Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,125
questions
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Estimate for the operator $A A_D^{-1}$
Let $O\subset\mathbb{R}^d$
be a bounded domain of the class $C^{1,1}$
(or $C^2$
for simplicity). Let the operator $A_D$
be formally given by the differential expression $A=-\mathrm{div}g(x)\nabla$
...
2
votes
0
answers
77
views
Trace class operators
There is a notion of trace class operator in a Hilbert space.
Is there a notion of trace class operator in arbitrary Banach space? locally convex space?
A reference will be helpful.
- 20.5k
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answers
62
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Weak derivative of $|u|^s$ $(s>1)$ [migrated]
I often come across instances in texts where people calculate the weak derivative of $|u|^s$ for $s>1$ as $s|u|^{s-1} \operatorname{sign}(u) \partial_x u$ for some $u\in W^{1,s}(\Omega)$.
However, ...
- 7
4
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153
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Is the Taylor map continuous?
(Skip to the bolded theorem below for my question, if you'd like)
Some context on asymptotic expansions and the Taylor map
In the setting of irregular singularities of meromorphic connections on the ...
- 350
1
vote
0
answers
59
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Weak$^\ast$ closure of a countably complete sublattice of the unit ball of $L^\infty(\Omega, \mu)$
This is a reframing of my previous question from a Banach lattice perspective: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity? The previous question ...
- 1,146
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votes
1
answer
216
views
A certainty principle?
Let $\mathcal{A}$ be a unital $\mathrm{C}^*$-algebra with $\varphi\in\mathcal{S}(\mathcal{A})$ a state. Where
$$\sigma_\varphi(a):=\sqrt{\varphi((a-\varphi(a)1_{\mathcal{A}})^2)}\qquad (a\in \mathcal{...
- 799
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0
answers
73
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Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?
(Cross posted from Math StackExchange: Does weak $L^2$ approximation implies $L^2$ approximation under a condition similar to convexity?)
Assume $(\Omega, \mu)$ is a probability space. Consider a ...
- 1,146
4
votes
1
answer
221
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Examples of Borel probability measures on the Schwartz function space?
Let $\mathcal{S}(\mathbb{R}^d)$ be the Frechet space of Schwartz functions on $\mathbb{R}^n$. Its dual space $\mathcal{S}'(\mathbb{R}^d)$ is the space of tempered distributions.
Minlos Theorem as ...
- 2,331
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votes
0
answers
35
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Kirszbraun-like extension of periodic functions
Let $\Lambda \subset \Lambda' \subset \mathbb{R}^n$ be lattices. Let $f : \Lambda' \rightarrow \mathcal{H}$ be a $a$-Lipschitz function, where $\mathcal{H}$ is a finite-dimensional Hilbert Space. ...
- 1
-1
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137
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How to prove that the uniform limit of $C^k$ functions is $C^{k-1,1}$?
Already asked in SE but no response, I think it also reasonably belongs here.
https://math.stackexchange.com/questions/4829428/uniform-convergence-of-ck-functions
Basically what the title says, plus ...
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0
answers
89
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Self-adjoint operator with pure point spectrum
Suppose that A is a self-adjoint (possible unbounded) operator from a separable Hilbert space H to itself. I would like to know if the following statement is true:
A has pure point spectrum (i.e., the ...
1
vote
0
answers
102
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Are these conditions regarding products of consecutive terms in a sequence of positive numbers equivalent?
Assume $w_n$ is a bounded (weight) sequence of positive numbers. We want to consider products of consecutive terms in this sequence. For $i,j\in \mathbb{N}$, define $M_i^j = w_i w_{i+1}\cdots w_{i+j-1}...
- 151
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132
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How to prove $\operatorname{Id}-K$ is a proper map when $K$ is a $C^1$ compact operator?
Assume $X$ is a Banach space, $\Omega \subseteq X$ is an open set, $K\in {C}^{1}( \overline{\Omega}, X)$ is a nonlinear compact map, Is $\operatorname{Id}-K$ a proper map? I think maybe it has ...
- 21
3
votes
1
answer
160
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Weighted Lebesgue space with exponential weights: smoothing effect and properties
I am researching whether there are weighted Lebesgue spaces of the type
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...
- 563
3
votes
2
answers
158
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How to diagonalize this tridiagonal difference operator with unbounded coefficients?
Problem: I have a self-adjoint operator in $\ell^2(\mathbb{Z})$ which acts as
$$T g(x)=q^{-2 x -3/2} g(x+1)+(1+q) q^{-2 x-1} g(x)+q^{-2 x +1/2} g(x-1),$$
and I am looking to diagonalize it. The ...
- 1,660
3
votes
1
answer
112
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Spectra of products variously permutated
Let $x,y$ be elements of a Banach algebra $A$ and $\lambda\in\mathbb C\setminus\{0\}$. If $\lambda-xy $ is invertible, then $\frac1{\lambda}\big[1+y(\lambda-xy)^{-1}x \big]$ is clearly an inverse of $...
- 55.5k
1
vote
0
answers
141
views
Linear third order water wave pde admitting particular gamma factor solution. How do you understand evolution on vertical strip in complex plane?
I would like to understand a little bit about how to interpret and construct $1$-parameter gamma factors that are dynamical - that is they are particular solutions to linear PDE's. Some possible ...
- 141
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votes
0
answers
68
views
Closed form solutions to polynomial operator equations
To the best of my knowledge the problem at hand is a generalisation of monic matrix polynomials. Can a closed form solution to the following equation be found,
$$u_3A_3X^3B_3 + u_2A_2X^2B_2 + ...
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votes
0
answers
41
views
Reference needed for powers of semi-group generators
Let $\mathcal{L}$ be the infinitesimal generator of a Markov semi-group. I am looking for references that study powers of $\mathcal{L}$; i.e. $\mathcal{L}^n$, for $n\in\mathbb{N}$.
For example, if the ...
- 116
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votes
1
answer
209
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Can a non-free Whitehead group embed as a discrete subgroup of a normed space?
Every countable discrete subgroup of a normed space is isomorphic to the direct sum of the group of integers. I wonder whether it is possible to push this beyond such direct-sum (free abelian) groups ...
- 11.1k
3
votes
1
answer
139
views
Continuity of conditional expectation
Let $X$ be a compact metric space, $\mu$ a Borel probability measure on $X$ and $f: X \to \mathbb{R}$ a continuous function. Consider an increasing sequence of $\sigma$-algebras $A_n$ so that for all $...
- 31
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votes
0
answers
28
views
Measurability of the weak completion of an orthogonal representation
Let $G$ be a locally compact group and let $\pi$ be a strongly continuous orthogonal representation of $G$ in a real Hilbert space $H$. Denote by $E$ the real Hausdorff locally convex space obtained ...
- 335
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votes
0
answers
37
views
Constants in the entropy number of the Sobolev space
For a Sobolev space with $W^s(\Omega)$, where $\Omega\subset R^d$ is a compact space with smooth boundary, we know that the entropy number satisfies $e(\delta, W^s(\Omega, 1),\|\cdot\|_{L_\infty})\leq ...
- 39
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vote
0
answers
129
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What are the current open problems in dilation theory?
I just started doing my PhD in mathematics. My topic is unitary dilations of operators. I've been reading a lot of papers on that subject so far (especially about the dilation of $n \ge 3$ commuting ...
- 29
1
vote
1
answer
73
views
Is it true that $\xi \in \partial G (v)$ implies $\frac{\xi}{F'(\phi (v))} \in \partial \phi (v)$?
I am reading the introduction of Chapter 10 in the book Gradient Flows by Ambrosio and his coauthors.
As we have seen in Section 1.4, in the classical theory of subdifferential calculus for proper, ...
- 1,111
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votes
1
answer
152
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Does an uncountable convex combination of elements of a set lie in the convex hull of the set in finite dimension?
Suppose that $\mathcal{F}$ is a finite-dimensional vector space and that $C\subseteq\mathcal{F}$ is a convex subset of $\mathcal{F}$.
Is it true that an uncountable convex-combination of elements of $...
- 11
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votes
0
answers
35
views
Generator of contraction semigroup
Assume $A$ is a symmetric operator in a Hilbert space, which generates a contraction semigroup (a priori it is not known, whether this semigroup is self-adjoint). Is A then self-adjoint?
- 41
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vote
0
answers
65
views
$T$ trace, then $Tg(u)=g(T(u))$ for all $u$ on $W^{1,p}$
The trace operator $T$ is defined for bounded domain $U$ with $C^1$ boundaries as the linear, continuous operator
$T: W^{1,p}(U) \rightarrow L^p(\partial U)$
such that
$$
Tu=u\;\text{ on }\partial U
$$...
- 11
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votes
0
answers
57
views
How to calculate the minimum of this functional
Let $ n:[0,2\pi]\to S^2 $ and $ m:[0,2\pi]\to S^2 $ be two smooth regular curves on $ S^2 $. Assume that $ |n(0)-n(2\pi)|=2 $ and $ |m(0)-m(2\pi)|=2 $, i.e. the start and end points of $ n,m $ are ...
- 1,091
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votes
0
answers
51
views
Product of d-dimensional Legendre polynomials
Let $P_n:\mathbb{R}\rightarrow\mathbb{R}$ be the $d$-dimensional Legendre polynomials, that is they are orthonormal polynomials w.r.t the probability measure $\mu_d$ on $[-1,1]$ given by $\mu_d= \...
- 145
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votes
1
answer
126
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Operator Semigroup: Resolvent estimates and stabilization, a detail in the paper of Nicoulas Burq and Patrick Gerard
In Appendix A of the paper Stabilization of wave equations on the torus with rough dampings https://msp.org/paa/2020/2-3/p04.xhtml or https://arxiv.org/abs/1801.00983 by Nicoulas Burq and Patrick ...
2
votes
1
answer
78
views
Show $v(x,t) \in L^2([0,T];H^2(\mathbb{R}))$ when $v(x,t)$ is a transformation of a $L^2([0,T];H^2(\mathbb{R}))$ function
Context: I am reading a paper on Long-Time Asymptotics of the thin film equations, in which the authors consider the strong solutions of the thin film equation in 1-D and transform them using a time-...
- 23
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votes
5
answers
2k
views
Extracting a common convergent indexing from an uncountable family of sequences
Let $\mathcal{A}$ be some uncountable index set and $X$ be some separable reflexive Banach space.
For each $\alpha \in \mathcal{A}$, let
\begin{equation}
\{ x_n^{\alpha} \}_{n=1}^\infty
\end{equation}
...
- 2,331
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votes
0
answers
54
views
Chapter 2, Section 5 of Chavel's book “Eigenvalue In Riemann Geometry" is about the zero-point distribution of the derivatives of eigenfunctions
In Chapter 2, Section 5 of Chavel's book, regarding the Neumann eigenvalues of the Laplacian in space forms, how did Chavel determine that $T'_{l,j}$ has ($j-1$) zeros? I have consulted books on the ...
1
vote
1
answer
98
views
Inequalities involving entropy: quantum discord and mutual information
My question is inspired by the following paper of Olivier and Żurek but for this question to be self-contained I will recall all the necessary definitions: for a quantum state $\rho$ we define the ...
- 9,170
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votes
1
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160
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A variant of Hardy's inequality for "convolutions"?
Consider Hardy's inequality on $L^{2}(\mathbb{R}^3)$. This inequality states that:
$$\int_{\mathbb{R}^3} dx \, \frac{|\psi(x)|^2}{|x|^2} \le K \int_{\mathbb{R}^3} dx \, |\nabla \psi(x)|^2.$$
I want to ...
4
votes
1
answer
153
views
Compact-open Topology for Partial Maps?
I asked the same question on MathStackExchange a month ago and received no answer. I feel that this would be more suitable for MathOverflow.
Compact open topology is one of the most common ways of ...
- 1,007
2
votes
2
answers
178
views
Making sense of the limit $\lim\limits_{x \to y} T(x,y) $ for a tempered distribution $T$ on $\mathbb{R}^{2n}$
I already posted a similar question on MO and looked into the references therein.
However, I cannot find a satisfactory answer for my question..So I ask here again in a more refined form.
Let $T \in \...
- 2,331
1
vote
1
answer
112
views
Optimal constant comparing $f(1/2)$ and $\|f\|_2$ when $f$ is $t$-Hölder?
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
- 410
3
votes
0
answers
94
views
Metrizing pointwise convergence of *sequences* of functionals in a dual space
This question was asked by myself on the math stackexchange a few days ago. I thought I'd repeat it here:
Let $X$ be a normed, real vector space of uncountable dimension. Let $X^*$ denote the set of ...
2
votes
2
answers
130
views
Relating function value to $L^2$ norm in Holder space
Suppose that $f \colon [0, 1] \to \mathbb{R}$ is $k$ times continuously differentiable and Holder in the sense that for some
$t = k + \beta$, where $\beta \in (0, 1]$ and $k$ is a nonnegative integer ...
- 410
4
votes
1
answer
229
views
Jordan normal form for compact operators
This question should be standard, but I didn't find it in the books.
For a compact operator $T$ on a Hilbert space $H$, we know that every spectral value $\ne 0$ is an eigenvalue, that each ...
- 1,900
2
votes
0
answers
80
views
On dense subspaces of $L^p$-spaces of finitely additive measures
Let $\mu$ be a finite, finitely additive measure defined on the Borel $\sigma$-algebra of a real separable Hilbert space $\mathcal{H}$ with dual $\mathcal{H}^{*}$. Write $L^{p}(\mathcal{H},\mu)$ for ...
- 461
2
votes
0
answers
84
views
Self adjoint operators from energy functionals
It is known that the equation
$$
\Delta f = 0
$$
on some bounded domain $\Omega$ on $\mathbb{R}^n$ subjected to certain boundary conditions can be derived through the minimization of the Dirichlet ...
- 101
2
votes
1
answer
148
views
On spectral calculus and commutation of operators
Let $\mathcal{H}$ be a Hilbert space, $B\in\mathcal{B}(\mathcal{H})$ be bounded and self-adjoint and $A:\mathcal{D}(A)\to\mathcal{H}$ closed (but not necessarily self-adjoint or bounded). The ...
- 833
2
votes
0
answers
90
views
Orthogonal representation of free products of two groups
Suppose $A$ and $B$ are two countable, discrete, amenable groups. One definition of amenability tells us that there is a sequence of finitely supported, positive definite functions that converges to 1 ...
- 141
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votes
0
answers
85
views
Amenability of $\textrm{w}_0(A)$ for a $C^*$-algebra $A$
Let $A$ be a $C^*$-algebra with only finite dimensional irreducible representations. As in a previous question, let $\textrm{w}_0(A)$ denote the subspace of $\ell^{\infty}(A)$ consisting of all weakly ...
- 2,118
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votes
0
answers
68
views
Complemented C* Algebras
let $A$ and $B$ be unital separable commutative $C^*$ algebras, with $A\subset B$. Is it true that $A$ is complemented in $B$?
- 151
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votes
0
answers
68
views
How to define the Sobolev quotient space $H^s(Γ)/{\mathbb R}$
Let $\Gamma$ be the boundary of a Lipschitz domain $\Omega\subset \mathbb R^3$. Denote by $H^s(\Gamma)$ the usual scalar Sobolev space for $s\in\mathbb R$. I want to know the definition of the ...
- 63
2
votes
1
answer
207
views
Continuous path of unitary matrices with prescribed first column?
Consider a continuous curve $u \colon [0,1] \to \mathbb{C}^n$ where $u(t)$ is always a unit vector, $u(t)^* u(t) = 1$.
Question 1: Does there exist a continuous curve $U \colon [0,1] \to \mathbb{C}^{n ...
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