Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,354
questions
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3
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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
3
votes
0
answers
91
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The numerical range of a composition of two operators
For a problem I'm working on, I need the following implication. $A,B$ are two closed densely defined operators on a Hilbert space $H$. I'll be a bit vague about the setting, add assumptions at will as ...
- 8,775
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votes
1
answer
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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
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votes
0
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59
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Differential operator
One define the operator $T$ as :$$T: = (I - {{{\partial ^2}} \over {\partial {x^2}}}):H_0^1(0,L) \cap {H^2}(0,L) \to {L^2}(0,L)
$$ let $f \in H_0^2(0,L) \cap {H^4}(0,L)$. What can we say about ${T^{ - ...
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A question regarding mollifiers on Sobolev spaces on closed manifolds
Let $M$ be a closed Riemannian manifold and denote by $H^s(M), \, s\in \mathbb{R} $ the standard Sobolev spaces on $M$ defined using powers of $1+\triangle$. Let $J_n: \mathcal{D}'(M)\rightarrow \...
- 473
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answers
102
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Extension of the Gagliardo Inequality
The Gagliardo Inequality generalizes Fubini's Theorem: let $f_j$ be $d-1$ non-negative measurable functions over ${\mathbb R}^{d-1}$. Let us form the function
$$f(x)=\prod_{j=1}^df_j(\widehat{x_j}),$$
...
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3
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Integration of Weyl operators multiplied by quasifree state over a symplectic space
I am reading the book "An invitation to the Algebra of Canonical Commutation Relations" by Denes Petz. It is freely available for download here. In Chapter 9, he defines the Lebesgue measure on a ...
0
votes
1
answer
100
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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 ...
4
votes
1
answer
246
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First isomorphism theorem for maps between Hilbert modules?
Let
$X$ be a compact Hausdorff topological space,
$H,K$ be Hilbert modules over the $C^*$-algebra $C(X)$,
$T:H\rightarrow K$ be a bounded $C(X)$-linear map such that ran($T$) is a Hilbert module ...
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1
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0
answers
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$C^*$-algebras with non-trivial center
The center $Z(A)$ of an algebra $A$ is the set of all those elements that commute with all other elements. If $A$ is the algebra of compact operators on a Hilbert space $H$, then $A^{**}$ is the ...
- 4,301
4
votes
1
answer
364
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Continuous linear combination of continuously varying vectors?
Let ${\bf{e}}_1, {\bf{e}}_2, {\bf{e}}_3:[0,1]\rightarrow \mathbb{R}^3$ be continuous, $\mathbf{0}\neq \mathbf{v}\in \mathbb{R}^3$.
Suppose that the following condition (C) holds:
$$
\exists d>0: ...
- 81
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votes
0
answers
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Is flatness of Wigner Ville Distribution of error function in Fourier Approximation possible? Is it required?
For a real valued function $f(t)$ I want to check the information left, after taking a Fourier partial sum/integral. Let $\hat{f}$ be its Fourier transform and let $$e_{\omega}(t) = f(t) - \int\...
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votes
2
answers
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Completeness of an exponential family
The question is this: Does there exist an integrable function $f\colon\mathbb R\to\mathbb R$ such that $f$ differs from $0$ on a set of nonzero Lebesgue measure and
\begin{equation}
\int_{\mathbb R}...
- 118k
1
vote
1
answer
120
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On a weaker condition of summability for Fourier series
The Wiener algebra $W:=W(\mathbb{T}^n)$ on the torus is defined as the algebra of all continuous fonctions $f$ on $\mathbb{T}^n$ such that $(\widehat f(k))_{k\in \mathbb{Z}^n} \in \ell^1(\mathbb{Z}^n)$...
- 975
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votes
1
answer
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An inequality about embedding of cube into metric spaces
A k-cube in $X$ is a function $\psi:\{-1,1\}^k\to (X,d)$.
An edge of a cube is a pair of points $\{\psi(\epsilon_1),\psi(\epsilon_2)\}$ in $X$ such that $\epsilon_1$ and $\epsilon_2 $ differ in ...
- 1,245
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votes
1
answer
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An extremal problem
Let $f:[0,\pi]\to [0,\pi]$ be a diffeomorphism. How to prove that
$$P[f]:=\int_0^\pi \sin^2(x) \left(3+2 \frac{\sin^2(f(x))}{\sin^2 x}+(f'(x))^2\right)^2dx $$ attains its minimum for $f(x)\equiv x$?
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Singular value decomposition in two spaces (reference in Russian paper?)
Let $H$ be a Hilbert space and $X$ be a Banach space such that $H \cap X$ is dense in both.
Now, let $T$ be an operator such that $T: H \rightarrow H$ and $T:X \rightarrow X$ exists in the sense that ...
12
votes
1
answer
885
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On an Inequality of Lars Hörmander
Let $P(z)$ be a non-null complex polynomial in $\nu$ variables $z=(z_1,\dots,z_n)$ of degree $\mu$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq \mu} c_{\alpha} z^{\alpha},
\end{equation}
where as usual ...
- 628
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votes
0
answers
227
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Does the reciprocal of a polynomial define a tempered distribution when it is locally integrable?
Consider a complex polynomial in $n$ variables $z=(z_1,\dots,z_n)$:
\begin{equation}
P(z)=\sum_{|\alpha| \leq N} c_{\alpha} z^{\alpha},
\end{equation}
where as usual for every $\alpha=(\alpha_1,\dots,\...
- 628
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votes
0
answers
92
views
Multiplicativity of $\zeta$-function regularized determinant
Let $A$ be a selfadjoint elliptic differential operator on a compact manifold. In mathematical physics and differential topology one often defines its determinant using the $\zeta$-function ...
- 21.1k
1
vote
1
answer
132
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Chain of interior of closed set
It is well known that a topological space with asending chain condition for open subsets is called Noetherian. Is there any characterizations or a nice property for a Hausdorff topological space such ...
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9
votes
2
answers
775
views
$\zeta$-function regularized determinants
In (mathematical) physics in order to compute path integrals one often makes an infinite dimensional change of variables and uses infinite Jacobian as a purely formal expression. This step is done in ...
- 21.1k
4
votes
1
answer
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views
Abstract Definition of a Reproducing Kernel Hilbert Space
This is a very basic question about the definition of a reproducing kernel Hilbert space (RKHS).
It seems the standard definition of a RKHS is as a Hilbert space $H$ of functions on some set $X$ ...
- 1,327
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votes
0
answers
113
views
Converse on the rectifiability of products of rectifiable sets
Let $1\leq k\leq m$ and $1\leq l\leq n$ fixed integers, $\mathscr{H}^k$ the $k$ dimensional Hausdorff measure and $E\subset \mathbb{R}^m$. We say that :
(1) $E$ is $k$ rectifiable if there exists $C\...
- 1,588
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0
answers
335
views
Spectral mapping theorem
Rudin's book contains in chapter 10 a spectral mapping theorem for (self-adjoint) unbounded operators that respects the point-spectrum, in the sense that he shows $f(\sigma_p(T))=\sigma_p(f(T))$ for ...
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votes
1
answer
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Self adjoint operators in Kasparov-Modules
In Blackadars book in 17.4.2 it says that for each element $x \in KK(A,B)$ there is a Kasparov module $(E,\pi ,T)$ such that $T=T^*$. Now, the argument for that is that if $(E,\pi, T)$ is any Kasparov-...
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votes
2
answers
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views
Why are $\Gamma_0$ functions called this
It is very common to indicate with $\Gamma_0(A)$ the set of lower semicontinuous convex functions from $A$ to $(-\infty,+\infty]$ with nonempty domain. An example of usage of this notation can be ...
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votes
1
answer
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views
What is the span of the Haar system in the $L^\infty$ norm?
The Haar system in $[0,1]$ has a closed span in the $L^\infty[0,1]$ norm which contains all the continuous functions in $[0,1]$. In fact, it contains all the piecewise continuous functions with ...
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votes
0
answers
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views
Does this Sobolev-space like construction have a name?
Take $\Omega \subset \mathbb{R}^n$ arbitrary then define as $X$ the closure of $C^1(\Omega) \cap W^{1,1}(\Omega)$ w.r.t. the norm $f \mapsto \left\lVert f \right\rVert_{\infty} + \left\lVert \nabla f \...
- 295
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1
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411
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Extensions of completely positive maps
It is known that for a completely bounded map $\psi:A\to B(H)$ there exist completely positive maps $\phi_1,\phi_2:A\to B(H)$ such that
$$\Vert \phi_i\Vert_{cb}=\Vert \psi\Vert_{cb},$$
and the map $\...
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votes
2
answers
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views
Distribution that vanishes against approximated delta is zero
Suppose we have a Schwartz distribution $\phi$ on $\mathbb{R}^d$ such that $$ \forall x, \ \lim_{\lambda \to 0}| \langle\phi, \psi^{\lambda}_x \rangle| =0$$
where $\psi^{\lambda}_{x}=\lambda^{-d}{\...
- 377
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votes
2
answers
141
views
Equality of spectra of products of operators
Let $A$ be a linear operator between two Hilbert spaces. Let $A^*$ be its adjoint.
Question. Under what conditions the non-zero spectra of $A^*A$ and $AA^*$ coincide counting multiplicities?
In my ...
- 21.1k
3
votes
0
answers
125
views
Equivariant $K$-homology with $G$-compact support
Let $G$ be a discrete countable group and let $A$ be $\sigma$-unital $G$-$C^*$-Algebra. For a proper locally compact Hausdorff $G$-space $X$ the equivariant $K$-homology with $G$ compact support and ...
- 31
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votes
1
answer
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views
Spectra on different spaces
This is a method request: I am looking for techniques that allow me to investigate problems like this:
Let $T_1: \ell^1 \rightarrow \ell^1$ be a bounded operator with $\Re(\sigma(T_1)) \subset (-\...
- 295
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0
answers
141
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Continuity of the spectrum under weaker notions of convergence
Let $T:X\to X$ be a linear operator on a Banach space $X$.
We know that the spectrum of $T$ is an upper semicontinuous
function of $T$ for the uniform convergence: that is, if $T_n:X\to X$ is a ...
- 757
1
vote
1
answer
98
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Showing that $\phi$ is a Jordan morphism
I have asked the following question on M.SE here, but I have not yet received a response.
I do apologize of this is not the correct site to post it on - if so, please do let me know and I will remove ...
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0
answers
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Interesting stipulation about completely monotone functions
This question relates to a question I asked here. I thought of a well thought out generalization which appears to follow in the situations I've encountered it. I tried to generalize the answer ...
user78249
0
votes
0
answers
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in search of convergent daughter sequences
Let $\{f_n\}\subset L^1(\Omega,\mu)$, where $\mu$ is the Lebesgue measure, and $\Vert f_n\Vert_1\leq M$ and $\Vert Df_n\Vert_{1/2}\leq C$ uniformly in $n$.
Question. Is there a subsequence $\{f_{...
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votes
0
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249
views
Does AX+XA=0 have any non-trivial solutions?
Let $X$ be a continuous linear self-adjoint operator on some Hilbert space $H$ and for arbitrary compact operators $A$ we have: $XA+AX=0.$ Does this imply that $X=0$ or can there be non-trivial ...
- 295
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votes
1
answer
239
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Density in the Space of absolutely convergent Fourier series
It is possible to approximate a function $f$ on $[0,2\pi]$ by a continuous function whose derivative is zero almost everywhere (as can be seen here : https://math.stackexchange.com/questions/67334/...
- 125
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votes
0
answers
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textbook of measure theory abstracted as functional analysis [closed]
Background
I have studied intro functional analysis, probability theory based on measures, and some elementary connection between them e.g. that weak conversion of random variables correspond to weak*...
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votes
1
answer
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Resolvents of Schrodinger operators
In the free case one can compute the resolvents of the Laplacian $-\Delta$ in many cases explicitly, in the sense that they are given by an integral operator. Often, one uses the Hille-Yosida theorem ...
- 295
1
vote
1
answer
150
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Regularity of integral kernel
Let $\Omega \subset \mathbb{R}^n$ be some open set.
If, for all $\psi \in L^2(\Omega)$ and some fixed integral kernel $k \in L^2(\Omega\times \Omega)$ and $\ell>0$, it is true that both
$\int_{\...
- 295
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vote
0
answers
90
views
Riesz transform on almost periodic functions?
It is well established that the Riesz transform is well-defined for $f\in L^p(\mathbb{R}^d)$ via$$
\mathcal{R}_jf(x) = c_d\lim_{\epsilon\to 0}\int_{|x-y|>\epsilon}\frac{(x^j-y^j)f(y)}{|x-y|^{d+1}}\,...
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1
answer
232
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Are there fundamental solutions of the laplacian that decay rapidly?
The question
I consider the Laplacian $\Delta = \partial_1^2 + \partial_2^2 + \partial_3^2$ in $\mathbb{R}^3$. By the "standard" fundamental solution of the Laplacian, I mean the function
$$ \...
- 53
3
votes
0
answers
211
views
Defining a trace-class operator with a Bochner integral
I had asked this question previously on Math.StacheExchange but did not get an answer there in several months. This isn't strictly speaking research level mathematics but I hope it is sufficiently ...
- 131
1
vote
1
answer
462
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 ...
- 295
2
votes
0
answers
321
views
Trace class operators convergent series
On wikipedia it is mentioned that if we are on some (separable) Hilbert space $H$ and there is an ONB $(e_n)$ such that any compact operator $K$ can be written as
$$ K = \sum_{n,m =0}^{\infty} K_{n,m}...
- 295
6
votes
3
answers
1k
views
Orthonormal basis in $W^{1,2}([0,1])$
Consider the Hilbertspace $W^{1,2}([0,1])$ (i.e. Sobolev space) with the standard inner product which is defined by: $(f,g) = (f,g)_{L^{2}([0,1])} + (f',g')_{L^{2}([0,1])}$. Here $[0,1]$ is not ...
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votes
4
answers
855
views
Can a $W^{1,2}$ map from the disk to the circle restrict to a degree one map on the boundary?
The restriction of a continuous map $D^2\to S^1$ to $\partial D^2\to S^1$ must have degree zero. Is that statement true or false if the map is only $W^{1,2}(D^2;S^1)$ and continuous on $\partial D^2$?
...
