Questions tagged [fa.functional-analysis]

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

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1answer
58 views

Ultrabornological representation for the space of uniformly continuous functions?

Let $\{\omega_i\}_{i\in I}$ be a non-empty set of increasing (not necessarily strictly) continuous functions preserving $0$. Then, for each $i \in I$ define the space $$ C_{\omega_i}(\mathbb{R}^n,\...
3
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1answer
101 views

Banach embedding of finite dimensional spaces

Recall that: let $0<r<s<2$, then $\ell_r$ uniformly contains a subspace isomorphic to $\ell_s^m$, $m\ge 1$ (see [JS]). I am wondering whether are any result for the case when $r>s>2$? ...
4
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1answer
153 views

Known dense subset of Schwartz-like space and $C_c^{\infty}$?

After reading this question, which asked for some examples of commonly used (proper) dense subsets of $C_0^{\infty}(\mathbb{R}^n)$ with the $L^p$-norm I wonder. What are some "well-known" ...
7
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0answers
88 views

Potential p-norm on tuples of positive operators

This is a follow-up to this question on p-norms of tuples of operators. Consider $\left[\begin{matrix} A \\ B \end{matrix}\right] \in B(H)^2_+$, meaning $A,B\geq 0$, and define $$ \left\|\left[\begin{...
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3answers
203 views

Openness of the set of injective functions in $C(\mathbb{R})$?

Let $C(\mathbb{R})$ be equipped with the topology of compact convergence (or equivalently the compact-open topology). Then, is the subset $\left\{f\in C(\mathbb{R}): \text{$f$ injective} \right\}$ ...
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0answers
185 views

What is the asymptotics of the Fourier transform of $\exp(-x^4)$ for large wave numbers?

The Fourier transform of $\exp(-x^4)$ has an analytical expression, it's the difference of two generalized hypergeometric functions: $\int d x \ e^{-x^4} e^{ikx} = 2 \ \Gamma(\frac{5}{4}) \ _0F_2(;\...
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3answers
189 views

Optimizing the gradient norm on the unit sphere

Let $ \Bbb S^{d-1}=\{(x_1,\cdots ,x_d): x_1^2+ \cdots +x_d^2=1\}\subset \Bbb R^d$ be the unit sphere. Let $\nabla u= (\partial_{x_1}u,\cdots, \partial_{x_d}u)$ be the gradient of a function $u\in C_c^\...
4
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0answers
97 views

Algorithm/iterative procedure for constructing hypercyclic vectors?

Let $B$ be a separable Banach space and let $L:B\rightarrow B$ be a hypercyclic operator; here I use the definition of hypercyclicity given implicitly by Birkhoff's Transitivity Theorem: continuous ...
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0answers
40 views

Image restoration quality general lower bounds

A typical image restoration model posits that, starting from a true image $f = f(x,y)$, we observe $$ \tilde f = f \star h + n $$ where $\star$ is convolution, $h$ is the point spread function (caused,...
4
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1answer
149 views

Supremum over which sets makes $H^{\infty}$ non-separable?

It is known that the space $H^{\infty}$ of bounded holomorphic functions on the unit disk $D$ is non-separable with respect to the supremum norm $\|\cdot\|_{\infty}^{D}$. Let $E\subset D$ be connected ...
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0answers
35 views

Prove compactness of approximate (finite differences) solutions of heat equation

Let $\{u^h\}$ be the family of solutions of the discretized heat equations on the interval $[0,1]$ (uniform grid of size $h>0$). $$\begin{cases} u^h_t - \Delta_hu^h = 0\\ u^h(t,0) = u^h(t,1) = 0 \\...
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0answers
55 views

Differential equation

Consider $u = u(\phi,\psi)$ where $\phi = \phi(x)$ and $\psi =\psi(x)$ are both analytic function. The following equation $$\partial_x u - u\partial_x (\phi-\psi)=0$$ has a trivial solution $u(\phi,\...
2
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0answers
46 views

Sum of an operator with disconnected spectrum and a compact operator is strongly reducible

Let $H$ be a complex, infinite-dimensional, separable Hilbert space. Let $T \in B(H)$ be an operator with disconnected spectrum. In the introduction of the paper: Jiang, C.; Sun, S.; Wang, Z. (1997). ...
6
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1answer
233 views

Path integral as quantum mechanics on the tangent bundle

Let $X$ be a configuration space, a finite-dimensional manifold. By "quantum mechanics on $X$" I mean a linear evolution equation on complex-valued functions on $X$, determined by a ...
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0answers
29 views

Norm equivalences for Gaussian random functions (Cameron-Martin space)

Preliminaries Consider the Hilbert space $H :={L^2_{\text{per}}(\mathbb{R})}$ of Gaussian random functions, $2\pi$-periodic in $\mathbb{R}$. These random functions are drawn from a Gaussian measure $\...
4
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0answers
88 views

If $f:U_1\to\mathcal L^p(\mu;E_2)$ is Fréchet differentiable, can we say anything about the Fréchet differentiability of $u\mapsto f(u)(\omega)$?

Let $(\Omega,\mathcal A,\mu)$ be a $\sigma$-finite measure space, $p\ge1$, $E_i$ be a $\mathbb R$-Banach space, $U_1\subseteq E_1$ be open and $f:U_1\to L$ be Fréchet differentiable at $x\in U_1$, ...
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2answers
796 views

In infinite dimensions, is it possible that convergence of distances to a sequence always implies convergence of that sequence?

This is a cross-posted on MSE here. Let $(X,d)$ be a metric space. Say that $x_n\in X$ is a P-sequence if $\lim_{n\rightarrow\infty}d(x_n,y)$ converges for every $y\in X.$ Say that $(X,d)$ is P-...
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2answers
91 views

lattice suprema vs pointwise suprema

What is the difference between the lattice supremum and the pointwise supremum of a family of functions? I mean, given a family of real valued functions $\mathcal{F}$, is the function $\sup\mathcal{F}:...
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2answers
135 views

Analogue of decay of Fourier coefficients of a smooth function on $\mathbb{S}^1$

Let $\nu$ be the uniform measure on the unit circle $\mathbb{S}^1 \subset \mathbb{R}^2$, normalised so that $\nu(\mathbb{S}^1) = 1$. Suppose $\mu$ is a Borel probability measure on $\mathbb{S}^1$ ...
6
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1answer
158 views

Potential p-norm on tuples of operators

Consider $\left[\begin{matrix}A \\ B\end{matrix}\right] \in B(H)^2$. One can define $$ \left\|\left[\begin{matrix}A \\ B\end{matrix}\right]\right\|_p = \| |A|^p + |B|^p\|^{1/p}. $$ Q: Is this a norm? ...
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0answers
107 views

Uniqueness of solutions of Young differential equations

Consider the following one dimensional Young differential equation: $$ Y_t=\int_0^t Y_s dX_s,\quad t\in[0,1]. $$ Here the driving process $X$ is a bounded functions $[0,1]\to\mathbb{R}$, which is $\...
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0answers
83 views

Bounded weak and weak-$\star$ topologies and metrics

Let $X$ be a reflexive and separable Banach space. Let $(h_n)$ be a sequence dense in $\overline{B^*_1}$ (the closed ball in $X^*$ of radius $1$). Set $$ d(x,y) := \sum_{n=1}^\infty 2^{-n} |(x-y, h_n)|...
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0answers
69 views

Error rate implying regularity

My question is a bit general/vague. It is well known that the regularity of certain functions can be measured through the rate of decay of certain error quantity based on an approximation procedure (...
18
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2answers
573 views

What is known about the “unitary group” of a rigged Hilbert space?

Suppose that $(E,H)$ is a rigged (infinite dimensional, separable) Hilbert space, i.e. $H$ is a Hilbert space, and $E$ is a Fréchet space, equipped with a continuous linear injection $E \rightarrow H$ ...
6
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1answer
186 views

Continuity of eigenvectors

Let $\mathbb{C} \ni z \mapsto M(z)$ be a square matrix depending holomorphically on a parameter $z$ with the property that $\operatorname{dim}\ker(M(z)))=1$ for $z $ away from a discrete set $D \...
2
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1answer
89 views

$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$ have periodic solution $\iff\; \frac 1T \int_0 ^ T f (t) dt \in g (\mathbb R) $

I have a research work concerning the equation: $$x '(t) + g (x (t)) = f (t),\quad \forall t\in \mathbb R$$ f and g are defined and continuous in $\mathbb R$ and with values ​​in $\mathbb R$. ...
3
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2answers
126 views

Conditions for continuity of an integral functional

Let $X$ be a metric space, $\nu,\mu$ be Borel measures on $X$, $f:X\times \mathbb{R}\rightarrow [0,\infty)$ be a measurable function. Under what conditions is the integral functional $F_f$, defined ...
4
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1answer
136 views

Equivalence of families indexes of Fredholm operators

Let $F=F(H,H)$ be the space of bounded Fredholm operators in a Hilbert space $H$ with topology inherited from the norm operator topology, and let $X$ be a compact topological space. For a continuous ...
2
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0answers
88 views

Open subset of compact-open topology

Let $E$ be a Banach space, $X$ a locally-compact metric space, equip $C(X,E)$ with the compact-open topology. Let $F:E \rightarrow E$ and consider the induced map $F_{\star}(f):=F \circ f$ on $C(X,E)$...
3
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0answers
146 views

How to prove the following linearized operator is positive?

In $L^2(\mathbb{R}^d)$, let $Q$ be the solution to \begin{equation} -\Delta Q+\alpha^2 Q = |Q|^{2\sigma} Q, \end{equation} and $Q$ satisfies that it is positive, radial, and exponentially decaying (...
2
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1answer
144 views

If an estimate is false on $L^{1}$, then it is false for the $\delta$ distribution?

Let $u=\int e^{\dot{\imath}K(x,y)} f(y) dy$. My advisor told me that we can disprove an integrability estimate $$\|u\|_{L^p}\lesssim \|f\|_{L^{1}}\label{1}\tag{1}$$ by disproving it when $f=\delta$, ...
4
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1answer
130 views

Approximate Sobolev embedding

It is well-known in $H^2(\mathbb R^3)$ embeds into $L^{\infty}(\mathbb{R}^3).$ Now consider a function $u \in \ell^{\infty}(h\mathbb Z^3)$ and a grid of points $x \in h\mathbb{Z}^3.$ We then define ...
0
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0answers
154 views

Eigenvalues of convolution matrices

Let $h: \mathbb{R}\to \mathbb{R}$ be a smooth function. Fix $0\leq s_1\leq \cdots \leq s_m\leq 1$ and $0\leq t_1\leq \cdots \leq t_n\leq 1$. Construct $A\in \mathbb{R}^{m\times n}$ by letting $A_{i,j}:...
2
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0answers
73 views

Approximation in fractional Sobolev space

Assume $\Omega\subset \Bbb R^d$ is Lipschitz open set. Let $p\geq 1$ and $0<s\leq 1/p$. How to prove that $C_c^\infty(\Omega)$ is dense in $W^{s,p}(\Omega)$? Recall that, $$|u|^p_{W^{s,p}(\Omega)}= ...
1
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0answers
99 views

Viewing limit as a map

Question: Let $X$ be a set of functions from $\mathbb{R}$ to itself. Consider the subset $X_0$ of the sequences $(f_n)_{n=1}^{\infty}\in X^{\mathbb{N}}$ for which $$ f_{\infty}(x) = \lim\limits_{n \...
4
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2answers
241 views

Examples of amenable Banach algebras which have non-amenable subalgebra

I am looking for examples of amenable Banach algebras which have non-amenable subalgebra I know 1: Each amenable Banach algebra has a bounded approximate identity 2: If $I$ be a closed ideal in an ...
1
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0answers
66 views

Using Paley-Wiener Theorem to prove the decay of $G(x-y)$

This question is related to my previous one, where I was looking for some help to prove the decay of the lattice Green function: \begin{eqnarray} G(x-y) = \int_{[-\pi,\pi]^{d}}\frac{d^{d}k}{(2\pi)^{d}}...
2
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1answer
128 views

About the metrizability of the space of Probability measures $\mathcal{P}(S)$

It is often proved in Books that the space of Probability measures $\mathcal{P}(S)$ on a Polish metric space $(S,\rho)$ endowed with the weak/narrow topology induced by declaring it to be be the ...
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0answers
154 views

Infinite composition of continuous functions

Let $f_n:\mathbb{R}\rightarrow \mathbb{R}$ be a sequence of functions and define $F_n:= f_n\circ \dots\circ f_1$. Then $F_n$ is continuous. However, the pointwise limit need not be (consider Mateusz'...
8
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1answer
161 views

On hereditarily reflexive Banach spaces

It was proved by W.B. Johnson and H.P. Rosenthal [Studia Math. 43 (1972), 77–92] that every Banach space $X$ with $X^{**}$ separable is hereditarily reflexive: every infinite dimensional closed ...
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0answers
91 views

Logical axioms used in the construction of counterexamples to ISP

In many cases, some problems are either solved in an affirmative way, or in a negative way. However, in some cases it turns out that some logical axioms lead to a proof of a certain statement, while ...
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0answers
56 views

Empty Weyl/Fredholm spectrum of an operator on an infinite dimensional Banach space

Let $X$ be a complex infinite dimensional Banach space, and let $T \in B(X)$ be nonscalar. The Fredholm spectrum of $T$ is defined by: $$ \sigma_{\Phi} (T) := \lbrace \lambda \in \mathbb{C} : T- \...
4
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1answer
118 views

Concrete example of non-nuclear operator $E \to F$ and isometry $F \hookrightarrow G$ so that the composition $E \to F \hookrightarrow G$ is nuclear

DISCLAIMER: I posted the same question a week ago on Mathematics Stack Exchange. We know by an abstract argument that there exist Banach spaces $E$, $F$, $G$ and maps $E \to F \hookrightarrow G$ such ...
5
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2answers
449 views

Compact operator without eigenvalues?

Consider the operator $M$ on $\ell^2(\mathbb{Z})$ defined by for $u\in \ell^2(\mathbb Z)$ $$Mu(n)=\frac{1}{\vert n \vert+1}u(n).$$ This is a compact operator! Then, let $l$ be the left-shift and $r$ ...
1
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0answers
38 views

A weaker weak time derivative than the one arising from Gelfand triples?

Let $V \subset H \subset V^*$ be a Gelfand/evolution triple where $V$ is a reflexive, separable Banach space and $H$ is a Hilbert space which has been identified with its dual. In the context of ...
3
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1answer
91 views

Is $C(\mathbb{R}^n)$ is a DF-Space?

I recently have begun reading about DF-spaces and its clear to me that $C(K)$ is a DF-space for any compact subset (non-empty) $K$ of some $\mathbb{R}^D$ for finite D, since $C(K)$ is Banach. However,...
2
votes
2answers
256 views

A norm inequality for operators

Let $A,B,C$ be self-adjoint operators of $L^2(\mathbb{R}^n)$ ($A$ and $B$ unbounded), $A\geq 0$, $B \geq 0$, with $\sqrt{A} C$ and $\sqrt{B} C$ bounded. Is the following inequality true for some ...
2
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0answers
47 views

$\|(A_n-z)^{-1} - (A-z)^{-1}\|\to 0\;\Rightarrow\; \|e^{-tA_n}-e^{-tA}\|\to 0$ for general $C_0$ semigroups?

In short, the question is whether norm-resolvent convergence implies operator-norm convergence of the assocoated semigroups. More specifically, assume the following: The $A_n$ generate contraction ...
5
votes
1answer
116 views

An extension of Lomonosov Theorem

Let $H$ be a complex infinite dimensional separable Hilbert space. There are various extensions of the following well known result: Theorem (Lomonosov): Every nonscalar $T \in B(H)$ which commutes ...
7
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2answers
232 views

Which complete orthomodular lattices arise from von Neumann algebras?

Let $A$ be a von Neumann algebra. Then a classic observation is that the set of projections $\Pi(A)$ is naturally a complete orthomodular lattice. Question 1: Is the construction $A \mapsto \Pi(A)$ a ...

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