Questions tagged [fa.functional-analysis]

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

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Is a sign-preserving operator on $L^2$ a multiplication?

Let $T:L^2(\mu)\to L^2(\mu)$ be a linear and continuous operator, where $L^2(\mu)$ is the (real) $L^2$-space to some $\sigma$-finite measure space $(\Omega,\Sigma,\mu)$. $T$ is assumed to be sign-...
daw's user avatar
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Spectrum of a differential operator on $L^2(0, \infty)$

Let $A:H^n(0, \infty) \subset L^2(0, \infty) \to L^2(0, \infty)$ be the differential operator defined by $$Af:= \sum_{j=0}^na_j f^{(j)}$$ for all $f \in H^n(0, \infty),$ where $a_j \in \mathbb{C}$ and ...
Mike Van's user avatar
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2 votes
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Browder's Fixed Point Theorem in uniformly convex Banach spaces with non-identical image

This is in fact an exercise from Dirk Werner's book "Funktionalanalysis", but I do think that the result is quite interesting and up to now, I can only partly solve this problem. From the point of my ...
Peter's user avatar
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-1 votes
1 answer
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Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values

To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
Paul B. Slater's user avatar
0 votes
2 answers
281 views

Naming convention: Adjective for linear operators that are endomorphisms

If a matrix has the same number of rows and columns, we call it a square matrix. The analogous concept for linear operators would be operators with the same domain and range, i.e., endomorphisms. Is ...
Dominique Unruh's user avatar
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0 answers
73 views

Hausdorff distance restricted to linear subspaces

Let $V$ be a Hilbert space, $Q \subset V$ be convex and compact and $Q_n \subset V$ be convex and compact for $n\in \mathbb{N}$ such that $Q_n \rightarrow Q$ for $n\rightarrow \infty$ in Hausdorff ...
Steve's user avatar
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1 vote
1 answer
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Extension of pseudodifferential operators to Sobolev spaces

Let $\Omega$ be a open subset of $\mathbb{R}^{d}$ and define \begin{eqnarray*} \mathcal{S}^m(\Omega ) &=&\left\{ \begin{array}{c} a\in C^\infty(\Omega \times \mathbb{R}^{d}): \hbox{for all ...
Math's user avatar
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4 votes
2 answers
715 views

Decay of eigenfunctions for Laplacian

Consider the discrete second derivative with Dirichlet boundary conditions on $\mathbb C^n$. Its eigendecomposition is fully known: see wikipedia It seems like the largest eigenvalue $\lambda_1$ is ...
Yannis Pimalis's user avatar
7 votes
2 answers
1k views

The norm of tensor product operator on Lp spaces

Let $X, Y$ be two $\sigma$-finite measure spaces and $p,q\in [1,\infty]$. Let $T_1, T_2:L^p(X)\rightarrow L^q(Y)$ be two bounded linear operators. Then one can define a linear operator $$T_1\otimes ...
Chris's user avatar
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2 votes
1 answer
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Understanding a proof about limit of a sequence of open sets

We are reading a proof about the following limit \begin{equation}\tag{1} \lim_{n \to \infty} \sigma_1(T_n)= \sigma_1(T), \end{equation} where $T:D(T) \subseteq H \to H$ and $T_n:D(T_n) \subseteq H \to ...
Mike Van's user avatar
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13 votes
1 answer
392 views

Does the $\overline{\partial}$ operator have closed image?

Let $X$ be a complex-analytic manifold, not necessarily compact. Does $\overline{\partial} : C^\infty(X) \rightarrow \Omega^{0,1}(X)$ have closed image with respect to the Fréchet topology given by ...
Daniel Bruegmann's user avatar
5 votes
1 answer
256 views

Compact space $K$ without convergent sequence while $c_0$ is complemented in $C(K)$

Let $K$ be a compact Hausdorff infinite topological space and $C(K)$ the Banach space of continuous functions from $K$ in $\mathbb{R}$ with sup norm. It is known that $c_0$ is complemented in $C(K)$ ...
Aligomez's user avatar
1 vote
1 answer
102 views

Continuous Left-inverse of Dirac Lipschitz-Free Space

Let $X$ be a separable pointed metric space and let $AE(X)$ denote the corresponding Lipschitz-Free (or Arens-Eells space) over $X$. The point-evaluation map $\delta:X\mapsto AE(X)$ is injective ...
ABIM's user avatar
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Why does this PDE have a solution?

Let $\Sigma$ be a compact surface and denote by $\nu$ the unit conormal for $\partial \Sigma$. Let $$E = \left\{ \phi \in C^{2,\alpha}(\Sigma) : \int_{\Sigma} \phi \, \mathrm{d}A = 0 \right\} $$ and $$...
Eduardo Longa's user avatar
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1 answer
125 views

Product of sets with the Radon-Nikodym Property (RNP)

I have read that it is somewhat well-known that if two Banach spaces $X$ and $Y$ have the Radon-Nikodym Property (RNP), then their product $X\times Y$ also has the RNP. Does the above result ...
BigbearZzz's user avatar
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2 votes
1 answer
117 views

Twisted canonical commutation relations

I am dealing with universal C*-algebra generated by $x,y$ with the following relations: $xy = qyx$, $x^{*}y = qyx^{*}$, $y^{*}x = qxy^{*}$, $x^{*}x = q^2xx^{*} - (1-q^2)yy^{*}$, $y^{*}y = q^2yy^{*} - (...
Invincible's user avatar
1 vote
0 answers
183 views

What imaginations of Lebesgue spaces or other Banach spaces do people intuitively share?

At several occasions I heared people discussing about the „colors“ of Lebesgue spaces $L^p$: $L^2$ is red, $L^1$ is white, $L^\infty$ is black, and the other $L^p$ are blue or violett. Of course this ...
phantomias's user avatar
2 votes
0 answers
131 views

Linear independence of functions

Let $x_1,x_2,\ldots,x_n\in\mathbb{R}^d$ be points so that no one point is in the positive span of another. That is, there is no pair of points $x_i,x_j$ such that $x_i=\alpha x_j$ for a positive ...
mohi's user avatar
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11 votes
1 answer
305 views

Conceptual explanation for the appearance of entropy in $\frac{d}{dp}\|x\|_p$

For $x\in \mathbb{R}^d$, an elementary computation yields that $$\frac{d}{dp}\log \|x\|_p =\frac{1}{p^2}\sum_{i=1}^d \frac{|x_i|^p}{\|x\|_p^p}\log \frac{|x_i|^p}{\|x\|_p^p}=-\frac{1}{p^2}\operatorname{...
tmh's user avatar
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27 votes
4 answers
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Proofs of Young's inequality for convolution

For $1\leq p,q \leq \infty$ such that $\frac1p +\frac1q\geq 1$, Young's inequality states $\|f\star g\|_r\leq \|f\|_p\|g\|_q$ (we work on $\mathbf{R}^d$ here), where $1+\frac1r = \frac1p+\frac1q$. ...
Ayman Moussa's user avatar
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1 vote
0 answers
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Angle between Fleming-Viot type 3-particle system

Consider $(X^1,X^2,X^3)\in (0,\infty)^3$ with each particle starting at $1$ and moving independently according to Brownian motion until random time $\tau_1:=\min \lbrace t>0: X_{t-}^1\wedge X_{t-}^...
maliesen's user avatar
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2 votes
0 answers
86 views

Invariance under diffeomorphisms of the Hajlasz-Sobolev spaces

In this post it was shown that if $\Omega$ and $\Omega'$ are diffeomorphic non-empty open domains in some Euclidean space then the corresponding local Sobolev spaces are diffeomorphic with ...
ABIM's user avatar
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0 votes
0 answers
134 views

Compact embedding of the $\mathcal{C}^k$ norm on a compact Kahler manifold

Given a smooth complex valued function $f$ on a Kahler manifold $X$, we can define its $\mathcal{C}^k$ norm to be $\sum_{p+q \leq k, 0 \leq p \leq q} sup_{X}|\nabla^{p} \overline{\nabla^q} f|_g$, ...
archer's user avatar
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12 votes
1 answer
1k views

List of all known Riesz representation theorems

Due to the history and development of measure and integration theory and different mathematical schools, there is a huge variety and inconsistency of definitions for concepts like tightness of a ...
yada's user avatar
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3 votes
1 answer
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Different definitions of a relatively compact operator

(Cross-post from Math Stackexchange, where some work has been done in the comments) Let $T,K$ be unbounded operators on a Hilbert space $H$. I've seen the following definition of a relatively compact ...
Jannik Pitt's user avatar
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1 vote
1 answer
77 views

Reference request for (weak*) metrizability of a bounded space of signed Radon measures on a compact set

I know the following is true and I know how to prove it (cf. exercise 50 on page 171 in Folland, Theorem 7.18 in Folland), but per my adviser's instructions, it would be better to find a source to ...
Ben Ciotti's user avatar
6 votes
1 answer
525 views

Volume doubling, uniform Poincaré, counterexample

The Poincaré inequality and the volume doubling property are important notions related to heat kernel estimates. Pavel Gyrya and Laurent Saloff-Coste obtain the two sided heat kernel estimate of ...
sharpe's user avatar
  • 701
6 votes
1 answer
290 views

Idempotent functions on Sp(1)

The quaternion group $Sp(1)\simeq S^3$ can be understood as $(z,w)\in\mathbb {C}^2$ with $|z|^2+|w|^2=1$ where multiplication is defined by $(z,w)(t,s)=(zt-\bar{s}w,zs+\bar{t}w)$. Question: How do ...
BigM's user avatar
  • 1,543
2 votes
1 answer
777 views

Characterising closed range self-adjoint operators

Let $T:\mathrm{dom}(T) \subseteq H \to H$ be a densely defined, self-adjoint operator on a Hilbert spaces $𝐻$. In general the range of $T$ is not guaranteed to be closed. What tools are available to ...
Dave Shulman's user avatar
6 votes
1 answer
885 views

Products of functions in fractional-order Sobolev spaces

It is well known that $\|fg\|_s \lesssim \|f\|_{s_1} \|g\|_{s_2}$ for functions $f: {\mathbb R}^n \rightarrow {\mathbb R}$ under certain conditions on $s$, $s_1$, $s_2$ (i.e. $s_1$, $s_2 \geq s$ and $...
olih's user avatar
  • 81
5 votes
2 answers
5k views

Domain of definition of Laplace Operator on $L^2$

I'm trying to combine two ways of looking at the Laplacian $\Delta$ on $\mathbb R^n$ (and on other domains). Firstly, it is well known that this operator is essentially self-adjoint on $C_c^\infty(\...
Nathanael Schilling's user avatar
1 vote
2 answers
112 views

Question on Parabolic PDEs: Improvement of the bound ${\vert \vert v(\cdot,t)-v_0(\cdot) \vert \vert}_{L^p}\le \hat C t^{1/q}$

Let $M$ be a $C^3-$compact manifold and $v \in W^{2,1}_p(M\times [0,T])$ ($1\le p<\infty$) be the solution of: $\begin{cases} \partial_t v-\Delta_{M} v=f(v), \quad M\times [0,T]\\ v(x,0)=v_0,...
kaithkolesidou's user avatar
2 votes
1 answer
144 views

Fermat stationary point theorem - a generalization exists?

Let $f:E\to\mathbb{R}$ a functional (here $E$ is a normed vector space). Is it true that if $x_0\in E$ is a local minimum for $f$, then all the directional derivatives are 0? We have the derivative ...
Bogdan's user avatar
  • 1,330
0 votes
1 answer
266 views

Bilinear Strichartz estimates for the Schrodinger equation

Let $d\geq 2$, $I\subset\mathbb{R}$ a time interval and $t_0\in I$. Let $u,v$ be solutions to the free Schrodinger equation on $\mathbb{R}^d$, i.e. $(i\partial_t+\Delta)u=(i\partial_t+\Delta)v=0.$ Let ...
Capublanca's user avatar
2 votes
2 answers
614 views

Rate of convergence of mollifiers // Sobolev norms

Following up to the question raised here, I am searching for a reference (or a simple argument) to establish (in the whole space) the following (suggested) equivalence : Given $N_1$ and $N_2$ two (...
Ayman Moussa's user avatar
  • 2,575
11 votes
2 answers
992 views

Low-degree polynomial approximation of the piecewise-linear function $x \mapsto \max(x, 0)$ on an interval $x \in [-R,R]$

For $R > 0$, consider the piecewise-linear function $\sigma_R: [-R,R] \rightarrow \mathbb R^+$, defined by $\sigma_R(x) := \max(x,0)$. Question Given $\epsilon> 0$, find a "low-degree" ...
dohmatob's user avatar
  • 6,716
1 vote
1 answer
145 views

Endpoint in commutator estimate

Let $p\in(1,\infty)$ and $J^s=(1-\Delta)^{s / 2}$ with $s>0$. Then we have the following commutator estimate by C. E. Kenig, G.Ponce and L. Vega (1991 JAMS), \begin{equation} \left\|J^{s}(f g)-f J^{...
YT_learning_math's user avatar
2 votes
1 answer
241 views

On the limiting behaviour of Sobolev space functions

Let $k$ be an integer such that $k>n/2$, and let $H^k(\mathbb{R}^n)$ denote the usual Sobolev Hilbert space. Let $f,g\in H^k(\mathbb{R}^n)$. Is it true that $\displaystyle \lim_{R\rightarrow \...
Guilia S's user avatar
  • 105
6 votes
1 answer
319 views

Reference: Hajlasz-Sobolev Spaces with Values in a Metric Space

Let $(X,d,\mu)$ be a separable metric measure space on which every ball has positive but finite measure. I've come across the definition of a homogeneous Fractional Hajlasz-Sobolev spaces $M^{s,p}(...
ABIM's user avatar
  • 5,019
4 votes
0 answers
478 views

Quasi-compact operators on $\ell^1$

Assume that $T:\ell^1 \rightarrow \ell^1$ is a bounded linear operator, given by an infinite matrix $(a_{ij})_{i,j\in\mathbb{N}}$ so that $(Tx)_i = \sum_{j=1}^{\infty} a_{ij}x_j$. In particular, I am ...
user avatar
10 votes
1 answer
194 views

Verbal description, or terminology, for the ${\mathcal L}_p$-spaces of Lindenstrauss and Pelczynski

This question is intended for Banach-space specialists and so I will not repeat all the definitions here. My aim is to find out how the Banach space community refers to such spaces in discussions, and ...
Yemon Choi's user avatar
  • 25.5k
4 votes
0 answers
135 views

Max / Argmax of a function which includes sums of Gaussian CDFs and PDFs can surprisingly be approximated by a power law

Given $N\in\mathbb{N}$, I have been trying to calculate $m_N=\text{max}_{x\in\mathbb{R}^{+}}\chi_N(x), d_N=\text{argmax}_{x\in\mathbb{R}^{+}}\chi_N(x)$ for the function: $$\chi_N(x)=\frac{\sum_{i\in ...
Yaniv006's user avatar
6 votes
1 answer
343 views

Equivalence of $\sigma$-weak topology to another topology

Let $\mathcal H$ be a Hilbert space. Define a topology $\tau_1$ on $B(\mathcal H)$ by the family of seminorms $x\mapsto |Tr(xa)|,$ $a\in L^1(B(\mathcal H)).$ Here $B(\mathcal H)$ denotes the set of ...
A beginner mathmatician's user avatar
0 votes
2 answers
321 views

subspace topology and strong topology

Suppose $X$ is a locally convex space and $Y$ is a subspace of the strong dual of $X$, is the induced topology on Y equivalent to the strong topology $b(Y,Y')$ on $Y$? If this is not correct, then on ...
Richard Kim's user avatar
0 votes
4 answers
1k views

Does the Leibniz (product) rule hold for the spectral fractional Laplacian?

Does the Leibniz (product) rule hold in some sense for the spectral fractional Laplacian (at least in 1 dimension)?
user avatar
1 vote
0 answers
154 views

A criterion for abelian $C^*$ algebra [closed]

Let $A$ be a unital $C^*$ algebra such that for any two positive elements $x$, $y$ in $A$, whenever $x\leq y$ we have that $x^2\leq y^2$. Prove that $A$ is abelian.
A beginner mathmatician's user avatar
0 votes
0 answers
50 views

Smooth compactly supported function with good scaling with respect to the fractional Laplacian

Is there a smooth cut off function with compact support such that $\phi: \Omega \subset \mathbb R^N \to \mathbb R$, $\mathrm{supp} \phi \subset B_R(0) \subset \Omega$ and $$(-\Delta)^s \phi \le C R^{-...
user avatar
6 votes
2 answers
227 views

Continuity of a differential of a Banach-valued holomorphic map

Originally posted on MSE. Let $U$ be an open set in $\mathbb{C}^{n}$ let $F$ be a Banach space (in my case even a dual Banach space), and let $\varphi:U\to F$ be a holomorphic map. I seem to be able ...
erz's user avatar
  • 5,385
3 votes
2 answers
366 views

Is a bounded sequence of $H^1(\Omega)$ tight?

Assume $\Omega$ is a bounded subset of $\Bbb R^d$ and $ (u_n)_n$ is a bounded sequence of the Sobolev space $H^1(\Omega)$. Question: Can we say that $ (u_n)_n$ is tight in $L^2(\Omega)$ namely: ...
Guy Fsone's user avatar
  • 1,033
7 votes
1 answer
549 views

On the geometric Hahn-Banach theorem

Let $ X \subset \ R ^ n $ be a closed convex set and let $ L $ be a straight line such that $ X \cap L = \emptyset $. Does there exist a hyperplane containing $ L $ that does not intersect $ X $ ? ...
Pascal's user avatar
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