All Questions
9,958 questions
1
vote
1
answer
630
views
Stuck on a convergence argument in $H_0^1(\Omega)$.
I'm trying to verify that a functional I have satisfies the Palais Smale condition for appliction of the Mountain Pass lemma.
However I've encountered this step along the way which seems clear to me ...
11
votes
1
answer
642
views
Random walk origin return monotinicity
Consider a Markov chain on $\mathbb{Z}^d$ with transition kernel $P$ for adjacent vertices (non-diagonal). Essentially this is a $d$ dimensional random walk with the probability of a transition ...
- 17k
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 ...
CommunityBot
- 1
5
votes
2
answers
491
views
Is independence meaningful for commutative $C^*$-algebras?
I don't know very much about spectral theory so probably the answer to my question has a basic reference which I would appreciate.
Let's say I have two self-adjoint operators on a Hilbert space and ...
2
votes
2
answers
1k
views
description of functions of conditionally negative type on a group
Recall that a kernel conditionaly of negative type on a set $X$ is a map $\psi:X\times X\rightarrow\mathbb{R}$ with the following properties:
1) $\psi(x,x)=0$
2) $\psi(y,x)=\psi(x,y)$
3) for any ...
- 11.1k
7
votes
1
answer
362
views
Nonexpansive multi-valued maps in $\ell^2$
Let $C$ be a nonempty bounded closed convex subset, say the unit ball, of $\ell^2(\mathbb{N})$. Let $T: C\to 2^C$ be a map such that $T(x)$ is nonempty closed for each $x$, and that $$D(Tx,Ty)\le \|x-...
CommunityBot
- 1
8
votes
1
answer
2k
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Limits of Nilpotent and Quasi-nilpotent Operators in a $\mathrm{II}_1$-factor
A bounded operator $A$ in a Hilbert space is called nilpotent if there exists $n$ such that $A^{n}=0$. An operator is called quasi-nilpotent iff
$$
\limsup_{n\to\infty}{ \|A^{n}\|^{1/n}}=0.
$$
Every ...
- 3,626
2
votes
1
answer
508
views
Fractional integration lemma
Hello everyone.
I am trying to establish a fractional integration lemma of this form.
For $\alpha\geq 0$, and
$1\leq p,q<\infty$ and $0\leq \frac{1}{q}-\frac{1}{p}=\frac{\alpha}{d}$
or $1\leq p,...
CommunityBot
- 1
6
votes
1
answer
404
views
Unique preduals up to (nonisometric) isomorphism?
It's well known that there are Banach spaces which has a unique isometric predual-- for example, any von Neumann algebra. As other questions on here (for example, Isomorphisms of Banach Spaces ) ...
CommunityBot
- 1
0
votes
2
answers
1k
views
Inverse Fourier transform of class of infinitely differentiable function with compact support
For which $f \in S(R^n)$, the Schwartz class, $\hat f \in D(R^n)$ ?
CommunityBot
- 1
3
votes
2
answers
642
views
Localization of Laplacian eigenfunction on the unit square?
Let A be the unit square, $\{u_k\}$ is the set of all L2-normalized Laplacian eigenfunctions with Dirichlet boundary condition. Is it true that for any open subset V, $C_V = \inf\limits_k \int\...
- 13k
-1
votes
1
answer
1k
views
relation between inclusion and embedding [closed]
Assume that $X$ and $Y$ are two Banach spaces, now we have that $X$ is included in $Y$, in the sense that $\forall a\in X$, we have $a\in Y$. Then can we get that $X$ is embedded in $Y$, namely, $\...
- 331
4
votes
2
answers
1k
views
Projective Banach spaces
Injective Banach spaces, with morphisms as contractive linear maps, have been classically studied (and are $C(K)$ spaces with $K$ Stonian). But what about projectives?
So $P$ will be projective if ...
- 5,743
2
votes
3
answers
358
views
Almost Orthogonal Vectors given a Unitary Operator
Let $\mathit{H}$ be a (real or complex) Hilbert space and $U:\mathit{H}\rightarrow\mathit{H}$ be a unitary operator. What conditions can be placed on $U$ to guarantee a sequence $v_n$ such that $|v_n|...
- 54.2k
2
votes
2
answers
1k
views
Characterization of closed subspaces of $ L^2(R)$
Natural way to find an example of banach spaces is to look at closed subspaces of Banach spaces. Initially, It was really hard to find examples of closed subspaces of $L^2(R)$. Then, my professor gave ...
- 60.5k
1
vote
2
answers
1k
views
Convergence of eigenvectors
Let $T$ be a compact operator on $l^2$. Let $T_n$ be finite rank operators and $T_n \to T$ in the operator norm. Is it true that the eigenvalues and eigenvectors of $T_n$ converge to eigenvalues and ...
- 3,504
3
votes
1
answer
588
views
orthonormal basis of eigenvectors for laplacian on a concave polygon
I am interested in the Laplace operator $\Delta$ on a concave polygon.
When the polygon is convex, it is known that $\Delta: H^2(\Omega) \rightarrow L^2(\Omega)$
is boundedly invertible. In addition, ...
- 3,343
1
vote
2
answers
294
views
inequality of norms [closed]
Let $X$ and $Y$ be two Banach spaces with norms $\|\|_X$ and $\|\|_Y$ respectively. If $Z=X\times Y$ is also a Banach space with norm $\|\|_Z$ then what is the relation between $\|\|_X,\:\|\|_Y$ and $\...
- 18.7k
2
votes
0
answers
520
views
Eigenvector of infinite matrix
I consider the system of reaction-diffusion PDEs in a ball
with Robin boundary condition.
It is a Steklov eigenvalue problem
(see G Auchmuty (2004) "Steklov eigenproblems and the representation
of ...
- 31
4
votes
1
answer
688
views
Subgroups of U(M_n)
can any subgroup of the unitary group of full matrix alg $M_d(\mathbb{C})$ be approximated on finite
sets by a finite subgroup?
i.e. is the following True or false?
Let $n, d$ be positive integers ...
CommunityBot
- 1
0
votes
1
answer
643
views
Is this (interpolation) inequality right?
Suppose that $\Omega$ is a bounded domain in $\mathbb{R}^3$, $F$ is bounded in $L^\infty (\Omega \times (0,T))\cap (\cap_{k=1}^\infty L^{5/3}(0,T;C^k(\bar{\Omega})))$.
Question: Can we say that $F$ ...
- 3,584
12
votes
1
answer
329
views
Ideals in smooth subalgebras of C*-algebras
Let $B$ be a $C^{*}$-algebra and $\mathcal{B}$ a dense *-subalgebra stable under holomorphic functional calculus and $C^{1}$-functional calculus for selfadjoint elements. Also, $\mathcal{B}$ is a ...
- 25.8k
5
votes
1
answer
410
views
Is the unitary group of $l^2(A)$ with the strict topology contractible?
Let $A$ be a $C^*$-algebra with countable approximate unit. Let $\mathbb{K}$ denote the compact operators on a separable Hilbert space. Mingo and later Cuntz and Higson have shown that the unitary ...
- 2,203
2
votes
0
answers
156
views
Holomorphic stability of inverse limit of pre-$C^*$-algebras
Let A be a C*-algebra and let At be a set of dense *-subalgebras of A, stable under holomorphic functional calculus on A, which are also Banach algebras complete with respect to the norms ||$\cdot$||t....
- 613
2
votes
3
answers
538
views
General theory for p-normed spaces
Hello,
in functional analysis and operator theory, you encounter several (at first glance, at least) similar constructions of normed spaces that can be indexed with some $ p \in [1,\infty]$, and ...
CommunityBot
- 1
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
3
votes
0
answers
251
views
What is the origin of the metrization problem for compact convex sets?
The following is an ``old question in analysis:''
Is it true that every perfectly normal compact convex subset of a locally convex topological vector space is metrizable?
Here perfectly normal means ...
- 3,547
17
votes
5
answers
7k
views
A counter example to Hahn-Banach separation theorem of convex sets.
I'm trying to understand the necessity for the assumption in the Hahn-Banach theorem for one of the convex sets to have an interior point. The other way I've seen the theorem stated, one set is closed ...
- 696
5
votes
2
answers
329
views
Looking for substitutes for co-free modules in a topological setting
I should say that I'm not a category theorist or an abstract algebraist, so maybe this will be very pedestrian. I have the following, somewhat vague question:
I have categories C and D, a ...
- 18.7k
9
votes
1
answer
2k
views
Rate of convergence of smooth mollifiers
How does one figure out/prove the rate of convergence (in some norm) of mollifiers given a function bounded in some other norm (say Sobolev space, Besov space)? Also, is there a dimensional analysis ...
- 61.9k
4
votes
2
answers
580
views
An analogue of Hilbert-Schmidt theorem for multilinear forms
Let $H$ be a (the) real separable Hilbert space. The Hilbert--Schmidt theorem says that a compact self-adjoint operator $A$ has an eigenfunction expansion. Instead of operator, we can think of a ...
- 1,612
6
votes
1
answer
1k
views
Must a surjective isometry on a dual space have a pre-adjoint?
Background: Let $X$ be a Banach space. We know a linear map $h$ is a surjective isometry of $X$ if and only if its adjoint $h^*$ is a surjective isometry of $X^*$.
In general, a linear map $g:X^* \...
- 7,329
3
votes
1
answer
436
views
When does a mother wavelet generate a frame?
This question is about conditions on a mother wavelet that generates a countable familily of child wavelets via scaling and translation, that are both necessary and sufficient for the child wavelets ...
3
votes
1
answer
181
views
Reference request - spectral radius formula for linear transformations in char p
I am finishing up a paper and I would like to be able to quote a theorem that does what
is said in the title. To be specific let me introduce some notations:
${\bf F}$ is a local field of ...
- 3,702
1
vote
0
answers
477
views
A norm ratio inequality
Let $y,z\in(0,1)^n$ satisfy $||y||_1 = ||z||_1=1$.
Then
$$
\frac{||z||_3}{||z||_2} \le
K_n
||z/y||_\infty
\frac{||y||_3}{||y||_2}
$$
where $z/y\in\mathbb R^n$ is the coordinate-wise quotient of $z$ ...
- 6,541
13
votes
1
answer
404
views
Self map of unitary group
Let $H$ be a Hilbert space and let $u_1 \in U(H)$ be a unitary operator on $H$. Consider the self-map $w: U(H) \to U(H)$ which is given by
$$w(v) := v^2 u_1 v^{-1}.$$
Since $U(H)$ is connected, there ...
- 61.9k
7
votes
1
answer
823
views
On a decomposition of L^1(G)
[EDITED by Y. Choi - I have attempted to paraphrase the original question into something a bit terser and more precise; if this is not what the original poster intended, they should make corrections ...
- 5,743
8
votes
1
answer
2k
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Recent progress on Bochner-Riesz conjecture
Consider the family of operators $T_\delta$, $\delta \geq 0$, defined on $\mathbb{R}^n$ by
$
\widehat{T_\delta f}(\xi) = (1-|\xi|^2)_+^\delta \widehat{f}(\xi).
$
($(1-|\xi|^2)_+^\delta$ are known as ...
- 13k
2
votes
1
answer
194
views
Is there any result discribing the value of the correlation of a measurable function of `$X$` and itself: `$corr(f(X),X)$` ?
Let $X$ be a random variable, and $f$ a measurable function. Is there any particular relationship between the expression of $f$ and $corr(f(X),X)$?
BACKGROUND
The background of asking the value of $...
CommunityBot
- 1
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
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
0
votes
1
answer
474
views
Hilbert space having all norms (and seminorms) continous.
Suppose I have a Hilbert space $H$ such that every seminorm on $H$ is continuous with respect to the inner-product induced norm. Is $H$ necessarily finite-dimensional? If not, is there an easy ...
- 171
0
votes
1
answer
915
views
Can you interpret this divergent integral?
In this ArXiv paper by Wilk and Wlodarczyk (published in Physical Review Letters), equation 16 has essentially the following definition of a function:
$$\text{f(x)=}\frac{c}{2Dx^2}\exp[\int^x_0 \frac{\...
- 5,721
1
vote
1
answer
367
views
An integral which is related to Biharmonic extension
In my research, I need to evaluate an integral:
$$\int_{R^{3}}\frac{y^{3}}{(|x-\xi|^{2}+y^{2})^{3}}\log(|\xi^{2}|+\frac{1}{4})d\xi$$
where $x\in R^{3}$, $y\geq0$. Moreover, I want to see whether it ...
- 221
1
vote
0
answers
174
views
Eigenvalues of a Parametrized Family of Linear Functions
Suppose that we have a family of linear functions $L(\alpha) : \mathbb{R}^n \rightarrow \mathbb{R}^n$, where $\alpha$ is a positive real number.
For each $\alpha$, it is given that $L(\alpha)$ is a ...
- 111
7
votes
1
answer
1k
views
dependence of eigenvalues on parameters
Let $f$ be a positive real-analytic function on the closed unit disk. Consider the eigenvalue problem $\Delta \phi = \lambda f \phi$,
with $\phi = 0$ on the boundary. There exists a sequence of ...
- 9,007
1
vote
1
answer
254
views
Extending linear operators to multi-linear ones
Suppose we are given a linear operator $L$ on a Banach space $X$. Is there any way to extend $L$ to a multi-linear operator $\mathcal{L}$ in such a way that
$$\mathcal{L}(x_1, x_2^*, \ldots, x_n^*) = ...
- 73.2k
18
votes
1
answer
1k
views
Commuting unitaries
Is the following true:
For every unit vectors $x_1,..., x_n$, $y_1,..., y_n$ in $\mathbb{C}^k$
there exist a Hilbert space $H$, unitary operators $U_1,...,U_n$ and $V_1,...,V_n$ in $B(H)$ and unit ...
- 4,705
1
vote
0
answers
237
views
Variation of a function
There are probably some of you guys who already know some of the terms that I am going to use so in order to be not so boring I will put the definition to the end.
Let $f$ be a piecewise expanding ...
- 11
6
votes
1
answer
355
views
Why is the dimension of Gaussian variables is bounded by the dimension of the space?
I'm looking at a probabilistic proof of a local version of Dvoretzky's theorem in Pisier's manuscript "Probabilistic Methods in the Geometry of Banach Spaces."
For each $\epsilon >0$ there is a ...
- 1,302