Questions tagged [unbounded-operators]
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11 questions from the last 365 days
-1
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0
answers
33
views
Do the domains of the two square roots of a positive (unbounded) operator coincide?
Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...
- 419
1
vote
0
answers
32
views
Reference request - Fourier multiplier of vector valued function
I would like to understand the concept of multiplier for vector valued functions and find appropriate references for the multiplier theorems out there.
For instance say that we would like to express $\...
- 111
3
votes
0
answers
95
views
Commutator of $A\otimes I$ and $I \otimes B$ vanishes?
Consider two Hilbert spaces $H_1$ and $H_2$, and $A$, $B$ unbounded operators on $H_1$, $H_2$ respectively. $(A \otimes I)$ is classically defined as the closure of the operator defined on the set of ...
- 31
5
votes
2
answers
148
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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
35
views
Operator-form correspondence without lower semiboundedness
When dealing with a normal unbounded operator $A$, it is often useful to frame questions about the operator in terms of questions about the associated form $\omega,$ which has domain $D(|A|^{1/2})$ ...
- 109
3
votes
0
answers
69
views
Perturbation of one-parameter groups of unitary operators
Let $H$ be a Hilbert space and let $h$ be a fixed, densely defined, possibly unbounded, self-adjoint operator on $H$. Letting $B(H)$ denote the space of all bounded operators on $H$, it is well ...
- 2,263
2
votes
0
answers
111
views
Everywhere-defined unbounded operators between Banach spaces
In this post, it is said that there are no constructive examples of everywhere-defined unbounded operators between Banach spaces; every example furnished must use the axiom of choice. This seems like ...
- 151
2
votes
0
answers
261
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When is an unbounded averaging operator on $\mathbb{R}\to \mathbb{R}$ closed?
Let $\{a_n\}_{n=1}^\infty$, $a_n\in \mathbb{R}$. Consider the following linear operator $A$ on functions $f:\mathbb{R}\to \mathbb{R}$:
$$(Af)(x) = \sum_{n=1}^\infty a_n f(x+n)+ \sum_{n=1}^\infty a_n f(...
- 20.2k
8
votes
1
answer
480
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On commutator of bounded operators
Let $\mathbb H$ be a Hilbert space and let $\mathcal B(\mathbb H)$ be the bounded operators on
$\mathbb H$. Let $J,K\in \mathcal B(\mathbb H)$ such that
$
J=J^*, K=-K^*.
$
Then the commutator $[J,K]$ ...
- 16.2k
2
votes
2
answers
155
views
"Completeness" for weak convergence of unbounded closed operators on a separable Hilbert space $H$
Let $H$ be a separable Hilbert space with the inner product $\langle, \rangle$ and $\{ T_n \}$ be a sequence of unbounded closed linear operators with a common dense domain $D \subset H$ such that $...
- 3,477
3
votes
2
answers
280
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Question about the Bessel operator
For $\nu>-1$ denote by $\{\lambda_{k,\nu}\}_{k\in\mathbb{N}}$ the succesive positive zeros of the Bessel function of the first kind $J_{\nu}$. The Bessel operator is given by
\begin{equation*}
L_\...
- 421