Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,305
questions
7
votes
0
answers
386
views
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 ...
4
votes
0
answers
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_{...
0
votes
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}...
3
votes
1
answer
137
views
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 ...
3
votes
0
answers
147
views
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*}
...
0
votes
0
answers
233
views
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 $$D_b(X)$ is said to vanish at infinity if for each $\varepsilon$>0 ...
11
votes
2
answers
427
views
On dense embedding of Banach spaces
Disclaimer: When I came up with this question yesterday, I suspected it to be trivial (trivially true or trivially false). Then it kept me awake several hours tonight... (I still hope, though, this is ...
11
votes
1
answer
438
views
Is the spectrum of a "self adjoint" operator real on $\ell^p$?
There might be an obvious answer to the question, but it doesn't come to mind.
Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all ...
6
votes
1
answer
126
views
Equivalence of antiderivative in L1 sense and in the usual sense
We say that$\ f$ is differentiable w.r.t to $L_1$ if there exists a$\ g$ such that:
$$
\lim_{h\to 0}\left\Vert\frac{f(x+h)-f(x)}{h} - g(x)\right\Vert_1 = 0
$$
where $\Vert \cdot \Vert_1$ is the $L_1$ ...
2
votes
0
answers
92
views
Tightness of Hilbert-space-valued arrays
Let $\mathcal{H}$ be a separable Hilbert space. Assume we have some triangular array $W_{n,j}, j=1, \ldots ,n $ of $\mathcal{H}$-valued random elements with $\mathbb{E} \Vert W_{n,j} \Vert_{\mathcal{H}...
2
votes
1
answer
100
views
Resonances for Schrodinger operators with radial potentials
Let $V\in L^{\infty}(\mathbb{R}^3)$ be a radial, compactly supported potential, and consider the Schrodinger operator $H:=-\Delta + V$ on $L^2(\mathbb{R}^3)$. Let $\psi$ be a resonance for $H$, i.e. a ...
1
vote
0
answers
101
views
Commutator estimates for $-(-\Delta)^s$, with $s \in (1,2)$
I'm currently trying to work with the non-local operator given by
$$
(-\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,
$$
where $f :\mathbb ...
4
votes
2
answers
980
views
A topological vector space $X$ is separable if its dual space $X^*$ is separable?
Let $(X,\tau)$ be a topological vector space such that the associated dual space $X^*$ is separable. Can we say that $X$ is separable ?
I know that this property is valid for Banach spaces but for ...
3
votes
0
answers
155
views
On Pitt's inequality (weighted Fourier inequality)
One of Pitt's Theorem (from "Theorems on Fourier Series" by H R Pitt, 1937) states that for an integrable periodic function $F$ over $[-\pi,\pi]$,
$$
\sum_{n=1}^{\infty} |a_n|^q n^{-q\lambda} \leq K(...
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 $...
0
votes
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 ...
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....
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})...
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 ...
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 ...
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
$$
\bigcup_{n \in \mathbb{N}} ...
7
votes
0
answers
241
views
Isometries on the unit sphere
Suppose that $X$ and $Y$ are two Banach spaces, $S_{X}$ and $S_{Y}$ their unit spheres, and $f$ an onto isometry between $S_X$ and $S_Y$. Does it follow that $X$ and $Y$ are isometric?
2
votes
0
answers
144
views
Integral estimate for the solution of the heat equation
Let $u$ be a solution of $\partial_t u - \Delta u =f$ with initial data $u(0,x) = 0$ on $\mathbb R^N$. How do one prove the following inequality?
$$
\int_0^T \int_{\mathbb R^N} \phi(f)(- \Delta) u(...
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 ...
2
votes
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 $\...
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 ...
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 ...
0
votes
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)...
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
$$ \Big| \sum_{k\neq ...
2
votes
1
answer
265
views
Interpolation of product spaces
Suppose that $X_{\theta}$ is an interpolation space between the Banach spaces $X_0$ and $X_1$. Let $\mathcal{B}$ be another Banach space.
Is it true that $X_{\theta}\times\mathcal{B}$ is an ...
7
votes
1
answer
118
views
Containment of $c_0$ in projective tensor products
Let $X$ and $Y$ be Banach spaces and denote by $X\hat{\otimes}_\pi Y$ the projective tensor product.
Question:
If $X\hat{\otimes}_\pi Y$ contains an isomorphic copy of $c_0$, must then $X$ or $Y$ ...
5
votes
3
answers
2k
views
Functional derivatives on Banach spaces
Physicists often use functional integrals and I'm trying to make sense of it in more precise terms. As you can see here, the functional derivative in Physics is defined in terms of Taylor expansions. ...
2
votes
0
answers
86
views
First Dirichlet eigenvalue below second Neumann eigenvalue?
Let $\Omega$ be a bounded domain in $\mathbb R^n $ with smooth boundary.
I was wondering if there exist any known conditions on $\Omega$ such that the 1st Dirichlet eigenvalue of the (positive) ...
2
votes
1
answer
180
views
Integral transformation, Laplace-like
Is the following integral transformation of $f$ known (for suitable $f$ and $s\in\mathbb{C}$)?
$$
\int_1^\infty f(t) \frac{e^{-ts}}{1-e^{-ts}}dt
$$
It resembles somewhat the Laplace transformation.
...
0
votes
1
answer
161
views
Commutator of translation invariant operators on $L^2(\mathbb{R})$
I have a question concerning the commutator of translation invariant operators on $L^2(\mathbb{R})$.
Recall that $S:L^2(\mathbb{R})\to L^2(\mathbb{R})$ is translation invariant if $Su_t=u_tS$ for all $...
1
vote
0
answers
53
views
Standard definition: vector-valued essential support
Let $f \in L^p(\mathbb{R}^n,\mathbb{R}^m)$. If $m=1$ then the essential support of $f$ is a mainstream definition; see here for example. However, when $m>1$ is the following definition used?
$$
\...
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} \...
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
$$
d(f,g)...
1
vote
1
answer
134
views
Understanding a family of Sobolev-type inequalities
I am reading Aspects of Sobolev-Type Inequalities by professor Laurent Saloff-Coste, where I found a claim on page 66 claiming the following:
Denote the following inequality as $S_{r,s}^{\theta}$: $\...
3
votes
1
answer
233
views
Reference request: completion of Banach norm on sum
Let $X_1,X_2$ be Banach subspaces of a locally convex space $X$. Then the subset
$$
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\},
$$
a is ...
3
votes
0
answers
61
views
Boundedness of $\chi_{\{f_n=0\}}$ in the BV norm
Let $f_n \in H^2(\Omega) \cap C^0(\bar \Omega)$ be a sequence of functions that are uniformly bounded in $H^2(\Omega) \cap C^0(\bar \Omega)$ on a smooth bounded domain $\Omega \subset \mathbb{R}^n$ ...
1
vote
1
answer
205
views
Density and the projective tensor product
Let $X$ be a locally convex space (over $\mathbb{R}$), $D\subset X$ be dense, $B$ be a Banach space (again over $\mathbb{R}$) with Schauder basis $\{b_i\}_{i =1}^{\infty}$. Is the set
$$
D^+\...
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 ...
-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 (...
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-...
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)])...
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 ...
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{...
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 ...
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 ...