# Questions tagged [fa.functional-analysis]

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

8,411
questions

1
vote

0
answers

47
views

### Integration with respect to Haar measures normalised over a subspace

Coming from physics I have come across the following integral over a haar measure (for $U$ unitary as an example) for something I am trying to determine for my work
$\int_{\mathcal{U}(d)} \frac{\...

1
vote

0
answers

29
views

### A certain property of positive-semidefinite infinite matrices

In this answer I concluded with this:
For which arrays $\big(\sigma_{ij}\big)_{(i,j)\in\mathbb N^2}$ [of real numbers] whose every upper-left corner is positive-semidefinite does line $(1)$ above ...

1
vote

0
answers

38
views

### "N-waves" and Hamilton-Jacobi equations

Let us consider the Burgers equation
$$u_t + (u^2)_x = 0$$
In
Liu, Tai-Ping; Pierre, Michel, Source-solutions and asymptotic behavior in conservation laws, J. Differ. Equations 51, 419-441 (1984). ...

1
vote

0
answers

26
views

### Minimal entropy conditions for conservation laws: an overview

Combining Theorem 2.3 and Corollary 2.5 of this paper gives that, for a strictly convex conservation law
$$u_t + f(u)_x = 0,$$ satisfying the entropy condition
$$\eta(u)_t + q(u)_x \le 0$$ in the ...

0
votes

0
answers

35
views

### Spectral perturbation theory of discrete spectra in presence of continuous spectrum

This is a 2 part question:
1). I am looking for a (hopefully accessible to beginning grad student who knows matrix perturbation theory) reference for doing concrete calculations of perturbed discrete ...

9
votes

1
answer

244
views

### Maximal ideals of the ring $\mathbb C \{T\}$

Consider the Banach $\mathbb C$-algebra
$$
\mathbb C \{T\} = \left\lbrace \sum_{i \geq 0} a_i T^i : \sum_{i \geq 0} |a_i| < \infty \right\rbrace
$$
With the norm given by $\| \sum a_i T^i\| = \sum |...

4
votes

1
answer

107
views

### Is the Borel lemma projection a smooth principal bundle?

Consider the Fréchet spaces $C^\infty(\mathbb{R},\mathbb{R})$ and $\mathbb{R}^\infty$, and the continuous linear map
$$
J\colon C^\infty(\mathbb{R},\mathbb{R}) \to \mathbb{R}^\infty
$$
returning the ...

4
votes

0
answers

38
views

### Criteria for tightness of Gaussian measures on Banach spaces

In Bogachev's book "Gaussian Measures" (Example 3.8.13) sufficient conditions for the (uniform) tightness of a sequence of centered Borel Gaussian probability measures on a separable Hilbert ...

4
votes

2
answers

136
views

### Is this an $L^p-L^{\infty}$ operator?

Let $1\leq p <\infty$ and let $P^{\prime}$ denote its conjugate exponent. Consider the following operator on Schwartz functions:
$$Tf(x)=\int_{0}^{\infty}t^{\frac{n}{2 p^{\prime}}-1}e^{-t}
\int_{|x-...

3
votes

0
answers

115
views

### Equality from the Grothendieck inequality

I asked the following question on math.stackexchange.com but have not received any response. So I would like to try my luck here.
This question is related to the Grothendieck inequality. Let field $\...

1
vote

0
answers

33
views

### Under what conditions a continues linear map maps a closed subspace to a closed subspace

Are there natural conditions that ensure that a continuous linear map $\phi:V\to W$ between TVS maps any closed subspace $L\subset V$ to a closed subspace in $W$.
It is obviously satisfied if $W$ is ...

0
votes

0
answers

74
views

### Integral over $\Bbb C$ [closed]

Is this correct? Let $a\in\Bbb C^*$ and $f\in L^1(\Bbb C^2)$
$$\int_{\Bbb C^2}|f(a(z,w))|dz dw={1\over |a|^2}\int_{\Bbb C^2}|f((z',w'))|dz' dw'$$
where $dz$ is the usual measure Lebesgue on $\Bbb C$
...

2
votes

0
answers

30
views

### How to get Bakry Emery Criterion $ \Phi'(t)=\frac{d}{dt}\int \Gamma_1(P_t f)d\pi=-\int\Gamma_2(P_tf)d\pi? $

I am reading Bakry Emery Criterion https://terrytao.wordpress.com/2013/02/05/some-notes-on-bakry-emery-theory/.
Let a function $H\in C^2(R)$ and define the infinitesimal generator:
$$
Lf=\Delta f-\...

3
votes

0
answers

37
views

### The embedding of a Banach lattice in an ultrapower

Given a Banach space $X$ and a non-trivial ultrafilter $\mathcal{U}$ on a set $I$, the ultrapower $X_\mathcal{U}$ is defined as the quotient of $\ell_\infty(I,X)$ by the closed subspace $N_\mathcal{U}(...

2
votes

0
answers

81
views

### Existence of analytic function in disk algebra [closed]

Does there exist an analytic function $f\in A(\mathbb{D})$, where $A(\mathbb{D})$ is the disk algebra, such that $f(0)=0$ and the real part of $f(z)$ is strictly positive?

3
votes

1
answer

221
views

### Extremely disconnected or extremally disconnected?

In the context of Banach space theory, what is the correct terminology: extremally disconnected or extremely disconnected. Looking through the internet I have met using both extremely and extremally ...

1
vote

0
answers

60
views

### When can we characterize Sobolev space $W^{2k,p}(\Omega)$ only via the Laplacian-like terms

One of the characterizations of the fractional Sobolev space $W^{s,p}(\mathbb{R}^n)$ uses the Fourier transform $\mathcal{F}$:
$f \in W^{s,p}(\mathbb{R}^n)$ iff $f$ is a tempered distribution such ...

1
vote

0
answers

24
views

### Grassmann algebra with an infinite number of generators and distributions

I was reading Fermionic Functional Integrals and the Renormalization Group by Feldman, Trubowitz and Knörrer and I got really confused about some concepts. On pages 19 and 20, the authors start ...

1
vote

1
answer

34
views

### Is $\sup\{\|A\widehat{k}_{\lambda}\|: \lambda\in\Omega\}=\sup\{\|A^*\widehat{k}_{\lambda}\|: \lambda\in\Omega\}$? where $A$ is an operator on RKHS

A functional Hilbert space $\mathscr H=\mathscr H(\Omega)$ is a Hilbert space of complex valued functions on a (nonempty) set $\Omega$, which has the property that point evaluations are continuous i.e....

2
votes

0
answers

29
views

### Green function of a 2D exterior domain

Consider solutions of the laplace equation
\begin{equation}
\begin{split}
-\Delta u=f, \ \ u|_{\partial D}=0,
\end{split}
\end{equation}
where the domain $D\subset \mathbb{R}^2$. If $D$ is bounded ...

1
vote

0
answers

100
views

+50

### An estimate for the first eigenfunction of the fractional Laplacian and a kind of "maximum principle"

Let $\Omega$ be a bounded Lipschitz domain. Let $u$ be the first eigenfunction of the fractional Laplacian
$$
(-\Delta)^s u = \lambda u \ \text{ in } \Omega, \quad u = 0 \ \text{ in } \mathbb R^n \...

2
votes

0
answers

118
views

+100

### Rate of uniform approximation by piecewise constant functions

Definitions and Notation:
Fix a positive constant $M>0$ with positive integers $m,n$ and the standard orthonormal basis $e_1,\dots,e_n$ of $\mathbb{R}^n$.
For every positive integer $N$, define the ...

0
votes

0
answers

56
views

### Regularity of map from $L^p$ to $\mathcal{W}_p$

Let $(X,d)$ be a polish metric space, fix a probability measure $\mathbb{P}$ on $(X,d)$ belonging to the Wasserstein $\mathcal{W}_p(X,d)$ for some fixed $p\in [1,\infty)$. Denote the Borel $\sigma$-...

0
votes

0
answers

68
views

### Type III von Neumann algebra

Let $\mathcal M$ be a type $III$ von Neumann algebra. Is it true that for all $n\geq 1,$ $\mathcal M$ contains a copy of $M_n$ as a von Neumann subalgebra? By Theorem 9.24 of https://mathweb.ucsd.edu/~...

0
votes

0
answers

60
views

### Matrix algebra as sub algebra of von Neumann algebra

Let $\mathcal M$ be a infinite dimensional von Neumann algebra which is not abelian. Suppose it is known that $\mathcal M$ does not contain type $I_n$ factors von Neumann subalgefbras for all $n\geq N....

1
vote

0
answers

144
views

+100

### Proving a sign rule for $f_{2n}$

If $t_{1},...,t_{n}$ are real numbers, consider the set of indexed linear operators $T(t_{1}),...,T(t_{n})$ on a Hilbert space $\mathcal{H}$ and define its ordering by:
$$\pi[T(t_{1})\cdots T(t_{n})] :...

1
vote

0
answers

39
views

### Sherman-Davis type inequalities for non-negative operator in a Hilbert space with trivial kernel

Recently I read Rupert L. Frank's paper "Eigenvalue Bounds for the Fractional
Laplacian: A Review". For a domain $\Omega\subset\mathbf R^n$, there are two different definitions of ...

0
votes

0
answers

234
views

### How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic?

Let $\langle;\rangle$ be the usual scalar product on $L^2(\Bbb R^2)$.
How to prove that the function $\lambda \mapsto \langle T_{\lambda}f; g\rangle$ is holomorphic on $\Bbb C^+_*=\{z\in\Bbb C:\text{...

0
votes

0
answers

15
views

### Is $\phi(t)=\|P(w+td)-w\|_X/t$ nonincreasing if $X$ is "only" a uniformly smooth and uniformly convex reflexive Banach space?

For a Hilbert space $X$ it is known that the function $\phi(t)=\frac{1}{t}\|P(w+td)-w\|_X$ with $t>0$ is nonincreasing. Here, $P:X\to C$ denotes the projection operator and $w \in C, d \in X$ are ...

2
votes

0
answers

44
views

### Explicit estimates on summability kernels

A "summability kernel" is a sequence of functions $k_n:[0,1)\to \mathbb C$ such that
$$ \int_0^1 k_n(t) \mathrm d t =1,$$
$$ \int_0^1 |k_n(t)| \mathrm d t =O(1),$$ with an implied constant ...

6
votes

1
answer

147
views

### Square-root lattices: where do they appear?

As an experimental physicist working on crystallography I'm often dealing with the reconstruction of an object from intensity data that emerge from an imaging device. In mathematics the problem is ...

1
vote

0
answers

53
views

### A statement on completeness of complex exponentials

I'm currently reading a paper by Olevskii on almost integer translates: https://www.sciencedirect.com/science/article/pii/S0764444297878731
In this paper the author considers for a given sequence $\{ \...

0
votes

0
answers

26
views

### On invariant subspaces of a nonautonomous heat semigroup

Consider the one dimensional heat equation with a spacetime dependent smooth conductivity coefficient $\sigma(t,x)>0$ on $[0,T)\times (0,1)$ for some $T>0$ subject to source terms $F$, that is ...

0
votes

0
answers

32
views

### The inversion of the Laplacian transform Pazy's Book "semigroups of linear operators and applications to Partial differential equations"

This question has been posted on Math Stack Exchange but no reply, and so I have to put it here. My question is:
In Pazy's Book page 26, the author gives a proof of Lemma 7.1, the lemma 7.1 says that: ...

0
votes

0
answers

36
views

### Elliptic partial differential equations with Robin boundary condition and domain of fractional power of Robin Laplacian operator

This question has been posted on Mathematics Stack Exchange but got no response, and so I put it here.
When I read the paper "On the attractor for a semilinear wave equation with critical ...

1
vote

0
answers

58
views

### Function monotony between [0,T] and $L^2$

Let $\Omega\subseteq\mathbb{R}^N$ be a bounded and smooth domain. If $z:[0,T]\to L^2(\Omega)$ is a function in $H^1([0,T],L^2(\Omega))$ with the property that $z'(t)(x):=z'(t,x)>0$ a.e. on $\Omega$ ...

1
vote

0
answers

71
views

### On Riesz decomposition of Volterra operator

Let $T:L^2((0,1)) \to L^2((0,1))$ be the Volterra operator defined by
$$ Tf(x) = \int_0^x f(t)\,dt.$$
Given any $\lambda\neq 0$ it is well known that there exists some positive integer $n=n(\lambda)$ ...

3
votes

1
answer

142
views

### Boundedness of an extension operator

Let $d \ge 2$ be a positive integer. For $x=(x_1,\dotsc,x_{d-1},x_d)$, we write $x'=(x_1,\dotsc,x_{d-1})$. Let $\mathbb{H}^d=\{x=(x',x_d) \mid x_d>0\}$ denote the $d$-dimensional upper half-space.
...

0
votes

0
answers

79
views

### For a vector field $f$ and a measure $\mu$, does conservativity $\mu$-almost everywhere have a sense?

$\DeclareMathOperator\Jac{Jac}$
Let $\Omega$ an open star shaped subset of $\mathbb{R}^d$ and $f : \Omega \rightarrow \mathbb{R}^d$ a differentiable vector field. For $x \in \mathbb{R}^d$, we denote ...

6
votes

1
answer

494
views

### Spectrum of the complex harmonic oscilllator

Let
$$
H_\lambda=-\frac{d^2}{dx^2}+\lambda^2 x^2,\quad\lambda>0.
$$ It is known that the spectrum of $H_\lambda$ is the set $\{(2n-1)\lambda,n\in \Bbb N^*\}$. Now put
$$
(U_\mu \phi)(x)= e^{\mu\...

5
votes

1
answer

95
views

### Properties of non-integer powers of the Hodge Laplacian

Consider a complete smooth Riemannian manifold $(M,g)$.
I think that it is not difficult to prove that the $k$ Hodge Laplacian is essentially selfadjoint in the relevant $L^2$ space of $k$ forms, ...

0
votes

0
answers

17
views

### Estimation of a scalar product under the action of an isomorphism

Introduce the following assumptions and definitions:
$\Omega\subset \mathbb{R}^3$ a bounded Lipshitz domain,
$(L^2(\Omega))^3$ the space of vector square integrable functions,
$M$ a $3\times 3$ real ...

2
votes

1
answer

251
views

### Bounded operator on $L^2(\Bbb R^2)$

Let $\lambda\in \{z\in\Bbb C\mid \text{Re}(z)>0\quad \text{and}\quad \text{Im}(z)>0 \}$. Consider the operator $T_\lambda: L^2(\Bbb C)\to L^2(\Bbb C)$:
$$
f\mapsto T_\lambda(f)(z)=\int_{\Bbb C}f(...

4
votes

0
answers

72
views

### Weighted logarithmic Sobolev inequality

$\DeclareMathOperator\Ent{Ent}$The usual logarithmic Sobolev inequality says that
$$
\Ent_\mu(f^2)\leq C\int |\nabla f|^2 d\mu
$$
where the entropy
$$
\Ent_\mu(f^2)=\int f^2 \log\left( \frac{f^2}{\int ...

4
votes

0
answers

47
views

### Proof of density of smooth functions in $H(\operatorname{curl})$ with $L^2$ tangential trace

I am trying to understand the proof of the following statement that is presented in “Finite Element Methods for Maxwell's Equations” by Peter Monk and Yangwen Zhang. The original source of the proof ...

2
votes

1
answer

84
views

### Example of a compact operator that is not uniformly continuous

I want to find a Banach space $E$ and a compact operator $K:[0,1]\times E \rightarrow E$ (that is, $K$ maps every bounded sequence onto a sequence that converges up to a subsequence) satisfying the ...

0
votes

0
answers

75
views

### Concentration compactness lemma and the best Sobolev constant

It is well known that the best Sobolev constant can be achieved on $\mathbf{R}^n$. More precisely, we have the following theorem (A):
Let $\frac{1}{q}=\frac{1}{2}-\frac{1}{n}$, $$S=\inf\limits_{{u\in ...

8
votes

1
answer

193
views

### Why operator systems?

A $\mathrm{C}^*$-algebra $\mathcal{A}\subset B(\mathsf{H})$ is a norm-closed, self-adjoint subalgebra of bounded operators on a Hilbert space. If we then take a unital self-adjoint (possibly closed) ...

0
votes

0
answers

36
views

### Boundary conditions for first-order nonlinear system of PDEs

Consider the following system of PDEs for the dependent variables $x=x(u,v)$ and $y=y(u,v)$,
\begin{align}
E(u,v)\:x_v^2-2F(u,v)\: x_vx_u+G\:x_u^2&=\Delta^2\\
E(u,v)\:y_v^2-2F(u,v)\: y_vy_u+G\:y_u^...

4
votes

1
answer

125
views

### The complement of $L_1(0,1)$ in $L_1(0,1)^{**}$

Let $\mu$ be a finite measure, like the Lebesgue measure in $(0,1)$. It is well-known that $L_1(\mu)$ and its second dual $L_1(\mu)^{**}$ are Banach lattices, $L_1(\mu)$ is a projection band in $L_1(\...