All Questions
10,049 questions
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Does $L^p$ contractivity imply $L^p$ dissipativity?
Does $L^p$ contractivity of an operator semigroup imply the $L^p$ dissipativity of the operator ?
Thank you in advance !
- 1
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54
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On cyclicity of a module
Let $A$ be a $\text{ von Neumann algebra }$, $\mathcal{H}$ is a cyclic $A$ module, $G$ be a finite group acting on $A$, is $\mathcal{H}$ cyclic module over fixed point subalgebra of the action? ...
- 677
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78
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$p$-summing operators space is a Banach space
Let $X,Y$ be Banach spaces and $p\geq 1$. A bounded linear operator $T$ is called $p$-absolutely summing, if there is exist $K>0$, such that for all $n\in N$ and $x_1,\dots, x_n\in X$:
$$
\left(\...
- 1
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59
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Nests on Banach spaces and their duals
Let $X$ be a Banach space and $\mathcal{E}$ a nest on $X$.
Take $f\in X^{*}$ and suppose:
$N \in\mathcal{E}$ is the largest element of the nest so that $f \in N^\bot$
$N=\bigcap_{M>N}M$
Is there ...
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155
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Does the set of positive definite kernels on some set X form a ring?
Given some non-empty set $X$, does the set of positive definite kernels on $K_X$ form a ring with pointwise addition and multiplication. I am convinced it does not as surely if $k \in K_X$ then we ...
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63
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Coarea-like formula for BV functions (not their derivative)
Let $\Omega \subset \mathbb R^N$ and $f \in BV(\Omega)$. The coarea formula states that
$$Df = \int_{\mathbb R} D \chi_{\{f >h\}} \, dh.$$
Unfortunately, the formula
$$f = \int_{\mathbb R} \...
- 839
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153
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Equivalent Definitions of Gaussian Process?
The Gaussian process $\{X_t\}_{t \in T}$ ($T=[0,1]$ for example) is usually defined using its finite-dimensional distribution. I came across this statement many times: linear operator (not necessarily ...
- 103
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109
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Elliptic equation with Neumann boundary condition: RHS in $L^2$ implies solution in $L^\infty$?
Consider the homogeneous Neumann problem $$-\Delta u + ku = f$$ $$\partial_\nu u = 0$$
on a smooth, bounded domain $\Omega$.
If $f \in L^2(\Omega)$, do we obtain the regularity $u \in L^\infty(\...
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294
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If $A$ is a dissipative self-adjoint operator with spectral decomposition $(H_λ)$, then $e^{tA}x$ tends to the projection of $x$ onto $H_0$ as $t→∞$
Let $(T(t))_{t\ge0}$ be a strongly continuous contraction semigroup on a $\mathbb R$-Hilbert space $H$ with dissipative self-adjoint generator $(\mathcal D(A),A)$. In particular, $T(t)$ is self-...
- 167
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326
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Measurability of the heat semigroup in $L^\infty$
Let $S(t)$ be the $C_0$-semigroup generated by the Laplacian operator with Dirichlet boundary condition in $L^2(\Omega)$, where $\Omega$ is a bounded open subset of $R^n$.
It is known that $S(t)$ ...
- 1
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156
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Function classes with high Rademacher complexity
My question is two fold,
Is there any general understanding of what makes a function class have high Rademacher complexity? (Sudakov minoration would say that one sufficient condition for a class of ...
- 2,246
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172
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Compact operator
Let $k:[0,1]^2 \to [0,1]$ be a measurable function. Define $K:L^2([0,1])\to L^2([0,1])$ to be the operator:
$$
(Kf)(x) = \int_0^1\int_0^1 f(z) k(x,y) \mathbf{1}_{x\leq z\leq y} \ \mathrm{d}z \mathrm{d}...
- 31
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56
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linear functions/hyperplanes vs. convex functions/convex sets in Hilbert space
The simplest Hahn-Banach extension theorem in Hilbert space $X$ avoids the use of the axiom of choice by virtue of the Riesz representation theorem. But what about the version of the theorem where the ...
- 1,461
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105
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Spectrum of Dirac sequences
Let $\delta_n\in C^0_c(\mathbb{R})$ be a Dirac sequence approximating the Dirac delta "function" $\delta$ with support in $0\in \mathbb{R}$. Then, for each $n$ we have a compact operator $K_n:L^2(\...
- 99
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65
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$\overline{conv}(C)$, where $C = \{ e _{1}, \cdots e _{n} \}$, $e _{i} \in \ell ^{p, \infty}$ is diametral
Let $C = \{ e _{1}, \cdots e _{n} \}$, where each $e _{i}$ are unit vectors in $\ell ^{p, \infty}$, and $1 < p < \infty$. I want prove that $\overline{conv}(C)$ is diametral. My doubt is: $\...
- 23
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63
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Feller semigroups and fractional operators
Have Feller semigroups been used to investigate the properties of the Cauchy problem associated with the fractional Laplacian (just like they have been used to study local degenerate second order ...
user60665
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273
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Local "boundary comparison principle" for harmonic functions
Let $u$ be a positive solution of the elliptic equation $\mathcal Lu = 0$ on $B^+_1 \subset \mathbb{R}^n$ such that $u$ vanishes continuously on $\{x_n = 0\}$. To fix ideas, we may take $\mathcal L = ...
user123456
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117
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Harnack Inequality for uniformly elliptic PDE via constructing a singularity
I am trying to prove a Harnack inequality for a nonnegative subsolution $u \in H^1(B_2)$ to the PDE $\text{div}(A Du) \ge 0$, where $A = A(x)$ is uniformly elliptic. The proof outline I am following ...
- 1
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170
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Strong continuity (weak to strong) of $\langle Au,v\rangle=\int u^3 v dx$
I am currently trying to figure out the following. If I consider the Sobolev space $W^{1,p}_0$ is it possible to show that the operator given by
$$\langle Au,v\rangle=\int u^3 v dx$$
is strongly (weak ...
- 101
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115
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If two spheres are isometric, does there exist a bijective isometry $T:S\to S$ with $\|Tu-\alpha Tv\|_Y \leq \|u-\alpha v\|_X$ for all $\alpha>0?$
Let
$$(S,\|\cdot\|) = \{(x,y)\in \mathbb{R}^2: \|(x,y)\| =1\},$$
that is, $S$ is the collection of all norm one vectors in $\mathbb{R}^2$ with respect to the norm $\|\cdot\|.$
Question: Let $\|\...
- 623
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119
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Holder-Sobolev type inequality
Let $U$ be a bounded subset of $\mathbb{R}^n$. Let $p>n$. Let $W^{2,1}(U_T)$ be the Banach space of functions $u:U\times[0,T]\rightarrow\mathbb{R}$ with the norm $\|u\|_{W^{2,1}(U_T)}=\sum_{2s+|\...
- 307
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113
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Conditions for the embedding of the space $L^\infty(I, W^{1,2}(U))$ into $L^\infty(I \times U)$
Let $I$ be a compact interval of $\mathbb{R}$ and $U$ be a bounded subset of $\mathbb{R}^2$.
If $f \in L^\infty(I, W^{1,2}(U))$, what (non-trivial) condition ($L^p$-estimate on $f$ or decay-like ...
user123457
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169
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Is there an option to handle Neumann-series when it diverges? (using infinite-sized Carleman matrices)
(I asked this in MSE but did not find resonance, there is also a relation to an older discussion here on summability see here and a followup formulating an $\text{ais}()$ already here)
...
- 5,342
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170
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What functions can one try employing to fit an apparently doubly-periodic real function over $[0,1]$?
I have a cosine-like data curve over $x \in [0,1]$ that I can rather well-fit by
a function of the form $a \cos{2 \pi x} +b$. Although good, the fit is still lacking,
in that the residuals from the ...
- 723
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101
views
Reference Request: Egoroff Theorem for nets
Does there exist a generalization of Egoroff theorem for nets instead of sequences of functions?
- 5,405
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75
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Dense Egoroff theorem
Suppose that $f_n:X\rightarrow V$ is a sequence of continuous functions from a compact metric space $X$ to a Banach space $V$ and let $\mu$ be a Radon measure on $X$ and $\epsilon>0$ be given.
...
- 5,405
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93
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Regarding $\ell_p$ direct sums
I am reading this paper by S.H Karin titled Norm attaining operators and pseudospectrum.
In page 2 he gives the definition of $l_p$ direct sum of a family of Banach spaces as follows:
If $1\leq p< \...
- 175
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79
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Iterative methods for minimizing sequences
Let $\mathbb{X}$ be a Banach space equipped with some norm $||\cdot||_\mathbb{X}$ and $F:\mathbb{X}\to\mathbb{R}$ be some linear functional. Suppose we are given a set $A\subseteq\mathbb{X}$ which is ...
- 364
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87
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Uniform convergence in Hadamard derivatives
Let $T\colon X \to Y$ be a nonlinear operator between Hilbert spaces which is Lipschitz and is Hadamard differentiable. It satisfies
$$T(x+th)=T(x) + tT'(x)(h) + r(t)$$
where $r(t)=r(t,x,h)$ is the ...
- 73
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237
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Spectrum of a Hamiltonian on the real line
Consider the following linear (Hamiltonian) operator on functions on the real line $\mathbb{R}$
$$H\psi(x)=-\frac{d^2}{dx^2}\psi(x)+V(x)\psi(x).$$
Assume that $V$ is a smooth function and $V(x)\to +\...
- 21.8k
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72
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weakly amenable weighted sequence algebras
Let $v=(v_n)_{n\in\mathbb{N}}$ be a positive weight with $\inf_nv_n>0$ (for convenience we may take $v_n\geqslant1$). Then $\ell_{\infty}(v)$ is a Banach algebra with coordinate-wise multiplication....
- 375
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161
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Topologies corresponding to norm, SOT and WOT under duality
This is a question from MSE which has not received any attention so far.
Let $X$ be a Banach space with norm dual $X'$. (I am mostly interested in the case $X = \ell^1$.)
For a linear mapping $T : X \...
- 1,773
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113
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Verifying that a map to $L^2_{\text{loc}}$ is continuous
Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the ...
- 1,903
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977
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Weak convergence can imply strong convergence [duplicate]
In $\ell^1(\mathbb N)$, weak convergence implies strong convergence. Is there a classification of infinite-dimensional Banach spaces for which such a property holds true ?
- 16.2k
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60
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Solution of a functional equation with cosine transform
What are the functions verifying:
$$\int_0^{\infty} f(t) \cos(2\pi xt)=\lambda \frac{1}{x} f(\frac{1}{x})$$
With $\lambda$ a constant ?
(Functions $x^{-\alpha}$ with $0<\alpha<1$ are solutions ...
- 1,199
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324
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Adjoint of differential equation
Motivation: Consider the ODE
$$y'(t)=Ay(t)$$ then it is true that the flow satisfies $\Phi(t)y_0=e^{tA}y_0$ and the adjoint of the flow is a solution to the adjoint equation
$$y'(t)=A^*y(t).$$
I ...
- 83
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70
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A question about an irreducible ultra-power
Let $A$ be a Banach algebra and $E$ be an irreducible Banach $A$-module. Is there a countably incomplete ultra filter $\mathcal U$ on $\mathbb N$, the set of natural numbers, such that the ultra power ...
- 2,118
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299
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When convolution with exponential kernel is bounded
Let $g(t)=e^{-\omega t}$, $\omega>0$. What is, in terms of well-known function spaces, the space $X$, $L_{loc}^2(0,\infty)\subset X$, of all functions $f:\mathbb{R}^+\to \mathbb{R}^+$, satisfying
$...
- 395
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55
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Continuity of a composite function
Let $n=2$ or 3 and let $\Omega$ be a bounded domain of $\mathbb{R}^n$. Let $T>0$ and $f \in L^2([0,T],H^1(\mathbb{R}^n))$.
Is the mapping
\begin{equation}
\begin{array}{rcl}
C^0([0,T],C^1(\bar{\...
- 143
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58
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Dense integer translates of a real-valued function with unequal limits at infinity
This is a follow up on a Previous question. Let $W$ be the space of continuous functions $f:\mathbb{R} \rightarrow \mathbb{R}$ such that $$\lim_{x\rightarrow \infty} f(x)=0~\mbox{and}~\lim_{x\...
- 537
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194
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Estimate of a Sobolev function after a change of variables
Let $n>0$ and $\Omega$ be a bounded domain of $\mathbb{R}^n$. Consider a smooth enough mapping $\Phi$, from $\Omega$ into $\Phi(\Omega)\subset\mathbb{R}^n$, that is orientation-preserving and ...
- 143
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328
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Find the trace for some elements in group algebra
Let $K=\langle b,c,d\mid b^{2}=c^{2}=d^{2}=bcd=1\rangle $. Now we consider $$D=K*\mathbb Z/2\mathbb Z=\left\{a,b,c,d\mid a^{2}=b^{2}=c^{2}=d^{2}=bcd=1\right\}$$ where $*$ is the free product. Then we ...
- 407
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89
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Hausdorff methods of summation
From the book of Boss "Classical and modern methods in summability":
"The class of Hausdorff methods includes the Hölder, Cesaro and Euler methods. A large number of other matrix methods which play ...
- 109
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90
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criterions for polar set of Feller processes
Suppose $X_t$ is the solution to
$$
d X_t=b(X_t)dt+dL_t,\quad X_0=x.
$$
where $L$ is a rotational symmetric $\alpha-$stable process with $\alpha\in (0,1]$, $b$ is Lipchitz.
Assume $\Gamma\subseteq ...
- 515
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65
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Does $\{ x^* \circ \psi_t:x^*\in ext(E^*), t\in K \}\subset ext(X^*)$ hold?
Notations: Let $K$ be a locally compact Hausdorff space and $E$ be a real normed linear space.
Recall that $C_0(K,E)$ is the set of $E$-valued continuous functions $f$ on $K$ such that $f$ vanishes at ...
- 623
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57
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A question on order unbounded sequences in Banach lattices
Let $E$ be a Banach lattice. It is well-known that every norm convergent sequence in $E$ admits an order convergent subsequence and hence admits an order bounded subsequence. But it seems that a norm ...
- 3,343
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719
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Green's functions/fundamental solution to a non-constant coefficients pde
We already know the relationship between Green's function and solution to elliptic partial differential equation, i.e $$u(y)=\int_{\partial \Omega}u\frac{\partial G}{\partial n} ds+\int_\Omega G\Delta ...
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298
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Generalization of the Chinese remainder theorem
Let $A$ be a Banach algebra and $\{I_{\alpha}\}_{\alpha}$ be a collection of closed two-sided pairwise coprime ideals of $A$. Is the Chinese remainder theorem true for $A$ and $\{I_{\alpha}\}_{\alpha}$...
- 565
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467
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Intersection of two subspaces of a Hilbert space
Background:
Let $D$ be a Klein Four group and consider free product $Z/2Z\star D=<a,b,c,d|a^{2}=b^{2}=c^{2}=d^{2}=bcd=1>$. Now we consider group algebra generated by $Z/2Z\star D$ with inner ...
- 407
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0
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56
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Existence of a couple of functions solution of a differential equation (with additional constraint)
I would like to know if we can find a real function $v(x)$ and a complex function $f(x)$, such that they solve the following differential equation (with $\alpha$ a complex, $0<Re(\alpha)<1$):
$$...
- 1,199