we know ...
- 536
1
vote
1
answer
130
views
Resolvent difference of absolute values!
Let $T$ be a bounded operator. Then, the operators $\left\lvert T \right\rvert:=\sqrt{T^*T}$ and $\left\lvert T^* \right\rvert:=\sqrt{TT^*}$ are well-defined.
Is there a way to write
$$(\left\lvert ...
- 115
1
vote
1
answer
518
views
Interpolation between Schatten classes
I was wondering if there is an analogue to the classical Riesz Thorin theorem for Schatten classes. I suppose the answer is yes, since Schatten classes are so similar to $\ell^p$ spaces for which the ...
- 305
1
vote
0
answers
159
views
Generalized functional for solution of PDEs
Asked this on Math Stack Exchange awhile ago but it got ignored then deleted.
To solve a differential equation of one variable, you need constraints equal to the number of derivatives.
For a partial ...
- 131
1
vote
0
answers
122
views
Series and solution of $-\Delta u + \lambda u = f(x)$
Consider a bounded smooth set $\Omega \subset \mathbb R^n$ (for example, we can take a ball). Can we write down the solution of
\begin{align*}
-\Delta u(x) + \lambda u(x) &= f(x), & x \in \...
- 131
1
vote
0
answers
93
views
Relative boundedness of the adjoint
Let $X$ be a separable Banach space and $T_1:D(T_1) \subset X \rightarrow X$ and $T_2:D(T_2) \subset X \rightarrow X$ two closed operators with $D(T_2)\subset D(T_1)$ and $D(T_2^*) \subset D(T_1^*).$
...
- 33
1
vote
0
answers
211
views
Propagation of singularities and the Schrodinger equation
I always thought that the propagation of singularities theorem by Hörmander says (on $\mathbb R^n$ for a classical symbol $p(x,\xi)=\xi^2+V(x)$) that for a Schrödinger equation
$$(i \partial_t-p(x,D))...
- 11
1
vote
0
answers
76
views
Which sets support which spectra?
I know (and this is of course rather elementary) that an isolated point in the spectrum of a self-adjoint operator $T$ always belongs to the point-spectrum.
I would like to ask: Are there similar ...
- 173
0
votes
1
answer
419
views
Stone–von Neumann theorem?
The Stone–von Neumann theorem says that given two unitary groups on a Hilbert space $H$ satisfying the canonical commutation relations (CCR)
$$
U(t)V(s) = e^{-i st} V(s) U(t) \qquad \forall s, t
$$
...
0
votes
1
answer
104
views
Operator identity for convergent series
Let $T_i$ and $S_i$ be a sequence of bounded operators such that
$$ \sum_{k,i,j=0}^{\infty} S_j^* T_i^* T_i S_k$$ converges unconditionally in operator norm on some Hilbert space. The limit is then ...
0
votes
1
answer
268
views
Linear operator has one-dimensional kernel
Let $S_{\lambda}$ be a family of linear bounded operator on $L^2(\mathbb{R}^n)$ depending on some parameter $\lambda$, I have recently encountered several problems that dealt with the question whether ...
- 329
0
votes
0
answers
81
views
Differential operator and equivalence
Here is the problem:
I have a certain PDE and there is the nonlinear terme $h$, I have as data:
$f \in H_0^2(0,L)$,,,$g \in {H^1}(0,L)$ with ${g_x}(0) = {g_x}(L) = 0$
Now on consider the fnction $$h(...
- 617