All Questions
Tagged with traces fa.functional-analysis
26 questions
35
votes
2
answers
9k
views
tr(ab) = tr(ba)?
It is well known that given two Hilbert-Schmidt operators $a$ and $b$ on a Hilbert space $H$, their product is trace class and $tr(ab)=tr(ba)$. A similar result holds for $a$ bounded and $b$ trace ...
- 43.2k
31
votes
3
answers
5k
views
When is an integral transform trace class?
Given a measure space $(X, \mu)$ and a measurable integral kernel $k : X \times X \rightarrow \mathbb{C}$, the operator
$$ K f(\xi) =\int_{X} f(x) k(x,\xi) d \mu(x),$$
the operator $K$ is Hilbert ...
- 11.2k
13
votes
1
answer
724
views
Trace-class operator satisfies $\sum |\lambda_n|<\infty$?
Here's an "exercise" which I thought should be easy, but which I find myself unable to do.
Let $V$ be a Banach space.
Recall that an operator $f:V\to V$ is trace-class if it is in the image of the ...
- 43.2k
12
votes
0
answers
476
views
Are Sobolev trace spaces equal from both sides of the boundary?
Let $\Omega\subset\mathbb R^n$ be a bounded open set and $\Omega'$ the complement of its closure.
Assume $\partial\Omega=\partial\Omega'$.
Are the quotient spaces $W^{1,p}(\Omega)/W^{1,p}_0(\Omega)$ ...
- 8,086
11
votes
3
answers
1k
views
Boundedness of the derivative of the trace of an H^1 function
As a research preface, this question is linked to a problem of increasing magnetism in Ginzburg-Landau equations that I have distilled for the purpose of getting to the bottom of this technical matter....
- 882
10
votes
0
answers
225
views
Can the trace be computed in any Schauder basis?
I'm cross-posting this question from Math.SE, as it didn't get much attention there.
Let $H$ be a separable Hilbert space and $T \in L(H)$ a trace-class operator. It is well known that the trace of $T$...
- 233
7
votes
2
answers
647
views
If I multiply the coefficients of a trace-class operator with bounded complex numbers is it still trace class?
Suppose that $T \in TC(l^2( \mathbb{Z}))$ is trace class.
Consider its kernel $ T(i,j) = \langle e_i, T e_j \rangle $ where $ \{e_i\}_{i \in \mathbb{Z}}$ is an ONB for $l^2( \mathbb{Z})$. Now, ...
- 1,054
6
votes
1
answer
1k
views
Every self-adjoint trace class operator on $L^2$ has integral kernel
I have asked this question on MSE but did not receive an answer. I thought I could try it here.
Let $T$ be a self-adjoint trace-class operator on $L^2(\mathbb{R})$. Is is true that it can be ...
- 437
5
votes
1
answer
709
views
What is the trace of the integral operator $(\mathcal{L}f)(x)=\int_0^\infty (x \wedge y)f(y) \, d \pi(y)$?
Let $\pi$ denote a probability measure on $[0,+\infty)$ and let us assume that
$$m:=\int_0^\infty x \, \mathrm{d} \, \pi(x)<+\infty.$$
Let us consider the integral operator $\mathcal{L}$ on $L_2(\...
- 51
4
votes
1
answer
303
views
Which operators on the trace-class operators extend to operators on Hilbert-Schmidt operators?
Let $\mathcal{H}$ be a separable Hilbert space and let $TC( \mathcal{H})$, $HS(\mathcal{H})$ be the space of trace-class operators and Hilbert-Schmidt operators on $\mathcal{H}$. Recall that these ...
- 1,054
3
votes
1
answer
752
views
A distributional normal derivative for functions in $H^1(\Omega)$
Let $\Omega$ be a smooth bounded domain with $\partial\Omega = \Gamma$. I have read this.
For all $u \in H^1(\Omega)$ such that $-\Delta u = g \in L^2(\Omega)$ in distribution, we can define the ...
- 266
3
votes
0
answers
82
views
Compatibility between the source and the boundary condition for an Helmholtz-type equation
Let $\Omega$ an open, convex, bounded domain in $\mathbb{R}^3$, and let us fix also $z\in\mathbb{C}\setminus\mathbb{R}$. Given $\phi\in H^{3/2}(\partial\Omega)$, I would like to show the existence of ...
- 943
3
votes
0
answers
128
views
Dixmier traces, Wodzicki residue and residues of zeta functions
Let $M$ be an $n$ dimensional closed manifold and consider an elliptic, pseudodifferential operator $P$ of order $-n$. Here are some facts which I had learned so far:
1. There exists a density defined ...
- 9,330
3
votes
0
answers
74
views
Trace space of $\{ t^su \in L^2(0,\infty;X) \mid t^su_t \in L^2(0,\infty;Y)\}$ for $s \in (-\frac 12, \frac 12)$
Let $s \in (-\frac 12,\frac 12)$ and let $X=D(\Lambda)$ be a Hilbert space with $\Lambda$ the infinitesimal generator of a bounded semigroup of class $C^0$ in $Y$ (which is another Hilbert space), and ...
- 251
2
votes
1
answer
766
views
reference request: trace/lifting operator for $L^{\infty}$ data in bounded $\Omega\subset R^d$
I know the answer to my question must be somewhere in the literature. Probably in [Adams-Fournier], [Dautray-Lions] or something alike, but I don't have access to a library right now so I'll ask ...
- 5,380
2
votes
2
answers
241
views
A Characterization of the traces of functions in $W^{1,2}$
I have a question about the traces of functions in $W^{1,2}$.
Let $D$ be a connected open subset of $\mathbb{R}^d$.We denote $W^{1,2}(D)$ by
\begin{align*}
W^{1,2}(D)=\{f \in L^{2}(D,dx) \mid \...
- 721
2
votes
1
answer
379
views
Extensions in parabolic Hölder spaces
Let $\alpha\in ]0,1[,k\in\mathbb{N}.$Let $\Omega$ be a open and bounded subset or $\mathbb{R}^n$ of class $C^{k+\alpha}$.
As one could find in G.M. Troianello "Elliptic Differential Equations and ...
- 301
2
votes
0
answers
267
views
Necessary and sufficient conditions on kernels of trace-class operators
Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$?
We know that ...
- 143
2
votes
0
answers
320
views
Sobolev trace theorem
Set $Q:=\Omega\times(0,T)\subset\mathbb{R}^{n+1}$,
where $\Omega$ is knows as a bounded domain
with smooth boundary $\partial D$.
We choose any subdomain $D\subset Q$
with smooth boundary $\partial ...
- 375
1
vote
1
answer
155
views
Spectrum invariant under (generalised) transpose as operator on trace class operators
For matrices $A$ it is well known that the spectrum is invariant under transpose $\sigma(A^T) = \sigma(A)$. Furthermore, the spectrum of the adjoint matrix $\sigma(A^*) = \overline{ \sigma(A)}$ the ...
- 1,054
1
vote
0
answers
111
views
References on the partial trace
For the Hilbert space $H^N:=L((\mathbb R^{3})^N,\mathbb C)$, consider the projection operator $D: H^N\to H^N$ as follows :
$$D(\Phi):=\left(\int_{(\mathbb R^{3})^N}\overline{\Psi(x_1,\ldots, x_N)}\Phi(...
- 1,389
1
vote
0
answers
97
views
Show that for all $\delta$, we have $\|u\|^2_{\Gamma}\le c_\delta(\|u\|^2_{\omega(\delta)}+\|u\|_{\omega(\delta)}\|\nabla u\|_{\omega(\delta)})$
I am reading the article On existence of weak solutions of the Navier-Stokes equations in regions with moving boundaries from Fujita and Kato and at some point they use an argument I have some trouble ...
- 452
1
vote
0
answers
60
views
Is there a vector-valued trace such that $\text{tr}((L\otimes_π\text{id}_H)T)=LT$ for all $L∈\mathfrak L(H,\mathfrak L(H))$ and $T∈H\hat\otimes_πH$?
Let
$H$ be a separable $\mathbb R$-Hilbert space
$L\in\mathfrak L(H,\mathfrak L(H,\mathbb R))$
$T\in\mathfrak L(H)$ be nonnegative, self-adjoint and nuclear (trace-class)
Note that$^1$ $$\...
- 167
0
votes
0
answers
66
views
Taking trace of a tensor product of matrix-valued smooth functions on the thin diagonal
Let $V$ be a finite dimensional real / complex vector space and consider the space $L(V,V)$ of linear operators on $V$.
Fix $n \in \mathbb{N}$ and let $\mathcal{M}$ be the real / complex vector space ...
- 3,477
0
votes
0
answers
104
views
Necessary and sufficient conditions for $\mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}) \ge 0$ for all $t$
Let $A$ and $B$ be positive-definite matrices of the same size. For any $t \ge 0$, define
$$
u(t) := \mbox{trace}(A^{-1/2}e^{-tB} (AB+BA) e^{-tB}A^{-1/2}).
$$
Question. What are necessary and ...
- 6,853
0
votes
0
answers
216
views
Bound on integral of elliptic theta function
I need to prove that the following bound is true. I thought this might follow from the inversion property of the theta function, as the infinite sum in the integrand is precisely $\theta_3(0,\mathrm{e}...
