Questions tagged [fa.functional-analysis]

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

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What is the relationship between Hecke algebras and the enveloping algebra of Lie groups?

Here is the story as I see it. Let $G$ be an abelian locally compact group. Then the (spherical) Hecke algebra for $K=1$ is by definition the endomorphism algebra of $l^2(G)$ as a $G$-module, where ...
Tim Phalange's user avatar
4 votes
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107 views

Is this subspace of $B(\mathcal{H})$ known?

Let $\mathcal{H}$ be a Hilbert space. Suppose that I take a fixed ONB of $\mathcal{H}$ let us call it $\{ e_i \}_{i\in \mathbb{N}}$ and then I define \begin{align*} \|T \|_{D} = \sup_{l_i, m_i} \sum_{...
Frederik Ravn Klausen's user avatar
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1 answer
67 views

Friedrich's extension of the generator of a continuous time markov chaoin

Consider the infinitesimal generator $G$ of a Markov chain with state space $\mathbb{Z}$ such that it is symmetric with respect to a measure $\mu$ on $\mathbb{Z}$. Then, the operator $(G,C_c(\mathbb{Z}...
Ribhu's user avatar
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Hardy-Littlewood-Sobolev for "componentwise product" of Riesz kernels

Let $d\in\mathbb N$ and $0<\alpha<d$. Define the Riesz kernel $K_\alpha(x):=|x|^{\alpha-d}$, and the associated convolution operator $$K_\alpha f(x):=\int\frac{f(y)}{|x-y|^{d-\alpha}}~dy.$$ The ...
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Markov semigroups and resolvents, difference of continuity

Let $(E,d)$ be a locally compact separable metric space. We have a Markov process $X=(\{X_t\}_{t \ge 0},\{P_x\}_{x \in E})$ on $E$. For bounded measurable function $f$ on $E$, we define \begin{align*} ...
sharpe's user avatar
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The set of all functions which vanish at infinity is a subset of the set of all functions which have vanishing variation

Let $X$ be a coarse space, we define the following: $D_b(X)$ is the set of all bounded functions $f:X\rightarrow \mathbb{C}$ $f\in $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 ...
Hussain Rashed's user avatar
11 votes
2 answers
427 views

On dense embedding of Banach spaces

Disclaimer: When I came up with this question yesterday, I suspected it to be trivial (trivially true or trivially false). Then it kept me awake several hours tonight... (I still hope, though, this is ...
Jochen Glueck's user avatar
11 votes
1 answer
438 views

Is the spectrum of a "self adjoint" operator real on $\ell^p$?

There might be an obvious answer to the question, but it doesn't come to mind. Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
an_ordinary_mathematician's user avatar
6 votes
1 answer
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Equivalence of antiderivative in L1 sense and in the usual sense

We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that: $$ \lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0 $$ where $\Vert \cdot \Vert_1$ is the $L_1$ ...
Tomer's user avatar
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Tightness of Hilbert-space-valued arrays

Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}...
esner1994's user avatar
2 votes
1 answer
100 views

Resonances for Schrodinger operators with radial potentials

Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a ...
Capublanca's user avatar
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Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$

I'm currently trying to work with the non-local operator given by $$ (-\Delta)^{\frac{s}{2}}f(x)= c_s\text{P.V} \int_{-\infty}^\infty \frac{-f(x+y)-f(x-y)+2f(x)}{|y|^{1+s}} dy, $$ where $f :\mathbb ...
Kernel's user avatar
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2 answers
980 views

A topological vector space $X$ is separable if its dual space $X^*$ is separable?

Let $(X,\tau)$ be a topological vector space such that the associated dual space $X^*$ is separable. Can we say that $X$ is separable ? I know that this property is valid for Banach spaces but for ...
John nany's user avatar
3 votes
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155 views

On Pitt's inequality (weighted Fourier inequality)

One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$, $$ \sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
DSM's user avatar
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7 votes
2 answers
265 views

Completeness of coefficient functionnals

My questions is about Schauder bases and more specifically about coefficient functionals. Let $(x_n)$ be a Schauder basis of a Banach space $X$. Thus for all $x$ in $X$, $x = \sum f_n(x) x_n$. The $...
jvc's user avatar
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1 answer
415 views

Separability of an algebra is equivalent to separability of its spectrum

Let $A$ be a commutative C*-algebra. I would like to show that $A$ is separable (i.e. has a countable dense subset) if and only if the spectrum of $A$ (denoted by $\Omega(A)$) is separable. Notes ...
ned grekerzberg's user avatar
1 vote
1 answer
616 views

Injectivity of an integral operator

Consider the operator $$K:L^2(0,1)\rightarrow L^2(0,1) \\ u\rightarrow\int_0^1k(s,x)u(s)ds.$$ with $k\in L^2((0,1)\times(0,1)).$ I want to know under what assumption the kernel is reduced to zero. i....
Gustave's user avatar
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1 vote
1 answer
172 views

Representation of algebras as bounded nilpotent operators

Let $A$ be a real/complex algebra (just a real/complex vector space with a multiplication; none PI's are required). Let $\mathcal{H}$ be a real/complex Hilbert space. Let $\operatorname{B}(\mathcal{H})...
Math-Phys-Cat Group's user avatar
6 votes
3 answers
378 views

$u_t=Au+F(u)$ where $A$ is the infinitesimal generator of $C_0$-semigroup

I asked this question on Mathematics Stackexchange, but got no answer. In Pavel's book: Nonlinear Evolution Operators and Semigroups - Applications to Partial Differential Equations, we have the ...
Math's user avatar
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4 votes
1 answer
342 views

Every convex set is of locally finite perimeter

I need to prove that every convex subset of $\mathbb{R}^n$ is of locally finite perimeter. $E$ is of locally finite perimeter if there exists a vector-valued Radon measure $\mu_E$ s.t. the Gauss ...
A. Ninno's user avatar
2 votes
1 answer
97 views

Sobolev topology on essentially compactly supported Sobolev-"functions"

The locally convex space of essentially compactly-supported $p$-integrable "functions" $\operatorname{L}_{\mathrm{comp}}^p(\mathbb{R}^d,\mathbb{R})$ is defined as the set $$ \bigcup_{n \in \mathbb{N}} ...
ABIM's user avatar
  • 5,019
7 votes
0 answers
241 views

Isometries on the unit sphere

Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
Markus's user avatar
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144 views

Integral estimate for the solution of the heat equation

Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality? $$ \int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
user avatar
1 vote
0 answers
164 views

Polar decomposition of the Volterra integral operator

Repost of this Math.SE question due to a lack of answers (No one was able to help me find the closed form of $U_T$ and $|T|$ after two bounties). I also searched extensively online but couldn't find ...
ViktorStein's user avatar
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0 answers
116 views

Examples of groups admitting a proper $1$-cocyle for a bounded representation

A representation $\pi: G \to B(H)$ of a group $G$ on a Hilbert space $H$ is called bounded iff $\sup_{g \in G} \| \pi(g) \|_{B(H)} = C < \infty$. A $1$-cocycle with respect to the representation $\...
Adrián González Pérez's user avatar
0 votes
1 answer
321 views

Eigenvalues of an integral operator

Let $K\in L^2((0,1)\times(0,1))$ and consider the operator defined in $L^2(0,1)$ by $$Lu(x):=u(x)-\int_0^1K(s,x)u(s)ds.$$ What kind of assumption might I impose on $K$ such that $\lambda=1$ will be ...
Gustave's user avatar
  • 545
8 votes
2 answers
350 views

Metrizability of a topological vector space where every sequence can be made to converge to zero

This is a follow-up to this answer. If $E$ is a (real or complex) topological vector space, we say that a sequence $\{x_n\}_{n=1}^\infty$ in $E$ can be made to converge to zero if there exists a ...
J. van Dobben de Bruyn's user avatar
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0 answers
153 views

Representations of Banach algebras

If $A$ is a Banach algebra and $L$ a left ideal of $A$, consider the representation $T_{L}$ of $A$ into the algebra $B(A/L)$ of bounded linear operators on the quotient space $A/L$ defined by $T_{L}(a)...
Paul Visoianu's user avatar
7 votes
3 answers
639 views

A generalization of discrete Hilbert's transform (Montgomery's inequality)

In the paper "Hilbert's inequality", Montgomery and Vaughan proved that a generalization of the discrete Hilbert transform is bounded in $\ell^2$. The inequality reads as follows $$ \Big| \sum_{k\neq ...
an_ordinary_mathematician's user avatar
2 votes
1 answer
265 views

Interpolation of product spaces

Suppose that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$. Let $\mathcal{B}$ be another Banach space. Is it true that $X_{\theta}\times\mathcal{B}$ is an ...
Capublanca's user avatar
7 votes
1 answer
118 views

Containment of $c_0$ in projective tensor products

Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}_\pi Y$ the projective tensor product. Question: If $X\hat{\otimes}_\pi Y$ contains an isomorphic copy of $c_0$, must then $X$ or $Y$ ...
Johann Langemets's user avatar
5 votes
3 answers
2k views

Functional derivatives on Banach spaces

Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. ...
IamWill's user avatar
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2 votes
0 answers
86 views

First Dirichlet eigenvalue below second Neumann eigenvalue?

Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary. I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
Landauer's user avatar
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2 votes
1 answer
180 views

Integral transformation, Laplace-like

Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)? $$ \int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt $$ It resembles somewhat the Laplace transformation. ...
borntomath's user avatar
0 votes
1 answer
161 views

Commutator of translation invariant operators on $L^2(\mathbb{R})$

I have a question concerning the commutator of translation invariant operators on $L^2(\mathbb{R})$. Recall that $S:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is translation invariant if $Su_t=u_tS$ for all $...
Lenard Velasquez's user avatar
1 vote
0 answers
53 views

Standard definition: vector-valued essential support

Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used? $$ \...
ABIM's user avatar
  • 5,019
1 vote
1 answer
552 views

Fourier transform of a translation invariant operator on $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$

Consider the space $l^2(\mathbb{Z}) \otimes l^2(\mathbb{Z})$ with distinguished computational basis $e_i \otimes e_j $ and a group of translations $T_a$ defined by $T_a e_i \otimes e_j = e_{i+a} \...
Frederik Ravn Klausen's user avatar
2 votes
1 answer
722 views

Metrizability of topology of compact convergence

Let $X$ be a separable metric space and $Y$ be a second-countable $\sigma$-compact Hausdorff space. Then the compact-convergence (compact-open) topology on $C(Y,X)$ is metrizable with metric $$ d(f,g)...
ABIM's user avatar
  • 5,019
1 vote
1 answer
134 views

Understanding a family of Sobolev-type inequalities

I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following: Denote the following inequality as $S_{r,s}^{\theta}$: $\...
user153765's user avatar
3 votes
1 answer
233 views

Reference request: completion of Banach norm on sum

Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset $$ X_1+X_2 = \left\{ x\in X:\, x= \beta_1 x_1 + \beta_2 x_2 \, \beta_i \in \mathbb{R},\, x_i \in X_i \right\}, $$ a is ...
ABIM's user avatar
  • 5,019
3 votes
0 answers
61 views

Boundedness of $\chi_{\{f_n=0\}}$ in the BV norm

Let $f_n \in H^2(\Omega) \cap C^0(\bar \Omega)$ be a sequence of functions that are uniformly bounded in $H^2(\Omega) \cap C^0(\bar \Omega)$ on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ...
FavorExistingPopularTags's user avatar
1 vote
1 answer
205 views

Density and the projective tensor product

Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set $$ D^+\...
ABIM's user avatar
  • 5,019
1 vote
0 answers
39 views

Example of periodic semidifferentiable function without absolutely convergent Fourier series

Is there an example of a periodic continuous function that is semidifferentiable (i.e the left derivative and the right derivative exist at each point), but with a non-absolutely convergent Fourier ...
Phil-W's user avatar
  • 975
-1 votes
1 answer
285 views

Expressing the sum of two squared inner products more compactly: is it possible to lift the dimension? [closed]

Let $v_1,v_2\in\mathbb{R}^d$ be two fixed vectors, and $\langle \cdot,\cdot\rangle_{\mathbb{R}^d}$ be the usual Euclidean inner product in $\mathbb{R}^d$. My question is as follows. Is there an (...
hookah's user avatar
  • 1,096
6 votes
0 answers
339 views

Recent work on Pseudo-Laplacian and Pseudo-cuspform in the spirit of Riemann Hypothesis after the work of Bombieri and Garrett

( This is my first MO question . I'm totally inexperienced on MO so, forgive me for my mistakes .) Paul Garrett and Enrico Bombieri were (are?) Secretly Working on Pseudo-Laplacians and Pseudo-...
user avatar
1 vote
0 answers
195 views

Asymptotic of a functional as $x\rightarrow \infty$

Consider the following functional : $$ I(x,s) =\int_0^\infty\mathrm dy \frac{F(x + \mathrm iy, s) − F(x −\mathrm iy, s)}{\mathrm e^{2πy}-1}, $$ where $ F(z, s) = \dfrac{\sinh(\sin^2[π\Gamma(z)/(2z)])...
bambi's user avatar
  • 375
9 votes
1 answer
1k views

Dual space of continuous Banach-space-valued functions

Let $X$ be a Banach space and $K$ some compact Hausdorff space. I am interested in the dual space of the Banach space $$C(K; X) = \lbrace f: K \to X, \ f \text{ is continuous}\rbrace, \qquad \lVert ...
Yaddle's user avatar
  • 371
5 votes
1 answer
274 views

Generalization of minimal selection theorem

Consider a metric space $X$ and a set-valued map $F : X \to \mathbb{R}^{n}$. We define the minimal selection \begin{equation*} m(F(x)) := \arg\min \big\{ \lvert u \rvert : u \in F(x) \big\}, \end{...
node's user avatar
  • 329
1 vote
0 answers
44 views

Decomposition of the space of Radon measures with respect fractional harmonic capacity?

It is well know that there is a generalization of Lebesgue decomposition theorem in the following way: Any non negative Radon measure can be decomposed uniquely into the sum of an absolutely ...
Hheepp's user avatar
  • 361
0 votes
0 answers
46 views

Show that $\big(s(. |C_n)\big)_n$ is equicontinuous on $X^*$

Let $(X,\|.\|)$ be a separable Banach space with dual $X^*$. $\mathcal{P}_{wkc}(X)$ be the class of nonempty, weakly-compact and convex subsets of $X$. For any $C\in\mathcal{P}_{wkc}(X)$ we define ...
Wer Wer's user avatar

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