Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
0
votes
0answers
36 views
Non Normal operator [on hold]
Standard example for non normal operator is the shift operator. It is continous but the image of the left shift is not dense. Can we have an example of a non normal operator $A$ which is continuous ...
1
vote
0answers
108 views
Sum-epimorphisms and prod-monomorphisms
Sum-epimorphisms
A longer time ago I have introduced the bi-onto maps for the topological category. Let me formulate here its general categorical definition:
...
0
votes
1answer
68 views
How to minimize this sparse quadratic function?
There is a problem when I'm reading a paper.
Equation:
$min_p|p-p^*|^2+\alpha |R(p)|^2 + \beta |D(p)-\delta|^2$,
where $p, p^*, R(p), D(p), \delta$ are all $M\times N$ matrices, and $p^*, R(), D(), ...
0
votes
0answers
66 views
Inductive/Projective Limits of Topological Algebras
It is common to form inductive/projective limits of Banach/Frechet spaces in order to come up with natural topologies for common vector spaces. For instance,
For $k \ge 0$ and $K_n$ compact ...
0
votes
0answers
22 views
Interchange summation and differentiation [migrated]
I asked this question already on math.stackexchange, but did not receive any answers
see here
Let $f = \sum_{n=0}^{\infty} a_n e_n $ where $e_n$ are an ONB of $L^2[0,1].$
Now assume we have that ...
6
votes
0answers
219 views
A matrix trace inequality
The well-known Powers-Stormer inequality says the following: for positive semidefinite operators $A, B$, we have that $\mathrm{Tr}((A - B)(A - B)^\dagger) \leq \| A^2 - B^2 \|_1$, where $\| \cdot ...
2
votes
0answers
70 views
Multivariate Maximal Hilbert Transform
One way to define the maximal Hilbert transform of a function, $f$, is by
$$\mathcal{H}[f](x):=\underset{\varepsilon>0}{\sup}\;\int_{|t|\geq\varepsilon}\dfrac{f(t)}{x-t}dt,\quad x\in\mathbb{R},$$
...
11
votes
1answer
220 views
Ultraweak topology on B(X): Is the map X\otimes X* -> B(X)* isometric?
Let $X$ be a Banach space. Consider the map
$$
\alpha\colon X\hat{\otimes} X^* \to B(X)^*,
$$
defined one simple tensors as
$$
\alpha(\xi\otimes\eta)(a) = \eta(a(\xi)).\quad (\xi\in X, \eta\in X^*, ...
5
votes
1answer
143 views
Analytic perturbation of eigenfunctions
Consider a domain $\Omega_0 \subset \mathbb{R}^n$, and deformations of $\Omega_0$, called $\Omega_t$, obtained by a one-to-one mapping $x \mapsto x + t\varphi (x)$, where $\varphi$ is smooth. It is ...
3
votes
1answer
64 views
Composition of spectral measures
Let $f: \mathbb{R}\rightarrow \mathbb{C}$ be a measurable function, $H$ some Hilbert space and
$$ f_E := \int_{\mathbb{R}} f dE$$ for some spectral measure $E$.
Now, my question is: When do we have ...
1
vote
1answer
63 views
About preserving real-rootedness of multivariable polynomials
Say $f_i(z_1,z_2,..,z_m)$ are polynomials real rooted in the $z$s for a bunch of polynomials indexed by $i$. When can one say that $\sum_{i} p_i f_i(z_1,z_2,..,z_m)$ is also real rooted?
If ...
2
votes
1answer
128 views
Is this structure a Banach bundle?
Let $X$ be a Banach space. Put $Y=\{ \phi\in X^{*}\mid\;\; \parallel \phi \parallel\leq 1\;\; \&\;\; \phi \neq 0\}$ which is a locally compact Hausdorf space with the weak star topology.
...
1
vote
0answers
90 views
A continuous choice of invertible elements
Let $A$ be a simple unital $C^{*}$ algebra with invertible elements $G(A)$. Assume that $A^{*}$ is its dual space, which is equipped with the weak star topology.
Is there a continuous map ...
1
vote
0answers
46 views
Strong solution to parabolic equation without differentiability assumption on coefficient?
Consider on $(0,T)\times \Omega$, $\Omega$ a bounded domain
$$u_t(t,x) - a(u(t,x))\Delta u(t,x) = f(t,x)$$
$$u|_{\partial\Omega} = 0$$
where $a$ is real-valued and satisfies
$C_1 \leq a(r) \leq C_2$ ...
0
votes
0answers
40 views
Searching for conditions?
I have this operator $$Au(t)=\int_0^1 G(t,s) f(s,u(s)) ds$$defined from $H^1_{0}$ to $H_0^1$ and satisfy the problem: $$\begin{cases} -(Au)''(t)=f(t,u(t)), t\in[0,1]\\Au(0)=Au(1)=0\end{cases}$$
Where ...
-3
votes
0answers
56 views
Example of topological vector space [closed]
Can Someone provide me any link or research paper about the working of definition of topological vector space(open neighborhood definition) on R to become a vector space?
5
votes
1answer
83 views
Inequality of the norm of the convolution in $L^p(\mathbb{R}^n)$ with symmetric decreasing rearrangement?
Is it true that
$$
||f*g||_p \le ||\,|f|^* * |g|^*||_p\quad ?
$$
where $|f|^*$ and $|g|^*$ are the symmetric decreasing rearrangements of the functions $|f|$ and $|g|$. Under what conditions on $f$ ...
0
votes
0answers
73 views
Explicit formula for Bergman kernel on the unit ball
On page 173 in Krantz's book "Explorations in Harmonic analysis" in the proof of Lemma 7.1.21 there is a part that I really don't understand. What I don't understand is why is ...
1
vote
1answer
127 views
Multiplication of generalized functions
I would like to know if there is any associative algebra $( A(\mathbb{R}), +, \cdot )$ such that:
EDIT: Fan Zheng found an inconsistency in my requirements. Therefore I changed the function space in ...
2
votes
1answer
144 views
Asymptotic behaviour of eigenvalues
If you look at $-\Delta + q$ on the sphere in $\mathbb{R}^3$ for example and $||q|| < \infty,$ is there a way to asymptotically describe the behaviour of the eigenvalues? Probably they behave ...
2
votes
1answer
103 views
Compact radial Sobolev embedding $H^1_{rad}\hookrightarrow L^p$
I want to show:
Let $N\geq 2$ and $2< q <2^\ast$. Then the embedding \begin{align}
H^1_{\text{rad}}(\mathbb{R}^N)\hookrightarrow L^q(\mathbb{R}^N)
\end{align}
is compact.
I was able to show ...
1
vote
0answers
62 views
Weak (Sobolev) derivative and the Frechet derivative (chain rule) [closed]
Let $A: H^s(\Omega) \to H^1(\Omega)$ be a bounded linear map. Let $u \in H^s(\Omega)$. Let $f:\mathbb{R} \to \mathbb{R}$ be nonlinear and Lipschitz such that $f(u) \in H^s(\Omega)$.
Is it possible to ...
3
votes
2answers
179 views
Does this C*-algebra embed into a simple nuclear C*-algebra?
Let $\mathcal K$ denote the C*-algebra of compact operators, and fix an embedding $\phi_n:M_n(\mathbb C) \to \mathcal K$ for each $n\in \mathbb N$. Define the C*-algebra
$A := \{(a_n)_{n=1}^\infty ...
1
vote
0answers
43 views
Fractional Poincare inequality on closed manifold
Let $u \in H^{\frac 12}(M)$ on a compact closed Riemannian manifold. Can someone refer me to a source where the inequality
$$\lVert u - \bar u \rVert_{L^{2^*}} \leq C|u|_{H^{\frac 12}}$$
is proved, or ...
0
votes
0answers
17 views
Solution of non-linear Fredholm (Hammerstein) equation with non-degenerate kernel and reciprocal non-linearity
I have asked this question at MSE but got no response. I have rephrased it so that anyone who knows operator theory and integral equations could help me out... I faced a problem in physics which is a ...
0
votes
0answers
51 views
Recontruction of the weak topolgy from the scalar product on a subset of a Hilbert Space
Let $M$ be a set a let $K:M\times M\to\mathbb{C}$ be a positive definite kernel. By a version of Moore-Aronszajn Theorem, there is a unique (up to the unitary euivalence) Hilbert Space $X$, and a map ...
2
votes
1answer
111 views
Does $\int \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \to \Phi(u)$ imply that $f_t \to \delta_1$?
I'm looking at a family $(f_t)$ of densities of some continuous random variables and know that
$$\int_{-\infty}^{\infty} \Phi \left( \frac{u}{\xi} \right) f_t(\xi) \mathrm{d} \xi \xrightarrow{t \to ...
0
votes
0answers
39 views
Interpolation of Banach spaces: theta = 0, 1
Let $A_1 \subset A_0$ be Banach spaces with continuous embedding. Is $(A_0, A_1)_{i, \infty} = A_i$ for $i =0, 1$, with equivalent norms? Here, $(\cdot, \cdot)_{\theta,p}$ denotes the $K$-method of ...
1
vote
0answers
90 views
A Characterization of Closed Ideals in $C^{\infty}(\mathbb{R}^n)$
The space $C^{\infty}(\mathbb{R}^n)$ can be turned into a topological ring using the Whitney topology. Whitney's Spectral Theorem says that the closure of an ideal in this ring is the ideal of all ...
1
vote
0answers
24 views
Topological tensor products of spaces of holomorphic functions of slow growth
Let $X$ be a Banach space, $M$ be a complex manifold, and $\Omega$ a relatively compact domain in $M$. We consider the space $\mathcal{A}^{-\infty}(\Omega, X)$ of $X$-valued holomorphic functions of ...
0
votes
0answers
48 views
Hilbert scales of covariance operators
Assume we have 2 covariance operators(positive definite trace class) $S$ and $T$ on Hilbert space $\mathcal H$ with corresponding eigenpairs $\{e_j,\lambda_j\}$ and $\{f_j,\lambda_j\}$. Assume that
...
0
votes
0answers
63 views
Condition for boundedness in stationary phase theorem
I am trying to understand theorem 7.7.1 in Hormander's Analysis of linear partial differential operators, vol.1.
Let $K \subset \mathbb{R}^n$ be a compact set, $X$ an open neighborhood of $K$ and ...
2
votes
0answers
107 views
Generalising the $L^2$-Norm to Compact Quantum Groups
For a compact quantum group (in the sense) of Woronowicz, we have a noncommutative $C^*$-algebra replacing ${\mathbb C}(G)$. This $C^*$-algebra is endowed with a linear functional generalizing the ...
0
votes
0answers
52 views
Specific optimization problem solution procedures
Is there a standard procedure to solve following two optimization problems?
$$\mathsf{Problem\mbox{ }I}:\mbox{ }\min_{A\in\{0,1\}^{n\times n}:rk(A)=r}\mbox{ }\max_{R,S\in\Bbb R^{n\times ...
0
votes
0answers
28 views
Causal (Volterra type) differential equation with local Lipschitz condition
Consider the equation
$$
u'(t) = (Fu)(t)
$$
where $F \colon L^2(0,T;\mathbb R^n) \to L^2(0,T;\mathbb R^n)$ is a causal (Volterra type)
nonlinear operator. It means that the value of $(Fu)(t_0)$ ...
2
votes
0answers
44 views
Hilbert c*-module over approximately finite c*-algebra
Is there a construction of a Hilbert c*-module over an approximately finite c*-algebra using Hilbert c*-modules over finite algebras? How do we get a Hilbert c*-module over an inductive limit of such ...
1
vote
1answer
100 views
Identifying the weak limit of a gradient (Bochner spaces)
Let $X=L^2(0,T;L^2(\Omega))$ for an unbounded domain $\Omega$. Let $f_n, f:\mathbb{R} \to \mathbb{R}$ be functions with $f_n \to f$, $f_n(0)=f(0)=0$ and $f_n$ Lipschitz with Lipschitz constant ...
1
vote
0answers
74 views
How do functions operates in a Fourier algebra $A^{q}(\mathbb T)$?
We put , $A^{q}(\mathbb T)= \{ f\in L^{q}(\mathbb T): \hat{f}\in \ell^{q}(\mathbb Z) \}.$
By Helson-Kahane-Katznelson-Rudin Theorem, it follows that,
"Let $F$ be a function on $\mathbb C$ and if ...
2
votes
0answers
81 views
Equivalence of two non-degenerate Gaussian measures on Banach space
The motivation of this question is to show that two probabilities on
$C_{0}^{n}(0,1)$ (the space of continuous $\mathbb R^{n}$ valued process
on $[0,1]$ starting from zero) induced by two ...
1
vote
0answers
70 views
Normal points of an operator and discrete eigenvalues
Let $\mathcal{H}$ and $\mathcal{L}({\mathcal{H}})$ denote a separable Hilbert space and the set of bounded linear operators on it respectively.
As a graduate student entering the field of ...
1
vote
0answers
88 views
Limit of a simple function including a zero of the Riemann Zeta function
Lets consider :
$$F(x)= \sum_{n\in\mathbb{N}} n^{-s_0} e^{2i\pi nx}$$
This function is well defined for $x>0$ (Abel summation formula proves it) and I would like to show that if $s_0$ is a zero ...
0
votes
0answers
61 views
Must the Lebesgue measure of a $\rho$ - neighbourhood of an $(n-2)$ - dimensional set be at least $c\rho^2$?
The Lebesgue measure of a $\rho$-neighbourhood of a point in $\mathbb{R}^2$ is of course equal to $c\rho^2$. Similar such considerations in higher dimensions lead me to the following question:
Given ...
1
vote
1answer
120 views
Infinite sum of asymptotic expansions
I have a question about an infinite sum of asymptotic expansions:
Assume that $f_k(x)\sim a_{0k}+\dfrac{a_{1k}}{x}+\dfrac{a_{2k}}{x^2}+\cdots$
with $a_{0k}\leq \dfrac{1}{k^2}$, $a_{1k}\leq ...
3
votes
2answers
110 views
Positive definiteness of infinite tridiagonal matrices
I am interested in the following problem: I have an infinite symmetric tridiagonal matrix
$$
A=
\begin{bmatrix}
a_1 & b_1 & & & \\
b_1 & a_2 & b_2 & & ...
3
votes
0answers
49 views
Equivalence of fractional Sobolev space defined through Gagliardo norm and interpolation; dependence on the domain
Let $\Gamma$ be a smooth hypersurface in $\mathbb{R}^n$. We can define the fractional Sobolev space
$$X = \{ u \in L^2(\Gamma) \mid |u|_X^2 := \int_\Gamma \int_\Gamma \frac{|u(x)-u(y)|^2}{|x-y|^{n+1}} ...
4
votes
1answer
172 views
Is the space of Radon measures a Prohorov space?
Consider the spaces $C_c(\mathbb{R})$ of compactly supported continuous functions equipped with the inductive limit topology and the Banach space $C_0(\mathbb{R}) = \overline{C_c(\mathbb{R})}^{\, ...
5
votes
1answer
165 views
Are polynomials dense in holomorphic $L^p(\mathrm{Gauss})$ for $p < 1$?
Let $\mu$ be standard Gaussian measure on $\mathbb{C}^n$, i.e. $d\mu = \frac{1}{(2 \pi)^n} e^{-|z|^2/2}\,dz$, and fix $0 < p < 1$ (note carefully).
Suppose $g$ is holomorphic on ...
0
votes
0answers
78 views
Second variation of the domain functionals [closed]
The Eigen value $\lambda(t)$ which is characterised by the Rayleigh quotient (where $t$ is a scalar variable):
$$R(u,\Omega_t)= \frac{\int_{\Omega_t} |\nabla u|^2 dy }{\int_{\Omega_t} u^2 dy}$$
...
1
vote
0answers
190 views
On a property of Riemann Zeta function zeros
Lets consider the function : $$F(x) = \sum_{n=1} (xn)^{-s_0} e^{-nx} $$
with $s_0$ a zero of the Riemann Zeta function in the critical strip.
This sum is well defined for $x \in \mathbb{R}^{+*}$. It ...
1
vote
0answers
94 views
Eigenvalue of product of self adjoint compact operators
Suppose A is a self adjoint $m \times m$ real matrix with eigenpairs $\{e_j, \lambda_j\}$ such that $\lambda_j > \lambda_{j + 1}$. Let $B$ be another self adjoint real $m \times m$ matrix such that ...
