Questions tagged [fa.functional-analysis]

Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.

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6
votes
1answer
119 views

Are lattice operations in a Lipschitz space sequentially continuous in the weak* topology?

This is a follow-up on this (answered) question on math.SE, but involves a different topology. I think this time it is more appropriate for MO. I will repeat the background from the question cited ...
3
votes
1answer
191 views

Normal linear functionals on bicommutants of C*-algebras

I am going through the proof of the Sherman-Takeda theorem and Fillmore's book "A User's Guide on Operator Algebras" seems to have a nice approach, but something seems off to me: We need to ...
1
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0answers
55 views

The $w^{*}$-convergent sequences and the Mackey topology on $X^{*}$

Let $X$ be a Banach space. Recall that the Mackey topology $\mu(X^{*},X)$ on $X^{*}$ is the topology of uniform convergence on weakly compact subsets of $X$. Let $(x^{*}_{n})_{n}$ be a sequence in $X^{...
4
votes
2answers
714 views

Vector-Valued Stone-Weierstrass Theorem?

The standard statement of the Stone-Weierstrass theorem is: Let $X$ be compact Hausdorff topological space, and $\mathcal{A}$ a subalgebra of the continuous functions from $X$ to $\mathbb{R}$ which ...
2
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0answers
119 views

Does a spectral theorem exist for linear operator pencils?

I was wondering if a version of the spectral theorem (the projection valued measure case) holds for linear pencils of the form $$ A-\lambda B $$ where $A,B$ are self-adjoint on some Hilbert space $\...
0
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0answers
102 views

Creation and Annihilation operators in QFT - Part II

Following some suggestions on my previous posts, I'm trying to reformulate my question in a more specific way. This is a continuation of my original post. Since the mentioned post, I think I've ...
0
votes
1answer
91 views

Question/References on the Skorokhod M1 topology

Let $D(0,T)$ be the space of right continuous functions with left limits defined on $[0,T]$. Consider the Skorokhod M1 topology on $D(0,T)$, see e.g. S. Ledger, Skorokhod’s M1 topology for ...
2
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0answers
44 views

Hilbert transform on a Besov space

Consider the usual Hilbert transform of periodic functions $$H(f) = \frac{1}{2\pi}P.V.\int_{-\pi}^{\pi}\cot(\frac{x-y}{2})f(y)dy.$$ We know $H$ does not map $L^\infty$ continuously to $L^\infty$. Now ...
1
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0answers
33 views

Moduli of continuity and Wasserstein differentiability of functions between measures

Let $X=\mathbb{R}^n$; I am also interested in the general case $X$ is a metric space but for simplicity let's focus on Euclidean space. Let $\mathcal{P}(X)$ denote the space of Borel probability ...
1
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0answers
45 views

Multivarate “RKHS” Examples

I've been reading about RKHSs and Hilbert spaces of functions these days a bit these days and I haven't yet come across an example of a hilbert space $H$ whose elements are all functions $f:\mathbb{R}^...
3
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0answers
70 views

Doubt when calculating the S-transform of Hida differential operator

Assume we have a Hida test function $\varphi\in (\mathcal S)$, and $y\in \mathcal S'(\mathbb R)$. Define the Gateaux directional derivative of $\varphi$ (in the direction of $y$) by: $$D_y\varphi(x):=\...
1
vote
1answer
75 views

On a limit for the resolvent norm

Let $H$ be a complex, infinite dimensional, separable Hilbert space. Fix any two nonzero operators $A,B \in B(H)$ such that $B$ is not a scalar multiple of $A$. It is well known that: $$ \| R_A (z) \| ...
3
votes
0answers
58 views

Smoothness and decay correspondence for Laplace transform

For the Fourier transform, there are various theorems formalizing a correspondence between the smoothness of a function and the rate of decay of its Fourier transform. For example, if a function is $n$...
2
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0answers
102 views

Green's function for Robin boundary condition

Let $\Omega$ be a bounded subset in $\mathbb{R}^n$, $n \ge 3$, with smooth boundary $\partial \Omega$. Assume given $a^{ij} \in W^{2, p}(\Omega)$ with $a^{ij} = a^{ji}$, $f \in L^p(\Omega)$ and $g \in ...
3
votes
1answer
142 views

Sufficient condition for asymptotic-$\ell_{p}$ in terms of spreading models?

Let $(X,\|\cdot\|)$ be a Banach space with a Schauder basis and fix $p\in[1,\infty]$. Suppose that $X$ is asymptotic-$\ell_{p}$ with respect to this basis. It is known that the closed linear span of ...
1
vote
1answer
123 views

Function series of normal lower semi-continuous functions

For a real-valued $f$ on a topological space $X$, the upper limit of $f$ at $x\in X$ is defined as follows: $ f^{\ast }\left( x\right) =\inf \left\{ \sup \left\{ f\left( y\right) :y\in U\right\} :U\in ...
4
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0answers
238 views

Can this function be minimized?

Let $X$ be a locally convex TVS, and let $A$ and $B$ be convex and compact subsets of $X$ with $A \subset B$. Let $f: A \times B \to [0,\infty]$ have the following properties: (1) For all $b \in B$, $...
4
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0answers
92 views

What is the completion of $L^\infty$ in the dual of BV?

Every $f \in L^\infty([0,1])$ induces a continuous linear functional on BV via $g \mapsto \int f g \mathrm{d}x$. I believe $L^\infty([0,1])$ is also separable in BV$^\ast$, while BV$^\ast$ is not ...
0
votes
0answers
48 views

Integral inequality with Fractional Laplacian

Is the following inequality true $$ \int_{B_1(0)} f(x) (-\Delta)^\alpha u(x) dx - \frac{1}{|B_1(0)|}\int_{B_1(0)}(-\Delta)^\alpha u(x) dx \cdot \int_{B_1(0)}f(x) dx \ge 0 $$ for a strictly convex $f:\...
3
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0answers
134 views

When does an operator from $\ell_1$ to itself factor through $\ell_p$?

I would like to know whether a given operator from $\ell_1$ to itself, given by a matrix $A$, factors through $\ell_p$, for $p>1$.Does anyone know any references/results on this topic? I am ...
8
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0answers
139 views

History of the Lewis-Stegall theorem on factorization of representable operators

The following questions are about the history of a particular result in functional analysis, hence not "mathematical questions" per se; but I think they are relevant to the business of ...
3
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0answers
65 views

Constructing a norm inequality for a given functions [closed]

Let $F_1(x)=x\log^2(2+x)$ and $F_2(x)=x\log(2+x)$ be such that $$ F_2^{-1}\left(\int_0^{\infty} F_2(c/x \int_0^{t}g(s) ds)\right)\leq F_1^{-1}\left(\int_0^{\infty}F_1(g(x))dx\right). $$ I have been ...
6
votes
2answers
247 views

Non-sequential spaces in the wild

TLDR: What are examples of (function-)spaces that are not sequential? When does this matter? As a simple analyst, I am most happy if I can just work with sequences all the time. In most situations ...
4
votes
1answer
123 views

When a normal functional is restricted to a vn Neumann sub-algebra

I have already asked this question and no comment(s) received up to now. I am so curious to get feedback concerning the problem. Let $M$ be a vn Neumann subalgebra in $B(H)$. Let $f$ and $g$ be ...
2
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0answers
35 views

Fractional Laplacian and convolution $(-\Delta)^\alpha (u \ast \eta_\epsilon) = (-\Delta)^\alpha u \ast \eta_\epsilon$?

For $u \in L^\infty(\mathbb R)$ and $\eta_\epsilon$ mollifier, it is well-known that for the (distributional) derivative it holds that $(u \ast \eta_\epsilon)' = u'\ast \eta_\epsilon$. Is it also true ...
1
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0answers
53 views

Uniqueness for measure valued ode

Morning! Basically I'm working on a mean field scaling for some measure valued process (valued on $M_F(N)$). The limit turns up as a (deterministic) solution to a measure valued ODE. Let's say : $$ d\...
2
votes
1answer
97 views

Can we define geodesic in the space of compactly supported functions?

From Wikepedia, the definition of geodesic is stated as: A curve $\gamma: I\to M$ from an interval $I$ of the reals to the metric space $M$ is a geodesic if there is a constant $v\geq 0$ such that ...
4
votes
0answers
61 views

What are the complemented subspaces of $(\bigoplus\ell_q^n)_p$?

Bourgain/Casazza/Lindenstrauss/Tzafriri proved in their unconditional basis UTAP book (1985) that $\ell_1$ is the only nontrivial complemented subspace of $(\bigoplus\ell_2^n)_1$, and hence by duality ...
0
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0answers
41 views

A basis for $H^1([0,1]\times[0,1])$ with a desired property coming from the Poisson bracket

In the following denote: $\Omega:=[0,1]\times[0,1]$ $\mathcal{T}_h$ is the uniform triangulation of $\Omega$ as follows $n$ is the number of nodes on any side of the square so that $h=\sqrt{2}/n$. $...
0
votes
0answers
25 views

How to project a sequence on the univariate moment cone

Folowing closely Schmüdgen, K. (2017). The moment problem (Vol. 9). Berlin/New York: Springer., Chapter 10, Section 2, for a bounded interval $[a,b] \in\mathbb R$ and $m\in\mathbb N$, define the ...
1
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0answers
58 views

Toeplitz operators for other measures then Lebegue

In the standard setting there is a lot known about Toeplitz operators i.e that the compression of a multiplication operator restricted to the Hardy space. Are there any results when one has a ...
5
votes
0answers
67 views

Relationship between continuous vector fields and divergence measure fields in dimension $\ge 2$

Let $\Omega \subset \mathbb R^d$ with $d \geq 2$ (I am mostly interested in the case when $\Omega$ is the unit ball). A vector field in $L^p(\Omega,\mathbb R^d)$ is called a divergence measure field ...
3
votes
0answers
80 views

Rigorous stability analysis of infinite dimensional ODEs : How to bound the tails?

My question is about linear stability analysis of dynamical systems obtained by discretizing linear(ized) partial differential equations. Consider, $\dot{x}=Ax$, where $x$ is the infinite dimensional ...
2
votes
1answer
87 views

Non-zero, bounded, continuous, differentiable at the origin, compactly supported functions with everywhere non-negative Fourier transforms

Do there exist functions $F(x) \! : \, \mathbb R \to \mathbb R$ which are non-zero and bounded: $$ \mathrm {Range} (F) = [l, u] \, , \quad \mathrm {where} \quad l, u \in \mathbb R \land u > l \, ; \...
5
votes
0answers
134 views

Wightman reconstruction theorem-details of the proof

First of all forgive me if this question is not well suited for this forum: it is motivated by physics however after all my concerns are mathematical so I hope it would be appropriate to post it here. ...
1
vote
1answer
138 views

Optimization problem with definite integral inequality constraints

Question: How can we prove that there exists a real constant $c\ge 1$ such that the following inequality holds for all integers $d>1$ and all real numbers $r\in\left[1,\sqrt{d}\right]$? $$\int_{-1}^...
7
votes
1answer
694 views

Where does the Laplace transform come from?

The Gelfand transform on the commutative Banach *-algebra $L^1(\mathbb{R})$ is just the Fourier transform. Q. What can we say concerning the Laplace transform?
24
votes
3answers
952 views

Is there a 'certainty' principle?

Heisenberg's uncertainty principle is a restriction on which probability distributions can describe the position and momentum of a quantum particle. In mathematical terms it says that if $\psi\in L^2$ ...
3
votes
1answer
137 views

Extension of positive functionals II

This is a follow-up to Extension of positive functionals. Assume that $X=R^n$ with the canonical order (I indicate with $K$ the positive cone, $x \in K$ iff $x_i \ge 0$ for all $i$) and let $L:M \to R$...
2
votes
1answer
94 views

Question regarding the Wick tensor in white noise analysis

I have a question regarding the definition of Wick tensor in the framework of the white noise analysis. To put some context to the question we start with the following Gel'fand triple $$S(\mathbb R)\...
6
votes
3answers
216 views

Representation theorem for quadratic form on Hilbert space

I think my question is more suitable for Mathematics Stack Exchange than to MathOverflow but I've already posted two related questions there and I got even more confused, so maybe I can clarify things ...
0
votes
0answers
45 views

How to ensure that an operator has a certain simple eigenvalue?

For instance, consider the operator $\mathcal{L}$ defined in $L^2_{per}([0,L])$ with domain in $H^2_{per}([0,L])$, given by $$\mathcal{L} :=-A\partial_x^2-3\varphi^2+1-2(\varphi', \partial_x)_{L^2}\...
14
votes
0answers
366 views

strong topologies on $C_c^\infty$

UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
7
votes
1answer
151 views

The double dual of the unitization of a $C^*$-algebra

I am studying the proof that if $A$ is a $C^*$-algebra such that $A^{**}$ is a semidiscrete vN algebra, then $A$ has the completely positive approximation property (CPAP). I was able to handle the ...
1
vote
0answers
120 views

Property $(\mathcal{L}(\phi),\phi)\geq 0$ about a operator $\mathcal{L}$

Consider the operator $\mathcal{L} : H^2(\mathbb{T}_L) \subset L^2(\mathbb{T}_L) \longrightarrow L^2(\mathbb{T}_L)$ given by $$\mathcal{L} = -\omega \partial_x^2+3\varphi^2-1,$$ that is $$\mathcal{L}(...
5
votes
0answers
85 views

Extension of positive functionals

Let $X$ be a function space as $C(K)$ or $L^p$, with its usual norm and order, that is $f \le g$ if and only if $f(x) \le g(x)$ for a.e. $x$. If $M$ is a subspace of $X$ and $L:M \to \bf R$ is a ...
5
votes
0answers
215 views

Generalized convexity

Let $X$ be a vector space. The positive-homogeneous function $\|\cdot\|$ is said to be a quasinorm if $\|x+y\|\le K(\|x\|+\|y\|)$, for some $K\ge1$; it is a norm if $K=1$. Question: 1. (terminology) ...
2
votes
0answers
97 views

Estimate involving Besov norm

When reading some old notes of my advisor on interpolation spaces, I bumped into a problem I can't quite wrap my head around. Here are the details. For $p\in(0,\infty)$ a $p$-variation semi-norm of a ...
4
votes
1answer
137 views

Consider a net of weak order units in a Riesz space converging in order to a weak order unit. Is there a tail whose infimum is a weak order unit?

Let $X$ be an extremally disconnected (the closure of an open set is open) compact Hausdorff space, and consider the Riesz space $C^\infty(X)$ of continuous functions from $X$ to the extended real ...
8
votes
2answers
241 views

Are (completely) positive maps approximated by normal (completely) positive maps?

Let $\mathcal{H}$ denote a Hilbert space and $B(\mathcal{H})$ denote the algebra of all bounded operators on $\mathcal{H}$. By recognizing the (Banach) dual of $B(\mathcal{H})$ with the double dual of ...