Questions tagged [fa.functional-analysis]

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

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63 views

The space of harmonic functions on an open set is infinite dimensional? [closed]

I want to prove that he space of harmonic functions on an open set $\Omega \subset \mathbb{R}^n $ , with $n \geq 2$, is uncountablely infinite-dimensional. I guess that I have to find a linearly ...
2
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1answer
59 views

Uniform integrability contradicts convergence to $L^2$ subspace

The following question was asked at https://mathoverflow.net/questions/361367/uniform-integrability-contradicts-convergence-to-l2-subspace : Let $V$ be a finite-dimensional subspace of $L^2(\...
2
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1answer
63 views

Approximation of vectors using self-adjoint operators

Let $T$ be an unbounded self-adjoint operator. Does there exist, for any $\varphi$ normalized in the Hilbert space, a constant $k(\varphi)>0$ and a sequence of normalized $(\varphi_n)$ such that $$...
1
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0answers
54 views

G-abelian systems

Let $(\mathfrak{A},\alpha,\phi)$ be a $C^*$-dynamical system made of a unital $C^*$-algebra, a $*$-automorphism and an extremal invariant (i.e. ergodic) state. Consider the covariant GNS ...
4
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0answers
56 views

Characterization of “PSD-Squared” Matrices

$\DeclareMathOperator\DNN{DNN}\DeclareMathOperator\CP{CP}$This question can be thought of as an offshoot of this MO question from a few months ago. Let $M_n(\mathbb{C})$ denote the set of $n \times n$ ...
2
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0answers
52 views

General construction of enveloping C*-algebra, left/right-regular representation, etc

In a number of contexts (e.g. groups, crossed products, groupoids, Fell bundles) there are similar constructions of enveloping C*-algebras and left/right-regular representations that incorporate ...
2
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0answers
41 views

Holomorphic semigroups vs analytic semigroups

Is there any difference between the two notions in the theory of semigroups? In the literature, we find some monographs use the farmer while others use the latter. I expect that they are always the ...
6
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3answers
707 views

Are nuclear operators closed under extensions?

Given $X_i, Y_i$ Banach spaces, $f_j, g_j, T_i$ bounded linear operators for $i=1,2,3$ and $j=1,2$. We have the following diagram $\require{AMScd}$ \begin{CD} 0 @>>> X_1 @>f_1>> X_2 ...
4
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1answer
121 views
+50

Riesz transform of fractional operators

I am interested in Riesz transforms linked to the fractional Laplacian and other fractional operators. I have been hunting down in the literature to find related results but I have not been able to ...
2
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1answer
47 views

Classes of Banach algebras (that aren't operator algebras) whose bidual comes from a “universal representation”

Are there any classes of (Arens regular) Banach algebras that are not operator algebras whose bidual comes from a “universal representation”, as in the case of C*-algebras?
3
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1answer
74 views

Generalized tensor-train decomposition

If $U\in\bigotimes_{k=1}^p\mathbb R^{n_k}$ and $U^{(k)}\in\mathbb R^{r_{k-1}}\otimes\mathbb R^{n_k}\otimes\mathbb R^{r_k}$ with$^1$ $$U_{i_1,\:\ldots\:,i_p}=\sum_{j_0=1}^{r_0}\cdots\sum_{j_p=1}^{r_p}U^...
1
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1answer
58 views

Eigenspace of Gaussian Markov operator

Consider the (one-dimensional) Gaussian distribution $Q := N(\nu,\tau^2)$ and the (Gaussian) Markov operator \begin{equation*} \begin{array}{rccc} R : & L_1(\mathbb{R},\mathcal{B}(\mathbb{R}),Q) &...
0
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0answers
38 views

Tensor contraction (vector-valued trace) on $\bigotimes_{i=1}^k\mathcal L(E_{i-1},E_i)$

If $E_i$ is a $\mathbb R$-vector space, then the vector-valued trace $\operatorname{tr}_{E_1}:(E_2\otimes E_1^\ast)\otimes(E_1\otimes E_0)\to E_1\otimes E_0$ (or tensor contraction) is the ...
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0answers
51 views

partial differential inequality [duplicate]

I want to prove that if $f\leq l$ and $\lim_{a\to \infty}\frac{f(a)}{l(a)}=1$ (we can also suppose that $\lim_{a\to \infty}f(a)=\lim_{a\to \infty}l(a)=1$) where $f$ is a positive continuous bounded ...
5
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2answers
175 views

Surjection in compact-open topology [closed]

Let $Z$, $X$ and $Y$ be topological spaces and let $f:X\to Y$ be a continuous surjection then is the induced map $g \to f\circ g$ from $C(Z,X)$ to $C(Z,Y)$ is continuous. But is it still a surjection?...
4
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4answers
171 views

On Köthe sequence spaces

I asked this a week ago at math.stackexchange, but without success. As far as I understand, there are several meanings of the notion of the Köthe sequence space, in particular, Hans Jarchow in his "...
4
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0answers
68 views

Smoothing continuous functions in metric space

Let $(X,\rho)$ be a metric space. For any $f:X\to\mathbb{R}$, define the local Lipschitz constant of $f$ at $x$ by $$ \Lambda_f(x) := \sup_{x'\in X\setminus\{x\}} \frac{|f(x)-f(x')|}{\rho(x,x,')} . $$...
2
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1answer
71 views

Extension of outer unit normal vector to interior

Suppose we have a bounded smooth domain $\Omega$ in $\mathbb{R}^n$, so there exists an outer unit normal vector field $\eta$ everywhere on the boundary. Can we extend it to the interior satisfying ...
1
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0answers
38 views

Operational quantities characterizing upper semi-Fredholm operators

An operator $T:X\rightarrow Y$ is said to be upper semi-Fredholm if its range is closed and its kernel is finite-dimensional. M. Schechter (1972) introduced a quantity $$\nu(T):=\sup_{\operatorname{...
1
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0answers
14 views

Show that a tensor-train is contained in a recursive sequence of subspaces

Let $p\in\mathbb N$; $n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$; $u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
4
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0answers
68 views

Equality in spectral inclusion theorem

I asked this question on Math SE but didn't receive any response. Let $(T_t)$ be a $C_0$-semigroup on a Banach space $X$ with generator $A.$ If $\lambda_0\in \mathbb{C}$ is such that $e^{\lambda_0 t}$...
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0answers
30 views

How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?

Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...
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1answer
44 views

Cores in the tensor-train decomposition

Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...
0
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0answers
32 views

A functional that occurs in Vlasov-Poisson equation

Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...
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0answers
49 views

Solving Problems of product Series [closed]

Is there any general method to solve various types of Product Series Problems including the pi product forms?
1
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1answer
64 views

Convergence of the regularized gradient of a Lipschitz function

Let $\varphi:\mathbb R^d\to\mathbb R_+$ be given as $$ \varphi(x) := \begin{cases} c\exp\big(1/(|x|^2-1)\big) & \mbox{if } |x|\le 1 \\ 0 & \mbox{otherwise}, \end{cases} $$ ...
3
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1answer
93 views

Is the closure of the ball of $1$-Lipschitz functions still equi-Lipschitz?

$\DeclareMathOperator\Lip{Lip}$Let $\Lip_0(\mathbb R^d)$ be the space of Lipschitz functions $f:\mathbb R^d\to\mathbb R$ vanishing at zero, i.e., $f(0)=0$, and equipped with the norm $\|f\|:=\|\nabla ...
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0answers
23 views

How to calculate score based on quantity and quality? [closed]

I want to give each user a correctness score based on number of times he/she was correct and total number of guesses that she made. One user might have 1 guess that was correct, I don't want to give ...
11
votes
1answer
293 views

Riesz–Markov–Kakutani representation theorem for compact non-Hausdorff spaces

Let $X$ be a compact Hausdorff topological space, and $\mathcal C^0 (X) = \{f:X\to\mathbb{R}; \ f \text{ is continuous }\}$. It is well known that for any bounded linear functional $\phi: \mathcal C^...
4
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1answer
111 views

Decay of Fourier coefficients of real analytic functions

I would like to have any suggestion/reference to the following question. I have a differential operator $\mathcal{L}$ with discrete spectrum defined on a a suitable Sobolev space on a domain $\Omega$, ...
0
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1answer
65 views

A time dependent variational problem coming from a second order linear PDE

Fix $u_0\in H^1(\Omega)$ and $f=f(x,y,t)\in L^2(\Omega\times [0,T])$ where $\Omega$ is a sufficiently smooth bounded domain in $\mathbb{R}^2$. Consider the problem of finding $u:\Omega\times[0,T]\to\...
4
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0answers
102 views

Continuous disintegration

Given a suitable Borel measure $\mu$ on a suitable topological space $X$ and a Borel function $\pi:X \to Y$, where $Y$ is another suitable topological space, the disintegration theorem gives a Borel ...
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0answers
40 views

Relationship between negative operator eigenvalues

Let $L>0$, $c \in (-1,1)$ and $\varphi \in H_{per}^{2}([0,L])$ be fixed. Define $w:= 1-c^2>0$. Consider the matrix operator $\mathcal{L}_{R}: H_{per}^{2}([0,L]) \times L_{per}^{2}([0,L]) \...
1
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1answer
43 views

Proof of universality of Toeplitz algebra

It is well-known that the Toeplitz algebra $\mathcal{T}$ (that I view as concrete subalgebra of $\mathbb{B}(\ell^2(\mathbb{N})$) is the universal algebra generated by an isometry, that is, for any $C^*...
2
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0answers
26 views

Extension Sobolev functions across of lower dimensional subset

This question may be well-known to experts, but I am trying to get myself a rigorous proof. Consider open set $\Omega=B^n_1(0)\setminus B_1^k(0)$ in $\mathbb{R}^n$. If function $u$ is in $H^1(\Omega)$,...
1
vote
1answer
92 views

A question on multiplicity of complex polynomial [closed]

This is not a research level question. But due to some reason I can't ask this question on Math Stack Exchange. So, I am asking this question here. By definition we know that we can measure the ...
2
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0answers
56 views

On norms of Boolean functions

Let $f: \mathbb{F}_2^n \rightarrow \{-1,1\}$ be a Boolean function, represented by a $N=2^n$ dimensional vector, $f \in \{-1,+1\}^N$. Define the Fourier transform of $f$ to be $\hat{f}$, where $$\hat{...
1
vote
1answer
41 views

Uniform boundedness of resolvents on the imaginary axis

Let $A \colon \mathcal{D}(A) \subset \mathbb{H} \to \mathbb{H}$ be a closed linear operator in a Hilbert space $\mathbb{H}$, which generates a $C_{0}$-semigroup. Suppose that in a $\varepsilon$-...
0
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0answers
31 views

How to find a sequence which converge to the fixed random variable

Suppose $\mathscr{A}$ is a convex set on $L^\infty(P)$, $\mathscr{B}$ is the closure of $\mathscr{A}$ under the topology $σ(L^\infty(P),L^1(P))$. For every $X\in\mathscr{B}$, can we find a sequence $\{...
2
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0answers
81 views

Example of a non-reflexive Banach space and two sequences

Let $(E,\mathcal {A}, \mu ) $ be a finite measure space and $X$ be a Banach space. The set of all Bochner-integrable functions from $E$ into $X$ is denoted by $\mathcal{L}_X^1$. If $X$ is reflexive, ...
0
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1answer
90 views

$ \overline{(A-A)}\cap\overline{B}(0,r)\text{ is weakly compact, }\forall r>0 $?

Let $X$ be a separable Banach space and $A$ is a subset of $X$ such that $$ A\cap\overline{B}(0,r) \text{ is weakly compact, } \forall r>0. $$ Can we say that : $$ \overline{(A-A)}\cap\overline{...
2
votes
1answer
71 views

Spectral representation of closed operators with finite spectral bound

Assume $A$ is a closed linear operator on a Banach space $X$ and is densely defined. Assume the spectral bound $s(A) = \sup\{Re\lambda: \lambda\in \sigma(A)\}$ is finite. For example, if $A$ is the ...
6
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0answers
111 views

Quotients of subspaces of $C(\alpha)$

A well known problem, attributed to H. P. Rosenthal, asks whether or not every quotient of $C(\alpha)$, $\alpha$ countable ordinal, is $c_0$-saturated. As it is known, $C(\alpha)$ are $c_0$-saturated ...
2
votes
1answer
66 views

On $s$-harmonic functions

Is this statement true? A bounded positive function $v\in C^{2}(\mathbb R^N)$ decaying to zero which is $s$-harmonic function ($s\in (0, 1)$) outside a ball behaves like $|x|^{2s-N}$ near infinity. ...
4
votes
1answer
90 views

Existence of an injective unbounded below operator

Given an infinite dimensional Banach space, can we always find an infinite dimensional Banach space $Y$ and an injective bounded operator $T:X\to Y$ such that $T$ is not bounded below? If $X^{*}$ is ...
0
votes
1answer
275 views

What is the relevant literature –if any– on real-valued functions on sets and their Boolean combinations? [closed]

As part of a project (https://arxiv.org/abs/2004.06745), I've constructed the following table, $\left( \begin{array}{ccc} \hline Constraint Imposed & Probability & Quasirandom Estimate \\ ...
7
votes
1answer
335 views

Basis vs Schauder basis in normed spaces

Following the conventions from Heil: "A Basis Theory Primer" and Albiac, Kalton: "Topics in Banach Space Theory", we might define a basis of an (infinite-dimensional) normed space $V$ as a sequence $(...
1
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0answers
66 views

What is the closure of this set in $H^1(\mathbb{R}^2)$?

I'm not sure that if this is a difficult question or not. I asked it on MSE and it hasn't been answered and so I thought I might ask it here: What is (or how can we describe) the closure in $H^1(\...
5
votes
1answer
230 views

Existence of injective compact operators

We know that if $X$ is a separable Banach space, then for every infinite dimensional Banach space $Y$, there exists an injective compact operator from $X$ to $Y$. My query is for every Banach ...
0
votes
1answer
153 views

$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\frac{(x-y)^2}{2\varepsilon}}l(y)dy\leq C\frac{1}{x}$

Let $l$ be a continuous bounded function ($l$ is not differentiable). I want to prove for $x$ large enough that $$\frac{\partial}{\partial x}\int_{\mathbb{R}}\frac{1}{\sqrt{2 \pi \varepsilon}}e^{-\...

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