Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
6,405
questions
-1
votes
0answers
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
votes
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
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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
votes
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 ...
-4
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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 $$\nu(T):=\sup_{\operatorname{...
1
vote
0answers
14 views
Show that a tensor-train is contained in a recursive sequence of subspaces
Let
$p\in\mathbb N$;
$n_k\in\mathbb N$ and $\left(e^{(k)}_1,\ldots,e^{(k)}_{n_k}\right)$ denote the standard basis of $\mathbb R^{n_k}$ for $k\in\{1,\ldots,p\}$;
$u\in\bigotimes_{k=1}^p\mathbb R^{n_k}...
4
votes
0answers
68 views
Equality in spectral inclusion theorem
I asked this question on Math SE but didn't receive any response.
Let $(T_t)$ be a $C_0$-semigroup on a Banach space $X$ with generator $A.$ If $\lambda_0\in \mathbb{C}$ is such that $e^{\lambda_0 t}$...
0
votes
0answers
30 views
How are the vector-valued trace and the unique linearization of $\mathfrak L(X,Y)\:\hat\otimes_π\:X→Y$ of $\mathfrak L(X,Y)×X→Y,\;(L,x)↦Lx$ related?
Let $X$ be a $\mathbb R$-Banach space and $X'\:\hat\otimes_\pi\:X$ denote the completion of the tensor product of $X'$ and $X$ with respect to the projective norm. The trace functional $\operatorname{...
0
votes
1answer
44 views
Cores in the tensor-train decomposition
Let $d_i\in\mathbb N$, $I_i:=\{1,\ldots,d_i\}$ and $u\in\mathbb R^{d_1}\otimes\mathbb R^{d_2}\otimes\mathbb R^{d_3}$. It's somehow clear to me that we may regard $u$ as a three-dimensional array (see ...
0
votes
0answers
32 views
A functional that occurs in Vlasov-Poisson equation
Let me share a functional that pops up in the analysis of the Valsov-Poisson equation (see the motivation below). At given time, the macroscopic mass density is $x\mapsto\rho(x)\ge0$. Assuming finite ...
-1
votes
0answers
49 views
Solving Problems of product Series [closed]
Is there any general method to solve various types of Product Series Problems including the pi product forms?
1
vote
1answer
64 views
Convergence of the regularized gradient of a Lipschitz function
Let $\varphi:\mathbb R^d\to\mathbb R_+$ be given as
$$
\varphi(x) := \begin{cases}
c\exp\big(1/(|x|^2-1)\big) & \mbox{if } |x|\le 1 \\
0 & \mbox{otherwise},
\end{cases}
$$
...
3
votes
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 ...
-2
votes
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
votes
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
votes
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
votes
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 ...
-1
votes
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
vote
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
votes
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
votes
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
votes
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
votes
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
votes
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
votes
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
vote
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^{-\...
