Questions tagged [fa.functional-analysis]

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

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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
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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). ...
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1 vote
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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 ...
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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 ...
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4 votes
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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 ...
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4 votes
2 answers
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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-...
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3 votes
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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 $\...
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1 vote
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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 ...
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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
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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-\...
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3 votes
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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}(...
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2 votes
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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 ...
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1 vote
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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
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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 ...
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1 answer
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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....
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2 votes
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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 ...
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1 vote
0 answers
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+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 \...
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2 votes
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+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 ...
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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 $\sigma$-...
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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/~...
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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
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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 ...
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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
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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 ...
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2 votes
0 answers
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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 ...
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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 ...
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1 vote
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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 $\{ \...
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0 votes
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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 ...
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0 votes
0 answers
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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 $\Omega$ ...
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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 $n=n(\lambda)$ ...
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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. ...
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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 ...
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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(\...
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