Questions tagged [fa.functional-analysis]

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

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2
votes
0answers
93 views

An integral transform and the Stone-Weierstrass theorem

For a bounded function $\operatorname{F}: \mathbb{R}_{\,\ge\ 0} \to \mathbb{R}$ (not necessarily non-negative), if $$ \int_{0}^{\infty}\frac{x^{k}\,s}{(s^{2} + x^{2})^{\left(k + 3\right)/2}\,\,}\, \...
0
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1answer
49 views

Some estimates on tensor norms

Denote $M_n$ to be $n\times n$ matrix. For $X\in M_n$ define $\|X\|_1:=\max\limits_{1\leq j\leq n}\sum_{i=1}^n|x_{ij}|$ and $\|B\|_2:=\max\{|\sum_{i,j=1}^nb_{ij}x_iy_j|:|x_i|=|y_j|=1,\ 1\leq i,j\leq n\...
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votes
0answers
40 views

Density of a graded algebra

I'm trying to prove the following proposition: If $ v \in V $ and $ Y \in \mathfrak{so} (V) $ then $[\dot\mu(Y), B(v)] = B(Yv)$. By definition $[\dot\mu(Y), B(v)] = \dot\mu(Y)B(v) - B(v)\dot\mu(Y).$ I ...
3
votes
2answers
144 views

Vanishing convolution between density and compactly supported function

Find a pair of functions $f,g:\mathbb{R}\to\mathbb{R}$ such that: $f$ is smooth and compactly supported (say, on $[0,1]$ but this isn't crucial), $g(x)>0$ for all $x\in\mathbb{R}$, $\int g(x)\,dx=...
1
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2answers
73 views

'Partial boundedness' of continuously parametrised power series

Let $a_1, a_2, \ldots : D\rightarrow\mathbb{R}$ be a sequence of continuous functions, with $D$ a compact metric space. Suppose that the function $f : D\times[0,\infty)\rightarrow\mathbb{R}$ given by $...
2
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0answers
52 views

Convergence of random operators

I'm a statistician not versed in functional analysis and operator theory. I wish that I might not find a wrong place for my question. All my questions are trivial in the scalar time series case, but ...
4
votes
1answer
232 views

Derivative of trace

Consider two positive-semi definite matrices $T_1, T_2$ of unit trace. Let $T(\lambda):=T_1 + \lambda(T_2-T_1)$ be the convex combination of the two. We then study $f(\lambda) := \operatorname{tr}(T(\...
3
votes
1answer
84 views

Uniform boundedness principle for almost surely converging sequence of operators

I'd like to do the following: I consider a separable Banach space $X$ with a probability measure $\mu$ on the Borel $\sigma$-algebra $\mathcal B(X)$. Additionally, I have a sequence of measurable, ...
-1
votes
0answers
82 views

Lower bounds on norm of convolution operator

This seems like too easy of a problem to not have an answer, but I've been stymied so far: Given $\phi\ge 0$, $f\ge 0$, I am interested in lower bounds of the form $\Vert\phi*f\Vert \ge C\Vert f\Vert$,...
3
votes
1answer
56 views

G.L. l. u. st. for subspaces of Banach spaces with an unconditional basis

A Banach space $X$ has Gordon-Lewis local unconditional structure (G.L. l. u. st.) if for every finite dimensional subspace $E$ of $X$, the inclusion operator $i:E\to X$ factors through a finite ...
7
votes
1answer
175 views

Can the intersection of a unitary and an irreducibly represented injective $C^*$-algebra be $\{0\}$?

Let $\mathcal{A}$ be an injective $C^*$-algebra irreducibly acting on a Hilbert space $\mathcal{H}$, and let $\phi$ be a completely positive idempotent from $\mathbb{B}(\mathcal{H})$ onto $\mathcal{A}$...
6
votes
1answer
107 views

Mittag-Leffler function

Let the Mittaq-Leffler function be defined by the expression $$ E_{\mu,\nu}(z) = \sum_{k=0}^{\infty} \frac{z^k}{\Gamma(k\mu+\nu)}\quad \text{$\mu>0$ and $\nu\in \mathbb R$}$$ Now let $n\in \mathbb ...
2
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0answers
102 views

A slight generalization of Skorokhod's representation theorem

Let $f:\mathbb{R}^p\rightarrow\mathbb{R}^q$ $(p,q\geq 1)$ be a continuous function and $(X_n)_{n\geq 1}$ a sequence of random values on $\mathbb{R}^p$ such that $f(X_n)$ converges in law to a random ...
1
vote
0answers
29 views

Extending the Dirac operator on an open subset of a manifold and preserving positivity

Let $M$ be a spin manifold and $U\subseteq M$ an open ball. Let $D$ be the Dirac operator on $M$ with respect to some Riemannian metric $g$, acting on sections of the spinor bundle $S\to M$. Suppose ...
1
vote
1answer
126 views

$K$-convex Banach spaces

Let $X$ be a Banach space. We say that $X$ contains $\ell_1^n$'s uniformly iff for all $n\in\mathbb N$ there exist subspaces $X_n\subseteq X$ with $d(X_n,\ell_1^n)\leq \lambda$ for some $\lambda\geq 1$...
1
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2answers
86 views

Show convergence of net associated to GNS-triplet associated to state on a $C^*$-algebra

Let $A\subseteq B \subseteq B(H)$ be an inclusion of $C^*$-algebras where $H$ is some Hilbert space. We have the following conditions: B is a von Neumann algebra with $A'' = B$. The inclusion $A \...
1
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0answers
39 views

Regarding commutative C* subalgebra for every element [closed]

Let $A$ be a complex unital $\mathrm C^*$-algebra. Let $a\in A$ be any element. Can there be a possibility that there exists no commutative $\mathrm C^*$-subalgebra containing $a$. Or asking in the ...
0
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0answers
98 views

Determine the norm of a continuous linear operator $T:L^1[a,b]\to L^1[a,b]$ [closed]

I have encountered the following exercise: Let $K(x,y)$ be a measurable function on $[a,b]\times [a,b]$. The function $I:y\in [a,b] \mapsto \int_a^b |K(x,y)|\, \text{d}x \in [0,+\infty]$ belongs ...
3
votes
3answers
94 views

Bounded $r$- variation function with a dense set of local maximum values

This is a sharpening of the following problem: $C^1$ function with a dense set of maximum values. Problem set up: Let $f \colon [0, 1] \to \mathbb R$ be a function on the unit interval. We say $y \in \...
0
votes
0answers
15 views

Conditionally negative definite kernels and operators defined by bijections

In short: I want to know if a linear map given by a bijection on a set is bounded for the norm induced by a CND kernel, under certain hypotheses. Let $X$ be a countable set and $\kappa:X\times X\to\...
0
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1answer
66 views

Reference request: Is if possible to estimate the local behaviour of the solution of $\nabla \cdot a(x) \nabla f=g$ via constant coefficients?

Consider the divergence form uniformly elliptic operator $\nabla \cdot a(x) \nabla$ where the coefficient $a$ are smooth and bounded and $D$ is a bounded and smooth domain of $\mathbb R^d$ $$ \begin{...
17
votes
7answers
1k views

Nonseparable counterexamples in analysis

When asking for uncountable counterexamples in algebra I noted that in functional analysis there are many examples of things that “go wrong” in the nonseparable setting. But most of the examples I'm ...
3
votes
1answer
117 views

A convolution type singular integral operator with log

Define a convolution type operator $T_m$ by $$T_m(f) = p.v.\int_\mathbb{R}f(x-y)\frac{\log^m|y|}{y}dy.$$ Here $m\ge0$ is an integer. Consider $f \in H^s (s > 0)$ which is the usual Sobolev space. ...
13
votes
1answer
493 views
+250

Smoothness of distance function to a compact set

Fix a non-empty compact subset $K\subseteq \mathbb{R}^n$ and let $d_K(x):=\min_{z \in K} \,\|z-x\|$ be the map sending any $x\in \mathbb{R}^n$ to its distance from $K$. Suppose that: $K$ is regular : ...
3
votes
2answers
175 views

What is the distribution of the following limit?

Assume $x \in \mathbb{R}$. We already know that $$\lim_{\epsilon \to 0+} \frac{1}{x-i\epsilon} - \frac{1}{x+i\epsilon} = 2\pi i \delta_x.$$ Here $\delta_x$ denotes the Dirac distribution. If we ...
3
votes
0answers
78 views

Fourier transform without characters (Eigenfunctions of an operator)

Let's consider a very simple problem in quantum mechanics: We have, in $\mathbb R,$ a potential barrier of the form $$ V(x) = V_0 \mathbf 1_{[-a,a]}(x), $$ where $\mathbf 1_{[-a,a]}$ denotes the ...
2
votes
0answers
41 views

Eigenvalue estimates for kernel integral operator for Laplace kernel on unit-sphere in high-dimensions

Let $d$ be a large positive integer and let $S_{d-1}$ be the unit-sphere in $\mathbb R^d$ and let $K_\gamma:S_{d-1} \times S_{d-1} \to \mathbb R$ be defined by $K_\gamma(x,x') = e^{-\|x-x'\|_2^\gamma}$...
1
vote
0answers
115 views

For which Banach spaces is the self composition operator Lipschitz?

Let $X\subseteq \{f|f:D\rightarrow \mathbb{R}^n\}$ be a Banach space, with at least all polynomials on $D$ contained in $X$, where $D\subseteq \mathbb{R}^n$ is open and bounded. Let $U\subseteq X\cap \...
2
votes
1answer
127 views

Opposite $C^*$ algebras

$\DeclareMathOperator\op{op}$Let $A$ be a $C^*$-algebra. We know that $A$ admits a natural operator space structure, namely the operator space structure induced by any faithful $*$-representation of $...
2
votes
0answers
55 views

Reference request for type of specific integral equation in two variable:

Consider the following integral equation: $$\int_0^\infty K(t,y)\phi(t,x)dt=0$$ Here, $K(t,y)$ is a trigonometric kernel and $\phi(t,x)$ is monotonic wrt $x$ ( for fixed $t$). I want to find the ...
5
votes
0answers
118 views

Lavrentiev phenomenon between $C^1$ and $C^2$

Does there exist a (onedimensional) functional that exhibits the Lavrentiev phenomenon between $C^1$ and $C^2$ that is $ F(y)=\int_a^b f(t,y(t),y'(t))\,dt \hspace{1cm}$ or possibly $ \hspace{1cm} F(y)=...
2
votes
0answers
35 views

Necessary and sufficient conditions on kernels of trace-class operators

Question: Let $K \in L^2(R^n\times R^n)$. Are "explicit" necessary and sufficient conditions known such that $K$ is the kernel of some trace-class operator $A \in TC(L^2(R^n))$? We know that ...
2
votes
0answers
55 views

Convergence of operator in norm resolvent sense and their eigenvectors

Let $\{T_n\}_{n=1}^\infty$ and $T$ be (unbounded) self-adjoint operators and $T_n\to T$ in norm resolvent sense, that is, for some $z\in \mathbb{C} \setminus \mathbb{R}$, $\|(zI- T_n)^{-1}- (zI- T)^{-...
2
votes
0answers
75 views

Another question about asymptotic models in Banach spaces

The array $(x_{i}^{k})_{i=1,k\in\mathbb{N}}^{\infty}$ of normalized $M$-basic sequences in a Banach space $X$ is itself called $M$-basic if, for every $k\leq i_{1}<i_{2}<\ldots$, the diagonal ...
13
votes
1answer
280 views

Existence of a translation-invariant basis of $\ell^2$

This question is heavily inspired by this other one, but is meant to be a hopefully more accessible variant of it (and I think slightly more natural). I give four equivalent formulations of the same ...
3
votes
0answers
136 views

Reflexive subalgebras of $B(X)$

Let $X$ be a reflexive Banach space, and let $B(X)$ denote the set of all bounded linear operators $X\to X$. Does there exist a subalgebra $A\subseteq B(X)$ with the following properties? A is unital....
15
votes
2answers
866 views

How to generalize the various vector calculus theorems to distributions?

Here is a list of vector calculus identities; in the proof of these identities, we all assume that these functions are $𝐶^𝑘$ in an open set, and we usually use these identities to calculate ...
0
votes
0answers
24 views

Embedding theorems for fractional Sobolev spaces $W^{s,p}(\Gamma)$ where $\Gamma$ is closed piecewise $C^1$ curve in $\Bbb R^2$

I am interested in embedding theorems for the fractional Sobolev space $W^{s,p}(\Gamma)$ where $\Gamma$ is closed piecewise $C^1$ curve in $\Bbb R^2$ such as the boundary of a triangle or rectangle. ...
17
votes
3answers
665 views

Existence of translation-invariant basis on $C_c(\mathbb R)$

Consider the space $C_c(\mathbb R)$ of complex-valued continuous functions of compact support. This is a vector space over $\mathbb C$, and I am not considering any topology, so the question is ...
0
votes
1answer
25 views

Green function of symmetric stable process in dimension 1 and 2

Are the results in this paper on the Green function of a symmetric stable process available also in space dimension $d =1$ and $d=2$? The main theorems here are stated only for $d \ge 3$.
2
votes
1answer
156 views

Reference request for semilinear PDEs in dimension 2

I am interested in the study of the (semi-linear, I suppose) equation $$\begin{cases}-\Delta u(x,y)+q(x)u(x,y)+h(x)=f(u(x,y)-kx),\;\;(x,y)\in\Omega,\\ u=g,\;\;\;\text{on }\partial\Omega.\end{cases}$$ ...
4
votes
0answers
176 views

Must this subset of a C*-algebra vanish?

Let $X$ be a compact, Hausdorff, topological space and let $B$ be a unital C*-algebra containing $C(X)$, and sharing units with the latter. In other words $$ 1_B\in C(X)\subseteq B. $$ For each $...
3
votes
2answers
229 views

Are equicontinuous function dominated by a continuous function?

Let $f_n: [0, 1] \to \mathbb R$ be an equicontinuous sequence of functions. Does there exist a continuous function $f$ that dominates $f_n $ in the following sense? We say $f$ dominates the sequence $...
0
votes
0answers
99 views

Extension of a Hilbert basis

The picture below is taken from this paper: http://real.mtak.hu/22877/. The authors claim that the basis of $H^2(\Omega) \cap H^1(\Omega)$ denoted by $\lbrace w_i \rbrace _{i \geq 1}$ can be extended ...
3
votes
0answers
126 views

Density of signed measures in dual space

Let $B$ be a Banach space of functions on a Radon space $X$. By the Hahn-Banach theorem, we know that the canonical evaluation map is isometric. That is, for every $f \in B$, we have $$\|f\| = \sup_{\...
5
votes
0answers
59 views

Homotopy, contraction mapping and the inverse function theorem on Banach spaces

We all know the "open mapping" part of the inverse function theorem on Euclidean spaces can be proved by either contraction mapping (iterations), or homotopy methods (degree theory / ...
3
votes
2answers
270 views

Convex set with no interior contained in hyperplane?

Let $K$ be a convex set in a normed space $X$. Assume that $int(K)=\emptyset$ (norm topology). Must $K$ be contained in some (affine) hyperplane? It's fairly easy to see that this is true in $ℝ^n$, ...
2
votes
2answers
175 views

Decay estimate of Fourier transform of a compactly supported function

Assume $f(x), x \in \mathbb{R}$ is a function with a compact support such that its Fourier transform $\hat{f}(\xi)$ has a decay rate $$\hat{f}(\xi) \lesssim \frac{1}{|\xi|^\gamma + 1}$$ for some $\...
0
votes
1answer
59 views

Interpolated Sobolev norm inequality

Let $\Omega \subset \mathbb{R}^n$ be a bounded Lipschitz set, and let $W^{k,p}(\Omega)$ denote the usual Sobolev space with $k \in \mathbb{N}$ being the order of the derivatives and $p \in [1, \infty)$...
1
vote
1answer
102 views

Young's convolution inequality for weighted norms

Young's convolution inequality states that, for $1/p+1/q=1/r+1$ ($1\leq p,\, q, r\leq \infty$), we have $$\lVert f * g \rVert_r \leq \lVert f\rVert_p \lVert g\rVert_q.$$ It is implicit here that the ...

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