# Questions tagged [fa.functional-analysis]

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

9,253
questions

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### Looking for examples of kernels with scalar Pick property but not the complete Pick property

I am studying Pick Interpolation and Hilbert Function Spaces by Agler and McCarthy.
A kernel $k$ on a set $X$ is said to have $M_{s,t}$ Pick property whenever $x_1,x_2, \ldots , x_n \in X$ and $W_1, ...

9
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1
answer

577
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### Prove J.L. Lions’s Lemma without using Fourier transform

When I read the book Linear and Nonlinear Functional Analysis with Applications, I came across J.L. Lions's Lemma (the book doesn't give a proof), which states
Let $\Omega \subset \mathbb R^n$ be a ...

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0
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65
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### On calculating the second quantization operator $\Gamma(A)$ of the Ornstein-Uhlenbeck operator $A$

Let $A$ be a self-adjoint operator on a Hilbert space , and let $d\Gamma(A)$ be the generator of the second quantization of $A$. Consider the following theorem from Segal's "Non-Linear Quantum ...

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### Absolute continuity of $t \to \lVert u(t) \rVert^2_{H}$ and Gelfand triple : are they equivalent?

Let $V$ be a separable Banach space and $H$ be a separable Hilbert space such that
\begin{equation}
V \subset H \subset V'
\end{equation}
and the inclusion maps are continuous with dense images. Here $...

-1
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1
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82
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### A question in functional analysis about selfadjoint operator [closed]

In Hilbert space $u$, Let $T_1$,$T_2$ is selfadjoint operator, if exit $c>0$ such that $cI\le T_1\le T_2$, prove $T_1$,$T_2$ have a bounded inverse operator and $c^{-1}I\ge T_1^{-1}\ge T_2^{-1}$.
I ...

1
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1
answer

170
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### Sufficient condition such that the set of zeros of an analytic function $f:\mathbb{R}^n \to \mathbb{R}$ contains only isolated points

Consider a real- analytic function $f: \mathbb{R}^n \to \mathbb{R}$. We know that zeros of $f$, roughly speaking, live in the low dimensional manifolds.
My question: Does a 'reasonable' sufficient ...

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79
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### The asymptotic behaviour of the Fourier transform of a certain class of radially symmetric functions

Fix $\theta\in (-\pi/2,\pi/2)$ and let $a>0$. Suppose that $f:\mathbb{C}\rightarrow \mathbb{C}$ is analytic in $S:=\{z\in \mathbb{C}: |\arg{z}|<\pi/2\}$ and
$$|f(z)|\sim |z|^{-a},\qquad |z|\to \...

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0
answers

48
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### Rescaling of cosine families

First of all, the best wishes for 2024. Recently, I got aware of cosine operator families (in the framework of evolution equations). It is well-known, that operator semigroups can be rescaled (see for ...

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1
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93
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### Convergence of ODE with uniform $L^\infty \cap L^1$ bound on nonlinearity

Consider the IVP
$$
\left\{
\begin{aligned}
\frac{d}{dt} \Phi_n(t,x) &= f_n(\Phi_n(t,x)) && \forall t \in \mathbf{R}_+ \\
\Phi_n(0,x) &= x && \forall x \in \mathbf{R}
\end{...

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2
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244
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### An integral inequality?

Let $v \in C^\infty(\mathbb R)$ such that $1 \ge v \ge 0$ and $\int_{\mathbb R} v \, dx = 1$.
I want to show that if
$$\int_{\mathbb R} v |v''|^2 \, dx < + \infty. \tag{$\star$}$$
then
$$ \int_{\...

1
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0
answers

78
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### For any $\beta>0$, there is a constant $c>0$ such that $\left\|(1-\Delta)^{\frac{\beta}{2}} f\right\|_{\infty} \leq c\|f\|_{C_b^\beta}$

For any $n \in \mathbb{Z}^{+}$, let $C_b^n\left(\mathbb{R}^d\right)$ be the class of real functions $f$ on $\mathbb{R}^d$ with continuous derivatives $\left\{\nabla^i f\right\}_{0 \leq i \leq n}$ such ...

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101
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### Find a function $f\geq 0$ such that $e^{-t[(x-\partial_x)\partial_x]^2} f$ is not non-negative for some $t\geq 0$

Consider the square of the Ornstein-Uhlenbeck operator $$A=[(x-\partial_x)\partial_x]^2=(x-\partial_x)\partial_x (x-\partial_x)\partial_x.$$ We know that $[(x-\partial_x)\partial_x]^2$ cannot be a ...

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### Characterization of the Picard's condition for integral equation

Picard's condition (Thm. 15.18, Kress et al. 1989) is essential to study the existence of a solution of a Fredholm integral equation of the first kind. Specifically, consider (the univariate case) the ...

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135
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### Isomorphic copies of $c_0$ in the projective tensor products

There exist Banach spaces $X$ such that the projective tensor product $X\mathbin{\hat{\otimes}}_\pi X$ contains an isomorphic copy of $c_0$ [BourgainPisier1983]. Moreover, $X$ is an $\mathcal{L}_\...

1
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1
answer

137
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### Unable to understand an application of Minkowski's inequality

Consider the following exerpt from the paper "Non-linear Quantum Processes" by Segal:
with the norm $\|F\|=\left(\int\|F(x)\|^p \, d x\right)^{1 / p}$, then the operator $T_1^{\prime}: F \...

7
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0
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152
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### Nontrivial examples of locally compact quantum groups

What are some families of locally compact quantum groups that are neither groups, duals of groups, compact, nor discrete?

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answers

116
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### Multidimensional weighted Paley-Wiener spaces are Hilbert spaces?

How to rigorously demonstrate that multidimensional weighted Paley-Wiener spaces are Hilbert spaces?
I am utilizing the exponential type definition established by Elias Stein in the book 'Fourier ...

7
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3
answers

832
views

### Using the Stone-Weierstrass theorem to solve an integral limit

The following question was posted on math stack exchange here but it got no answers
Let $c\in (1, +\infty)$ and $f \colon [0, c] \to \mathbb{R}$ be a continuous and monotonically increasing function ...

2
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0
answers

70
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### Converse of existence of minimizers

Let $(V,\|\cdot\|)$ be a real normed linear space. $V$ has the property that given any nonempty convex, closed subset $K$, there exists a unique $v_0\in K$ such that $\|v_0\| \leq \|v\|, \forall v\in ...

2
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933
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### On a deceptively tricky calculus problem

Motivation for this question: If the operators $B_i'$ satisfy an inequality, prove that $B_1'+\dots B_n'$ also satisfies the same inequality
Let $A$ be a non-constant operator acting on $C^...

3
votes

1
answer

139
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### Are the paths of the Brownian motion contained in a suitable RKHS?

Let $H_B$ be the reproducing kernel Hilbert space (RKHS) of the Brownian Motion $(B_t)$ on $[0,1]$. It is well known that with probability 1 the paths of $(B_t)$ are not contained in $H_B$.
But is ...

1
vote

1
answer

112
views

### Can functions with "big" discontinuities be in $H^1$?

How can I prove that the function:
$$u:\Omega\to\mathbb{R},\ u(x)=\begin{cases} 0, x\in\omega \\[3mm] v(x), x\in\Omega\setminus\omega\end{cases}$$ is not in $H^1(\Omega)$, knowing that $v\geq 1$ is ...

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149
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### AQFT from a Lagrangian

In physics, the fundamental description of physical theories frequently revolves around the concept of a Lagrangian. My expertise encompasses diverse algebraic formulations within the domain of ...

7
votes

0
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147
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### What is the current status of research on the von Neumann's inequality for $n \ge 3$?

Problem
Let $(T_1, \ldots, T_n)$ be a tuple of commuting contractions in Hilbert space $H$.
Does a constant $C_n \ge 1$ exist, for which it would be true, that:
$$\forall_{p \in \mathbb{C}[x_1, \ldots,...

3
votes

3
answers

826
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### Can these identities for the Euler-Mascheroni constant be proven?

I stumbled upon these 4 limit/integral identities involving Euler's constant aka gamma (~0.5772). They appear to be valid based on inspection but I have no idea how to prove them. In addition, I have ...

3
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72
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### Cylindrical Wiener processes or SPDE that can make use of Banach valued rough paths?

Rough paths theory has an often advertised perk that it mostly works for general Banach spaces. I am trying to think of some nice examples that actually use this feature, and am coming up stuck.
The ...

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0
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93
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### If we have $\mu_{xy}$, why can we only construct the spectral measure if $\| \mu_{xy} \| \le \| x \| \|y \|$?

Definitions
Representation
Let $X \subset \mathbb{C}^N$ and $\mathcal{A}$ be an algebra in $\mathcal{C}(X)$. Also, we denote $L(H)$ as the set of all linear operators on HIlbert space $H$.
We call $\...

5
votes

1
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193
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### Syndetic sets and Banach limits: reference request

First of all, let us give a few definitions. Suppose that $A$ is a subset of natural numbers. We say that $A$ is syndetic if there is a constant $M$ such that every set of $M$ consecutive natural ...

4
votes

0
answers

106
views

### Flatness of $C_0(S)$-module $L_\infty(S,\mu)$

Let $S$ be a locally compact Hausdorff space. By $C_0(S)$ we denote the space of continuous functions vanishing at infinity. Let $\mu$ be a finite Borel regular measure om $S$, then consider $L_\infty(...

3
votes

2
answers

426
views

### A problem about how dominated convergence is used in the analysis of variation

I'm reading Existence of solutions to a higher dimensional mean-field equation on manifolds and get stuck on Lemma6. When $\lambda>\Lambda_1$, with $\Lambda_1=(2 m-1) ! \operatorname{vol}\left(S^{2 ...

2
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0
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57
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### Is the Leray projection continuous with respect to the Frechet topology of smooth periodic vector fields in $3$ dimensions?

Let $\mathbb{T}^3:=(\mathbb{R}/\mathbb{Z})^3$ be the $3$-torus and $C^\infty(\mathbb{T}^3,\mathbb{R}^3)$ be the Frechet space of smooth periodic vector fields on $\mathbb{T}^3$.
By Helmholtz ...

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0
answers

124
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### Reference request: an introduction to nuclear spaces

I am looking for a short introduction to nuclear spaces and nuclear operators. I am interested in these spaces as they often arise in mathematically rigorous quantum field theories. I have read the ...

1
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0
answers

70
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### $f \in L^2(X\times Y,\mu \times K)$ for Kernel $K$, is the map $X \ni x \mapsto (f(x,\cdot),x) \in \bigsqcup_{x \in X}L^2(Y,\Sigma_Y,K_x)$ measurable?

Let $(X,\Sigma_X)$ and $(Y,\Sigma_Y)$ be two measurable spaces, let $\mu$ be a measure on $(X,\Sigma_X)$, and let $(K_x)_{x \in X}$ be a transition kernel from $(X,\Sigma_X)$ to $(Y,\Sigma_Y)$, that ...

4
votes

1
answer

122
views

### Embeddings of the maximal domain for the Laplacian

Let $\Omega \subset \mathbb{R}^n$ be a bounded smooth domain and $n \geq 2$. Consider the subspace of $L^2$-functions whose distributional Laplacian is also an $L^2$-function:
$$D = \left\{ f \in L^2(\...

0
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0
answers

50
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### Why is von Neumann inequality important for equivalence of $\forall_j \ T_j^n\rightarrow 0$ in A-topology and abs continuity of $(T_1,\ldots, T_N)$?

The whole theorem goes as follows:
Let $(T_1, \ldots, T_N)$ be a tuple of commuting operators in Hilbert space $H$ satisfying:
$$\exists_{M > 0} \ : \ \forall_{p \in \mathbb{C}[z_1, \ldots, z_N]} \ ...

8
votes

2
answers

603
views

### Question on whether, "An entire function, nowhere zero, has an entire logarithm," holds for matrix-valued entire functions as well

It is known that an entire function that is nowhere zero must be the exponential of another entire function.
Does this hold for matrix-valued functions as well? That is, given a matrix-valued entire ...

2
votes

1
answer

153
views

### 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=-\operatorname{div}g(x)\...

2
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0
answers

115
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### 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.

5
votes

0
answers

206
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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 ...

1
vote

0
answers

68
views

### 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
vote

1
answer

260
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{...

0
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0
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86
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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 ...

4
votes

1
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257
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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 ...

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0
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45
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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. ...

0
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0
answers

168
views

### 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 ...

0
votes

0
answers

118
views

### 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 ...

2
votes

1
answer

229
views

### 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}...

2
votes

0
answers

137
views

### 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 ...

3
votes

1
answer

188
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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
$$ \left\{ f\omega(x)\in L^p(\mathbb{R}^n):\|f\|_{L^p_\omega}=\int_{\mathbb{R}^n}|f|^p\omega^p(x)\,dx< \infty,\right\} $$
...

3
votes

2
answers

171
views

### 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 ...