All Questions
10,050 questions
1
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378
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Adjoint operators in LCS
Before my main question let me start with the following notions.
Let $X$ and $Y$ be locally convex spaces and let $T \colon X \rightarrow Y$ be a linear mapping. The adjoint of $T$ is an operator
$T^...
- 145
4
votes
0
answers
109
views
How fast is discrete-time diffusion on a continuous set?
This question is inspired by Joseph O'Rourke's beautiful answer to my previous question.
Let $\mathbb{S}^{d\times n}$ denote the set of real $d\times n$ matrices whose columns have unit norm and sum ...
- 7,633
2
votes
1
answer
208
views
Is there an elementary proof for preserving inequalities under the change of l_p metrics?
Here is what I mean exactly:
Let $A=(a_1,a_2)$ and $B=(b_1,b_2)$ be two points in the real plane (for simplicity, but general finite dimensions would also be nice), and define the $\ell_p$-metric as ...
2
votes
2
answers
303
views
Characterisation of positive elements in l¹(Z)
Consider the Banach $^* $-algebra $\ell^1(\mathbb Z)$ with multiplication given by convolution and involution given by $a^*(n)=\overline{a(-n)}$.
I would like to find nice necessary and sufficient ...
- 3,184
5
votes
0
answers
157
views
Containment of an element to an operator system
This question will probably appeal to people in operator systems theory as it is very much related. However, I'm interested in down-to-earth concrete systems with finite dimensional Hilbert space ...
- 315
8
votes
0
answers
349
views
Finding a dimension-free bound for a certain multiplier on Euclidean space
The following question is indirectly motivated by strong type maximal function estimates. Let $f\in L_{p}(\mathbb{R}^{n})$. For $\xi=(\xi_{1},\ldots,\xi_{n})\in\mathbb{R}^{n}$ define $m(\xi)$ so ...
- 141
2
votes
0
answers
327
views
Generalizations of Kato-Rosenblum theorem?
The Kato-Rosenblum theorem says that if $H_0, H$ are self-adjoint operators on a Hilbert space such that the difference $H-H_0$ belongs to the trace class, then the strong limit of $\exp(itH)\exp(-...
- 21
2
votes
1
answer
272
views
Contractions and spaces
Suppose $X$ is a closed subspace of an $L^{1}$-space and $X$ is isometric to another $L^{1}$-space. Then we know that $X$ is in the range of a contractive projection on the $L^{1}$-space. Is there any ...
- 95
1
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0
answers
369
views
Infinite internal direct sums of subspaces
Given a compact Hausdorff space $K$ such that $C(K)$ is of density $\omega_1$. Suppose that every copy of $c_0(\omega_1)$ in $C(K)$ is complemented. Let $\{Y_\alpha\colon\alpha<\omega_1\}$ be a ...
1
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0
answers
187
views
Injective modules over Fourier algebra
Is there any article on injective modules over Fourier Algebras?
Do we have anything about injectivity of $A(G)$ as a $A(G)$-bimodule?
- 71
2
votes
1
answer
413
views
General Sobolev Inequalities
In Partial Differential Equation by Lawerence Evan p284 there is this theorem stated:
Let $U$ be a bounded open subset of $\mathbb{R}^n$ with $C^1$ boundary. Suppose $u\in W^{k,p}$ then if $k>n/p$ ...
- 3,217
1
vote
1
answer
275
views
Shift operator that generates separable orbit
Suppose, that $f$ is bounded measurable function, $T_h(f)(x) = f(x+h)$ is the shift operator.
How to prove, that if the whole orbit $T_h(f):\, h\in\mathbb{R}$ has a dense, countable subset $T_{n_k}(f)$...
- 696
2
votes
1
answer
373
views
Strong measurability reference
I'm reading a book on Lyapunov Exponents by Lian and Lu in which they refer to strong measurability of operator-valued maps. They define this by saying an operator valued map $T:\Omega\to L(X,X)$ is ...
- 23.2k
3
votes
0
answers
385
views
Off-diagonal asymptotic expansion of the Bergman kernel on hyperbolic Riemann surfaces
Let $X$ be a compact Riemann surface of genus at least 2. Let $K$ denote the canonical line bundle, and $E$ be any vector bundle. Let $P^{(m)}$ be the projection map from the space $L^2(X,K^mE)$ of ...
user2529
2
votes
0
answers
242
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Core of divergence form operator with unbounded coefficient
Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of
$$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$.
I also assume that $a(x)...
- 617
1
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0
answers
460
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Topology for test functions [closed]
One naive way to define a topology on test functions ${\mathcal D}(\Omega)$ would be to exhaust $\Omega$ by compacts $(K_n)$ and to take the metric induced by the semi-norm system
$$
{\| f \|} _ {n} :=...
- 53
0
votes
1
answer
384
views
spectral measure
how to calculate spectral measure for a given normal operator for example right shift operator?
- 1
1
vote
0
answers
100
views
Conditions on a measure to satisfy certain relation on moments.
Suppose we have a measure $\mu$ on $\mathbb R_+$ such that $\forall s>-1$ $t^s\in L^1(\mathrm d\mu(t))$.
I'd like to impose some conditions on $\mu$ so the function
$$f:p\to \frac{\int_0^\infty t^...
- 173
0
votes
0
answers
65
views
Interpolation with time continuity
If $u(x,t)$ is a function depends on $x\in\Omega$ and $t\in[0,T]$. The following result could be found in L.C. Evans's book "PDE".
Suppose $u\in L^2(0,T;H_0^1(\Omega))$, with $u_t\in L^2(0,T;H^{-1}(\...
- 103
1
vote
0
answers
103
views
Generalized bilinear estimates
Hello. Let $ X^{s,b} $ be the Bourgain space generated by $ \tau - \xi^3 $. It is proved that, for $ s\in (-\frac{1}{2}, 0] $, we have
$$
\|(u^2)_x\| _{X^{s,b'-1}} \leq c \|u\|_{X^{s,b}} \|u\|_{X^{-...
- 425
2
votes
0
answers
262
views
A specific projection and compactness on the Bargmann-Fock space
Let $F_2$ be the Bargmann Fock space defined as the space of entire functions $f$ on $\mathbb{C}$ such that \begin{align*} \int_{\mathbb{C}} |f(z)|^2 e^{- |z|^2} dA(z) \end{align*} ($dA$ is just ...
0
votes
0
answers
137
views
$\mathcal{D}(0,T;V)$ is dense in $W(0,T)$
Where can I find a proof that $\mathcal{D}(0,T;V)$ (the space of $V$-valued compactly supported functions on $[0,T]$) is dense in the space $W(0,T)$, where $$W(0,T) := \{ u \in L^2(0,T;V) : u' \in L^2(...
- 113
1
vote
1
answer
224
views
Can symmetrizing a contraction increase the speed of convergence?
Dear community,
I have a problem which is very simple to state but seems to be hard to answer.
Statement of the problem
Let $f$ and $g$ be two symmetric, real functions in $n$ and $m$ variables, ...
- 199
1
vote
0
answers
104
views
On generalization of Wigner semi circle
I want to analyse noise model for a matrix M whose entries are not real numbers. The matrix is a collection of N permutation matrices of size nxn i.e, M is NnxNn. Because its a collection of ...
0
votes
0
answers
298
views
High dimensional beta integral (question following the previous post)
Hello,
This post is a question following the previous post. In one dimensional case, we have
$$
\int_0^x |y|^{1-\alpha} |x-y|^{1-\beta} d y = \frac{\Gamma(\alpha)\Gamma(\beta)}{\Gamma(\alpha+\beta)} |...
- 1,649
3
votes
0
answers
217
views
Is this integral operator about Stokes' Flow compact?
Consider the following integral operator $\mathcal{A}$ on [EDITED: continuous vector function $f=(f_i):\partial S\to{\mathbb R}^3$]:
$$
({\mathcal A}f)_j(x_0):=\int_{\partial S}\sum_{i=1}^3 f_i(x)G_{...
user14319
2
votes
0
answers
290
views
Consequence of Modified Young's inequality
Let $f\in L^1(\mathbb R^n)$. Define operator $T_f(g)=|f|\ast g$ for functions $g$ on $\mathbb R^n$. The set of measurable functions $f$ on $\mathbb R^n$, such that $T_f$ is bounded from $L^p(\mathbb R^...
- 415
1
vote
0
answers
400
views
Uniqueness of differential adjoint operator
I wonder if someone can help me on what is, probably, a simple question but is baffling me at the moment!
In standard texts on functional analysis, something like the following is written
Let $L\...
- 11
1
vote
1
answer
219
views
fourier transform of cumulative function
Hi
I've encountered a test that uses the cumulative value of a finite time series to deterime the data set's stationarity.
I would like to know the characteristics of this test in frequency space,...
- 11
4
votes
0
answers
140
views
When is $A^*A$ invertible for Banach space?
Let's consider a linear functional $A$ from smooth objects to smooth ones. It is first order operator in the sense that it extends to be a map from $W^{k+1,p}$ to $W^{k,p}$. Assume that we have $L^2$ ...
- 527
1
vote
0
answers
121
views
showing convergence of a function recursion relation
I have obtained (formally) a perturbative solution
$$
H(y) = \sum_{n=0}^\infty \delta^n H_n(y)
$$
to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...
- 351
1
vote
0
answers
466
views
Bounding point-wise maximum of the absolute difference of two convex functions
Let $\Delta: R \times R \rightarrow R_{+}$ be a positive and convex function (convex in, say, both the arguments) called the loss function.
Let $x \in R^d$. Moreover, let $H_1,...,H_r$ be sets of ...
- 11
2
votes
0
answers
366
views
Are affine continuous functions on Bauer sub-simplices of the probability measures given by integration over continuous functions?
Let $X$ be a compact (non-metrizable) Hausdorff space and $\mathcal{P}(X)$ the set of Radon probability measures with
weak-$*$ topology (weak topology induced by the continuous functions).
Consider a ...
- 888
0
votes
1
answer
319
views
Hilbert space automorphisms realized as induced by transformations of some base-spaces
Following question may be soft. Fix abstract hilbert space H and consider any automorphism A in banach-spaces sence (i.e. no conditions on metric). Call A is realizable if exist measure space $(X,\mu)$...
- 789
1
vote
0
answers
57
views
Looking for CDFs that I can integrate a particular transformation of
I need two CDFs $G$ and $\lambda$ with unbounded support such that I can integrate
$$ \int_{-\infty}^t \lambda(a(x+b))dG(x), $$$a>0,b\in\Re$. As far as I can tell, there exist no functions that ...
- 11
13
votes
0
answers
564
views
Symmetric (extended) Haagerup tensor product
Given a von Neumann algebra M, then the weak$^*$ (or extended) Haagerup tensor product of M with itself is the collection of $\tau\in M\overline\otimes M$ with $$\tau=\sum_i x_i\otimes y_i$$ the sum ...
- 18.7k
0
votes
2
answers
146
views
representation of compact supported distribution
Is this true?
Any compact supported distribution can be represented as finite sum of partial derivatives of functions.
- 9
1
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0
answers
202
views
Weak solution of a certain pde with integral term
Let us consider the following pde on the domain $(0,T)\times(0,1)$
$
\dot{p}(t,x)+v(t)p_{x}(t,x)+v'(t)\int_{0}^{1} \rho(t,s)p_{s}(t,s)\ ds=0
$
with initial data $p(0,x)=p_{0}(x)$ and boundary data $...
- 11
1
vote
1
answer
154
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Question about coercivity of a functional
Hi!
Let $(M,g)$ be a compact Riemannian manifold without boundary of dimension $2m$. Let
$$T:W^{2,2}(M)\rightarrow L^{2}(T^{*}M\otimes TM)$$
be a second order, linear, differential operator (...
- 1,727
0
votes
1
answer
142
views
A special Integral Kernel
Does there exist either one / general class of non-negative definite , symmetric Integral Kernel map satisfying the following properties ??
$f(x)=(Kg)(x)=\int_{\Omega}K(x,y)g(y)dy$
$K:L^2(\...
- 17
0
votes
0
answers
35
views
Approximate rank of the set formed by all delayed replicas of a bandlimited signals between 0 and T
Given a complex-valued signal with a certain delay $s(t-\tau)$ for which we sample $N$ instants
$$
\mathbf{s(\tau)}=\left[s(0-\tau),\ldots,s\left(\frac{N-1}{f_s}-\tau\right)\right]^T
$$
at Nyquist ...
- 345
0
votes
0
answers
164
views
Can we separate Toeplitz matrices for negative and positive eigenvalues?
Consider a Toeplitz matrix T which has both positive and negative eigenvalues. Can we prove that there exist two Toeplitz matrix T1 and T2 such that T1+T2=T and T1 has only one positive Eigenvalues ...
- 9
1
vote
0
answers
660
views
Fractional Fourier transform [closed]
Let $T: L^2(\mathbb{R}^n) \rightarrow L^2(\mathbb{R}^n)$ be the Fourier transform. Is there any reasonable definition of fractional Fourier transform (i.e. operator $A$ such that $A^{\alpha}=T$ for $\...
- 3,323
4
votes
0
answers
90
views
$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology
This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
3
votes
0
answers
356
views
Stability of convex sets w.r.t. integration over [0,1]
In the preprint on pages 19−20, first using Hahn−Banach, one proves
Lemma 38. For any closed convex set $U$ in a real Hausdorff locally convex space $E$ and for any Riemann integrable $\gamma:[0,1]\...
- 3,584
11
votes
0
answers
657
views
For which Lie groups is the convolution of any two nonzero integrable compactly supported functions nonzero?
The Titchmarsh convolution theorem implies that the convolution of two nonzero functions $f,g\in L^1(\mathbb R)$ with compact support is nonzero. There is a generalization of this theorem to the case ...
- 637
2
votes
0
answers
93
views
Inclusions between $L^p$ continuous functions and Triebel-Lizorkin spaces
Working in $\mathbb{R}^{d}$, consider on the one hand the space of continuous $L^{p}$ functions (let's use $V$ to denote this space), and on the other the family $\{ F_ {\alpha}^{p, q} \}_{\alpha, q}$ ...
- 21
5
votes
0
answers
160
views
reference for perturbation of projection result
Let $A$ and $B$ have the same rank and dimensions. If $P_A$ denotes the projection onto the range space of $A$, then
$$
\|P_A - P_B\|_2 \leq \|A - B\| \cdot \min (\|A^\dagger\|_2, \|B^\dagger\|_2).
$$
...
- 922
1
vote
0
answers
119
views
Boundedness of Riesz transforms.
The Riesz $R_i$ transform on $\mathbb{R}^n$ is defined by
$$ R_if(x)= \int_{\mathbb{R}^n} \frac{t_i-x_i}{\vert x-t \vert^{n+1}}f(t) dt$$
for a Schwartz function $f$ on $\mathbb{R}^n$. Can you please ...
- 583
1
vote
1
answer
359
views
Convergence of operators to the identity on Banach spaces
Let $U_\infty$ be a compact space, and let $U_r$ be an increasing family of compact subspaces whose closure is all of $U_\infty$. That is, $U_r \subseteq U_{r'}$ if $r \le r'$ and $U_\infty = \...
- 8,512