All Questions
10,049 questions
-1
votes
1
answer
246
views
Determine the singular values of a compact operator in terms of the eigenvalues of an alternating tensor product of operators
Let $H$ be a $\mathbb R$-Hilbert space, $A\in\mathfrak L(H)$ be compact and $$|A|:=\sqrt{A^\ast A}$$ denote the square-root of $A$. By definition, the $k$th largest singular value $\sigma_k(A)$ of $A$ ...
- 167
-1
votes
1
answer
114
views
Construct a probability function on the operator monotone functions, $g(t)=t g(t^{-1})$, fitting certain values
To immediately pose the question of interest to us, without first expanding upon its (quantum-information-theoretic) origin—we seek a univariate function $f$, for which we have the ("two-qubit ...
- 723
-1
votes
1
answer
77
views
Parseval frame, convergence of $\sum_{k=0}^\infty \left\|g_k\right\|$ [closed]
Let $\mu$ be a Borel probability measure on $[0, 1)$, and $\{g_k\}_{k=0}^\infty$ be a Parseval frame for $L^2(\mu)$. Does
$$\sum_{k=0}^\infty \left\|g_k\right\|$$
converges?
- 297
-1
votes
2
answers
440
views
Motivation for weak solution of a PDE (initial condition)
The following question came to me when reading the famous paper of ALT and LUCKHAUS: "Quasilinear elliptic-parabolic differential equations"
When looking at a (nonlinear degenerate) PDE like
$$ \...
- 1
-1
votes
1
answer
152
views
Question regarding to the basis of L^p space via compact self adjoint operators. ( eg: inverse of -laplacian )
Do eigenfunctions of inverse of elliptic operator (eg: Laplacian) form basis of $L^P(\Omega)$ ? For p=2 we know the answer is yes, I am looking for p>2.
More generally, is it true that eigenfunctions ...
-1
votes
1
answer
696
views
Can singular measures be viewed as vanishing distributions? (Answer No!)
Hello,
Here is my original question: let $\mu$ be a singular measure with respect to the Lebesgue's measure on $R$. Is it true that $\int \psi \mu(d x)=0$ for any test function $\psi\in C_c^\infty(R)$...
- 1,649
-1
votes
1
answer
2k
views
Absolute values and Frobenius norm [closed]
The Frobenius, or Hilbert-Schmidt, norm of an $n$ by $n$ matrix $A$ is defined as $\|A\|_2 = \sqrt{\sum_{i,j=1}^n |A_{ij}|^2}$. The absolute value of $A$ is the unique positive matrix $|A|$ satisfying ...
- 1
-1
votes
1
answer
311
views
A differential equation
let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
$$\frac{dg}{ds}+(v\...
- 1
-2
votes
1
answer
423
views
Brouwer's theorem 2.0? [closed]
Let $f\in C([0,1]^n,\mathbb R^n) $ with $[0,1]^n \subset f([0,1]^n)$
Is it true that $\exists x \in [0,1]^n, f(x) =x$?
- 4,074
-2
votes
3
answers
850
views
Books on analytic functions on Banach spaces over a non-Archimedean field
I'm looking for good textbooks on analytic functions on Banach spaces over a non-Archimedean field.
If you know one(s), please let me know.
- 1,169
-2
votes
2
answers
1k
views
Are there examples of compact infinite dimensional manifolds? [closed]
Are there known examples of compact infinite dimensional manifolds?
The word "manifold" is important.
- 73
-2
votes
1
answer
1k
views
holomorphic extension of a function [closed]
hi,
I have the following question: let $U \subset \mathbb{C}^{n}$ be some open set containing zero. let $\tilde{U} = U \cap \mathbb{R}^{n}$. assume we have a real-valued analytic function $f : \tilde{...
- 29
-2
votes
1
answer
241
views
Does a group representation being transitive on a basis imply irreducibility?
Let $G$ be an infinite discrete group and $\pi$ a representation of $G$ on the Hilbert space $H$.
Suppose that the group representation is transitive on an orthonormal basis $B = \{e_j\}_{j=1}^{\infty}...
-2
votes
1
answer
3k
views
Separability of continuous functions with compact support [closed]
Hi,
is the space $C_0(\mathbb{R}^m)$, $m \in \mathbb{N}$ of continuous functions with compact support separable? If yes: where can I find a proof for that?
Please note: this is not a duplicate of ...
- 9
-2
votes
1
answer
665
views
weak convergence
I know the following result is true in the case of strong convergence. But I don't know whether it is true in the case of weak convergence also.
Let $p>1$. Suppose that each $x_n$ is a non negative ...
- 779
-2
votes
1
answer
143
views
Relationship between noncommutative torus for different values of theta [closed]
Let $u,v\in B(L_2(\mathbb T))$ defined as $u(f)(z)=zf(z)$ and $v(f)(z)=f(ze^{-2\pi i\theta})$ for $z\in\mathbb T$ where $\theta\in\mathbb R\setminus\mathbb{Q}$. Denote the $C^*$ algebra generated by $...
- 1,879
-2
votes
1
answer
216
views
Inverse of Sobolev interpolation inequality : $\lVert u \rVert_2 \lVert \Delta u \rVert_2 \leq C\lVert \nabla u \rVert_2^2$?
If $u : \mathbb{T}^3 \to \mathbb{R}$ is a smooth function on the $3$-dimensional torus $\mathbb{T}^3$, I wonder it is possible to reverse the Sobolev interpolation inequality in the sense that
\begin{...
- 3,477
-2
votes
1
answer
1k
views
Weak convergent $+$ strongly convergent subsequence $\Rightarrow$ strong convergence? [closed]
Let $X$ be a Hilbert space containing functions defined over a bounded region $\Omega\subset \mathbb{R}^N$. Assume $f_n\in X$ converges weakly to $f\in X$, and also has a strongly convergent ...
- 395
-2
votes
1
answer
99
views
A question on the zeros involving the equation containing exponential factor [closed]
I recently encounter a puzzle that: how to show that for any constant $c_1,c_2,c_3,c_4 \in \mathbb{R}$ the equation
$$c_1 e^t+c_2e^{-t}+c_3 e^{\alpha t}+c_4 e^{-\alpha t}=0$$
has at most only one ...
- 353
-2
votes
1
answer
802
views
No Hilbert space can have countable Hamel basis without using Baire's Category theorem [closed]
I want to prove that no Hilbert space can have countable Hamel basis just using the fact that any finite dimensional subspace is closed (more specifically without using Baire's theorem). I saw a paper ...
- 317
-2
votes
1
answer
3k
views
Multiplying two Fourier series gives one Fourier series, but what are the new coefficients? [closed]
If I have $A(x)=B(x) C(x)$ (sine periodic from 0 to 1) rewritten as
$\sum_n A_n \sin(n \pi x)=\sum_m B_m \sin(m \pi x)\sum_p C_p \sin(p \pi x)$
is there any easier way to compute $A_n$ from $B_m,...
- 149
-2
votes
1
answer
118
views
Mismatch between equivalent definitions of the Bohr compactification of the reals
I feel I'm overlooking something very silly.
The Bohr compactification of $\mathbb R$ has two equivalent definitions.
The set of (possibly discontinuous) homomorphisms $\mathbb R \to \mathbb T$ under ...
- 1,955
-2
votes
1
answer
217
views
If a continuous function is differentiable at a point, is it differentiable in some neighborhood around that point? [closed]
This seems like it should be true but I was wondering if anyone could prove it. Thanks!
- 109
-2
votes
1
answer
138
views
Weak center is same as center for $C^{\ast}$-Algebra? [closed]
Let $A$ be a $C^{\ast}$-algebra. We say $A$ is weakly commutative if $ab^*c=cb^*a$ for all $a,b,c \in A$ and define weak center of $A$ as $$Z_w(A)= \{ v \in A : av^*c=cv^*a \;\forall a,c \in A \}.$$
...
- 1,115
-2
votes
1
answer
147
views
Asymptotics for certain integrals
I stumbled on the following problem, if you can see a way through it.
Let $x$ be a real variable and fix a real value $\frac14\leq\nu\leq\frac34$.
QUESTION. For $x\rightarrow0$, does there exist a ...
- 43.2k
-2
votes
2
answers
325
views
$f\in (W^{1,p}(\Omega)\cap C(\Omega) \cap L^{\infty}(\Omega))\setminus C(\bar{\Omega})$, $f=0$ on $\partial \Omega$ imply $f\in W^{1,p}_{0}(\Omega)$?
Q1:
Let $p\geq 1$, and let $f\in W^{1,p}(\Omega)\cap C(\Omega)$. Assume also
$f\in L^{\infty}(\Omega)$ and $f=0$ on $\partial \Omega$. Is it true that
$f\in W^{1,p}_{0}(\Omega)$ even if $f\notin C(\...
- 852
-2
votes
1
answer
146
views
a measure convolution equation
My question is:
Given a function $f$ in the Schwartz class, we are looking for a measure $\mu$ which is a solution of the convolution equation: $f = e^{-|.|^2/2} \ast \mu$, where $e^{-|.|^2/2}$ is ...
- 367
-2
votes
1
answer
80
views
Density property for Sobolev spaces
My question is as follows: is the space $ C_c^{\infty}(\mathbb{R}^3 \setminus \mathcal{C}) $ dense in $ H^1( \mathbb{R}^3) $ where $ \mathcal{C} $ is the circle $ \{(x,y,z) \in \mathbb{R}^3 \mid x^2 +...
- 1
-2
votes
1
answer
314
views
Series representation for $\log(|\zeta(\frac{1}{2}+it)|)$
(Question is short and straight-forward. )
What is/are "nice and non-trivial" series representation/s of $\log(|\zeta(\frac{1}{2}+it)|)$ ??
By "nice and non-trivial" I mean contains no ...
- 375
-2
votes
1
answer
158
views
About local maxima of multivariable polynomials
Lets say I have a real valued function which is writable as a polynomial in terms of Frobenius norms of a pair of matrices as in it is of the form, $f_B(A) = f(||A||_F^2, ||AB||_F^2, ||A^TAB||_F^2)$ ...
- 2,246
-2
votes
1
answer
193
views
Analysis of Sobolev spaces [closed]
I just wanted to know wthether the following is OK or not.
Let $X$ be $H_0^1(\Omega)\bigcap L^{\infty}(\Omega)$, thought of as a subspace of $H^1_0(\Omega)$ and endowed solely with the usual $H^1$ ...
- 157
-2
votes
1
answer
295
views
When does the adjoint operator map closed convex subsets to closed convex subset?
Let $T:X\rightarrow Y$ be a linear continuous map between Banach spaces $X$ and $Y$ and denote by $T':Y'\rightarrow X'$ the norm adjoint of $T$. Let $M\subseteq U'$ be a subset of
the unit sphere $U'$ ...
- 101
-3
votes
1
answer
315
views
Are the injective functions dense in $C([0,1]^n,\mathbb R^n) $?
Let $n\geq 2$. Are injective functions dense in $C([0,1]^n,\mathbb R^n) $ with the uniform norm?
- 4,074
-3
votes
1
answer
232
views
Function satisfying $f(x)^{f^{-1}(x)}=x^2$ with $ f^{-1}$ is a compositional inverse of $f$ and $f:\mathbb{R+}\to \mathbb{R+}$?
Let $f$ be a function such that :$f:\mathbb{R+}\to \mathbb{R+}$ and $f^{-1}$ is a compositional inverse of $f$ , I have tried to find solution of the following functional $f(x)^{f^{-1}(x)}=x^2$, I ...
-3
votes
2
answers
768
views
Question on Linear Operators
Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is:
$$ \forall v \in V \quad \...
- 741
-3
votes
1
answer
634
views
compactly supported harmonic functions [closed]
Do a significant class of compactly supported smooth functions u on Ω⊂Rn such that Δu≥0 exist?
Thanks!
- 25
-3
votes
1
answer
392
views
A generalization of Chebyshev's sum inequality
From some my previous questions here and here and well-known rearrangement inequality. I pose an inequality as follows and I am looking for the proof or a reference.
Inequality: Let $y=f(x,y)$ is ...
- 1,776
-3
votes
1
answer
76
views
Minimal norm problem with linear combination of translation operator to be estimated
Follow up question from this one
Suppose $X = L^2(G)$, where $G$ is some locally compact group. Let $x, y \in G$ I for fixed $n$ I am seeking for an operator $H \in B(X)$ of the form
$$
H = H(\alpha_1,...
- 115
-3
votes
1
answer
63
views
How to show $\lambda_i \in \sigma_A(x)$?
Let $\sigma_A(x)$ be the spectrum of $x$ in $A$, and linear functional $\phi$ satisfying $\phi(x)\in \sigma_A(x)$ for every $x \in A$, consider $p(\lambda)=\phi((\lambda e-x)^n)$, and denote its roots ...
- 33
-3
votes
1
answer
320
views
Is :$\frac{\Bbb d}{\Bbb d x}$ a chaotic operator in infinite-dimensional Hilbert space? [closed]
I proposed this question in SE but no answer ,may I have a problem in my question, I would like to know when $\frac{\Bbb d}{\Bbb d x}$ does chaotic operator in Hilbert space ?
Let $H$=$L^2(\mathbb R)$...
-3
votes
0
answers
43
views
Do the domains of the two square roots of a positive (unbounded) operator coincide? [closed]
Let $H$ be a Hilbert space and $D:\mathrm{Dom}(D) \to H$ a densely defined operator on $H$. We further assume that $D$ is closed and self-adjoint. If we further assume that $D$ is positive, then we ...
- 415
-3
votes
0
answers
68
views
Exercise generalizing (related to) Hölder's inequality
I came across this exercise and feel absolutely stuck:
Let $p, q, r \in (1, \infty]$ be such that $1/p + 1/q = 1 + 1/r$. Suppose that $F : \mathbb{R}^n \times \mathbb{R}^n \to \mathbb{R}$ satisfies ...
- 1
-3
votes
1
answer
451
views
Exponential decay of kernel
Let $A: \ell^2 \rightarrow \ell^2$ be a bounded operator given by
\begin{equation}
(Au)(\alpha) = \sum_{\beta}A(\alpha,\beta)u(\beta)
\end{equation}
where $\left|A(\alpha,\beta) \right|\le Ce^{-|\...
- 11
-4
votes
2
answers
530
views
Inverse square-law as a positive definite kernel?
Newtons law for gravity states that:
$$F_{12} = \frac{G m_1 m_2} {|x_1-x_2|^2}$$
The function :
$$k(x,y):=\exp(-| x-y|^2)$$
is known to be a positive definite function, called the RBF-kernel.
It ...
- 3,064
-4
votes
2
answers
286
views
Does the Laplacian commutes with the indicator function [closed]
We define the laplacian operator $\Delta$ with the Neumann boundary conditions on the space $H^2(\Omega)$, where $\Omega$ is an open set of $\mathbb{R}^n$ with a smooth boundary $\partial\Omega$, and ...
-4
votes
1
answer
200
views
How I can choose $(t_1,t_2,...,t_{r}) \in (0,1)^{r}$ such that $f^{(k)}\left(1-2\prod_{j=1}^{k}t_{j}\right)=0$?
Let $f:\mathbb{R} \to \mathbb{R}$ be a real analytic function. Assume that $f$ has simple trivial zeros at each nonpositive integer. Then, all the $k$-th derivatives $f^{(k)}$ of $f$ have necessarily ...
- 1,197
-4
votes
1
answer
370
views
Is delta function symmetric against real axis? [closed]
Is $\delta\left(a+bi\right)=\delta\left(a-bi\right)$?
I wonder whether Dirac Delta (as defined via Fourier transform) is symmetric against the real axis.
We can write Delta function as
$$\delta(z) = \...
- 10.1k
-4
votes
1
answer
144
views
Coordinate free computation of the second derivative of a functional [closed]
Let $F(g(f))$
be a functional that sends functions of a vector variable (from $n$-dimensional vector space) to $\mathbb R$.
$g$
is some function of scalar valued functions $f$.
I'm interested in a ...
- 1
-6
votes
1
answer
180
views
An analog of Anderson's result in C* algebra setting [closed]
Let $\mathcal{A}$ be a unital $C^{*}$-algebra and $S(\mathcal{A})$ denote the states space of $\mathcal{A}$.
For $a\in \mathcal{A}$ , define $W(a) =\{\phi(a):\phi\in S(\mathcal{A})\}$
It's known that $...
- 307