All Questions
10,050 questions
1
vote
1
answer
767
views
Is the set of all probability measures weak* closed?
Let $(\Omega,\Sigma)$ be a measurable space. Denote by $ba(\Sigma)$ the set of all bounded and finitely additive measures on $(\Omega,\Sigma)$ (see http://en.wikipedia.org/wiki/Ba_space for a ...
- 101
0
votes
1
answer
286
views
Irreducible subspaces of separable Hilbert space
A question about definition: Let $\mathcal{H}$ be a separable Hilbert space over $\mathbb{C}$, with $B(\mathcal{H})$ the bounded linear operators on it. What does it mean to have an irreducible ...
- 103
-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
2
votes
2
answers
422
views
non-Identity operator on a separable Hilbert space
Suppose $\mathcal{H}$ is a separable Hilbert space over $\mathbb{C}$ (countable dimensions) with inner product $\langle,\rangle$. Let $A$ be a bounded linear operator on $\mathcal{H}$, i.e, in $B(\...
- 103
9
votes
3
answers
2k
views
Generalizations and relative applications of Fekete's subadditive lemma
Fekete's (subadditive) lemma takes its name from a 1923 paper by the Hungarian mathematician Michael Fekete [1]. A historical overview and references to (a couple of) generalizations and applications ...
- 10.5k
11
votes
2
answers
2k
views
Spectrum of $L^\infty(X,\mu)$
Suppose that $(X,\Sigma,\mu)$ is a measured set with respect to $\sigma$-algebra $\Sigma$.
Suppose that $L^\infty(X,\mu)$ is the set of all $\mu$-equal bounded $\Sigma$-measurable functions on $X$. ...
2
votes
1
answer
213
views
Local convexity of C([a,b])
Let $C([a,b],\mathbb{R})$ denote the space of continuous functions from $[a,b]$ to the real numbers. For a function $f\in C([a,b],\mathbb{R})$ and $d\gt 0$, define
$$p_d(f) :=sup\{\lvert f(x)-f(y)\...
- 41
6
votes
2
answers
744
views
Unicity of a vector space frame's dual frame
The Wikipedia page on vector space frames gives a construction to find a dual frame for a given frame. Specifically, given a set of vectors $\{ e_k \}$ in a Hilbert space $\mathcal{H}$ such that for ...
- 2,358
0
votes
1
answer
220
views
Frames and completeness
Let $H$ be a separable Hilbert space.
A sequence $\{f_{n}\}$ is a frame for a separable Hilbert space $H$ if there exists $A,B>0$ such that for all $f$ in $H$
$$
A\|f\|^2 \leq \sum |\langle f, ...
- 71
5
votes
2
answers
686
views
Bounded linear functionals and representations
Suppose that $A$ is a unital C$^*$-algebra and that $\varphi: A \to \mathbb{C}$ is a bounded linear functional. Then there exists a Hilbert space $H$, a representation $\pi: A \to B(H)$ and vectors $\...
- 153
1
vote
0
answers
125
views
Isomorphisms of group extensions arising from antisymmetric forms
Let $V,W$ be topological vector spaces and fix continuous antisymmetric bilinear forms $\omega_1:V\times V\to \mathbb{R}$, $\omega_2:W\times W\to\mathbb{R}$. Since $\omega_1$ is a 2-cocycle (in fact ...
- 1,411
18
votes
1
answer
996
views
Existance of certain almost invariant functions related to amenability and piece-wise transformations
We would like very much to know the answer to the following question:
Let $\|\cdot\|$ be any norm on $\mathbb{Z}^d$ and let $W(\mathbb{Z}^d)$ be the group of all bijections of $\mathbb{Z}^d$ such ...
4
votes
0
answers
1k
views
The spectrum of a Markov Operator and Invariant Measures
Suppose I have a discrete-time Markov Chain (in an infinite dimensional state space $\Omega$) with Markov operator $P$, a linear operator on the space of bounded measurable functions on $\Omega$. (Or ...
- 509
1
vote
0
answers
465
views
Reference book for a stronger version of Goldstine theorem
Let me quote the proof of Goldstine theorem from wikipedia ( http://en.wikipedia.org/wiki/Goldstine_theorem ):
Given an $x^{**} \in B_{X^{**}}$, a tuple $(\phi_1, \dots, \phi_n)$ of linearly ...
- 311
5
votes
2
answers
356
views
$L^\infty$ properties of an infinite-dimensional Gaussian semigroup
Let $W$ be a separable Banach space and $\mu$ a Gaussian Borel measure on $W$ which is centered and non-degenerate. For $F : W \to \mathbb{R}$ bounded Borel and $t \ge 0$, let
$$P_t F(x) = \int_W F(x+...
- 29.8k
1
vote
0
answers
119
views
Particular types of basis on a normed vector space of finite dimension
Is it true that on every normed vector space $V$ of dimension $n$ there exists a basis of norm $1$ vectors $v_i$, such that $\|\sum_{i=1}^n\epsilon_iv_i\|\geq 1$ for all possible combinations of $\...
- 31
1
vote
2
answers
581
views
Lower bound for the eigenvalue
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
- 71
0
votes
1
answer
2k
views
weak convergence in Sobolev spaces and pointwise convergence?
I encounter a problem when reading Struwe's book Variational Methods (4th ed). On page 38, it is assumed that
$\|u_m\|$ is a minimizing sequence for a functional $E$, i.e. $E(u_m)\rightharpoonup I$...
8
votes
0
answers
1k
views
Strictly singular operators and their adjoints
This is a question I thought about a while back and figured I'd throw it out there to see if anyone has some insight that I am missing.
Let $X$ and $Y$ be infinite dimensional separable Banach ...
- 1,073
5
votes
2
answers
881
views
Fourier transform on locally compact quantum groups
I have read some articles on locally compact quantum groups and the Fourier Transform on them. I wonder why we define the Fourier transform as an operator valued functions from $L^1(\mathbb{G})$ to $L^...
- 71
0
votes
0
answers
150
views
$n$-th derivative of the prolate spheroidal function
For a given real number $c>0$ define functions $\left(\psi_{k,c}(\cdot)\right)_{k\ge0}$, as an eigenfunctions of the Sturm-Liouville operators $L_c$ defined
$$
L_c(\psi)=(1-x^2)\frac{d^2\psi}{dx^2}...
- 71
8
votes
2
answers
583
views
Does every operator from a Hilbert space to $L^0$ factor through a canonical one?
Let $A:H\to L^0(S, \mu)$ be a continuous operator from a Hilbert space to the space of (equivalence classes of) measurable functions on a probability measure space $S$ with convergence in measure. Let'...
- 4,010
3
votes
1
answer
352
views
Integral Equation with "convolution"
I've got the following problem I'm working on which is related to some of my research:
Solve:
$f(x) = \int_{-\infty}^x G(x,y)f(y)f(x-y)dy$
for f, given $G$ which has whatever smoothness ...
- 31
11
votes
0
answers
601
views
High-dimensional geometry: Top-down Vs. Bottom-up
There are several ways to leverage one's intuition from low-dimensional geometry to understand high-dimensional phenomena. For example, one can get a clearer picture of the behaviour of high-...
- 1,666
1
vote
0
answers
121
views
showing convergence of a function recursion relation
I have obtained (formally) a perturbative solution
$$
H(y) = \sum_{n=0}^\infty \delta^n H_n(y)
$$
to the following integro-differential equation ($\delta$ is a small constant, $\nu$ is a L\'evy ...
- 351
5
votes
2
answers
2k
views
Double dual of C*-algebra
Suppose, that $A$ is a $C^{\ast}$-algebra. It follows, according to Sakai's book, that double dual of $A$ is also $C^{\ast}$-algebra. I'm not quite sure if I understand the proof correctly. Author ...
- 111
1
vote
1
answer
190
views
cardinality of discontinuity curves of BV function
If the function $f:R\to R$ is of BV class then it has at most countably many discontinuity points (since it can be represented as a sum of two monotonic function).
I am interested to know whether the ...
- 13
2
votes
2
answers
181
views
convergence of the coefficients of lacunary series
I just want to find some standard reference to the following result: let $(a_k)_k$ be the sequence of coefficients of a lacunary Fourier series which converges to an $L_1(T)$ function in the sense of ...
0
votes
1
answer
905
views
Hölder continuity of uniform limit of piecewise constant functions
Consider a piecewise constant function $v: [a,b] \rightarrow \mathbb{R}$ defined by a finite partition $a=t_0 < t_1 < t_2 < ... < t_s=b$ of the interval $[a,b]$, and constants $m_1,m_2,...,...
- 81
1
vote
0
answers
153
views
The existence of the solution of the perturbed KdV Equation(semi-group operator)
Consider the perturbed KdV Equation$$u_t-6uu_x+u_{xxx}=\epsilon u,u(x,0)=f(x)$$where $f(x)=v(x,0)$，$v(x,t)$ is a soliton solution.$u$ satisfies the condition$u\to 0 $when $|x|\to\infty$
I want to use ...
- 61
1
vote
1
answer
4k
views
how to prove the range of a closed linear operator is closed ?
The closed range theorem tells us that given two banach spaces X,Y,and a closed densely defined linear operator T：$X \to Y$. We have the following equivalence $R(T)$ is closed in $Y \iff R(T^{*})$ is ...
- 1,644
1
vote
0
answers
271
views
When Pelczynski's property (V*) forces (V) in the dual space?
Recently, I have been asked if are there any sufficient conditions for a Banach space with Pelczynski property (V*) to have dual with property (V). Since it was a long standing open problem, I guess ...
2
votes
1
answer
1k
views
Are bounded functions L-1 compact?
Let $(X,\Sigma,\mu)$ be a finite measure space (i.e., $\mu(X) < \infty$). Let $\mathcal{F}$ be the set of $\mu$-measurable functions $f:X \to \mathbb{R}$ that are bounded in $[0,1]$, so that $0 \...
- 1,322
5
votes
3
answers
914
views
One-sided Cauchy principal value
What is the notion of a principal value of an integral when the singularity appears at one endpoint? Namely,
$ PV \int_a^b f(t) dt = ? $,
where the integral is convergent in the upper limit, but ...
1
vote
1
answer
201
views
Reference request for sums of Grothendieck spaces
I much appreciate your help with my previous question. Now, I'd like to ask about a reference. I need a fact asserting that if $p\in (1,\infty]$ (with emphasis on the case $p=\infty$) then the $\ell_p$...
- 125
30
votes
4
answers
4k
views
Elementary applications of Krein-Milman
This is a cross-post from MSE: Elementary applications of Krein-Milman. I'm starting to suspect that the question just doesn't really have a great answer, it's worth a try.
Recall that the Krein-...
31
votes
1
answer
2k
views
Szőkefalvi-Nagy's unitarizability theorem in the Calkin algebra?
Here's a research problem, which I think interesting.
Suppose that $t$ is an invertible element in the Calkin algebra $\mathcal{Q} = \mathcal{B}(\ell_2)/\mathcal{K}(\ell_2)$ which satisfies $\sup_{n \...
- 10.1k
11
votes
1
answer
504
views
Do ultrapowers of classical Banach spaces have unconditional bases?
I am trying to imagine (to some extent, of course) the geometry of ultrapowers of certain 'easy-to-handle' Banach spaces. Let me start with $X = \ell_p$, $p\in (1,\infty)$ or $X=c_0$.
Since the ...
- 125
0
votes
0
answers
272
views
L_2-norm representation
Let
$$
f^{\alpha}_+(x)=\frac{1}{\Gamma(\alpha+1)}\sum_{k\ge 0}(-1)^k{\alpha+1 \choose k}(x-k)^{\alpha}_+,
$$
where $\alpha > -\frac 12$.
I am wondering if one can get nice representation of $L^2$-...
- 71
7
votes
1
answer
331
views
States/functionals on crossed product C*-algebras
Let $A$ be a C*-algebra, $\alpha$ a strongly continuous automorphic action by a locally compact group $G$ on $A$, and consider the crossed product $A\rtimes_\alpha G$. I am looking for references ...
- 1,017
11
votes
1
answer
702
views
Kuiper's theorem via approximation
Kuiper's theorem says that the unitary group $U(H)$ of a separable infinite dimensional Hilbert space $H$ is contractible, if it is equipped with the norm topology.
Let's suppose, I do not know this ...
- 7,637
8
votes
2
answers
2k
views
when a pseudo-differential operators to be compact?
In the theory of Pseudo-differential operators,when a symbol $a(x,\xi)\in S^{0}$,then the operator $a(x,D)$ defined by$$a(x,D)u=\int{e^{ix\xi}a(x,\xi)\widehat{u}}d \xi$$ is $L^2$ bounded.$ $
My ...
- 1,644
1
vote
1
answer
887
views
How to explain the condition (C) in critical point theory?
Condition (C). The closure of any nonempty subset S of H on which f is bounded but on which $\|\nabla f\|$is not bounded away from zero, contains a critical point of f.
How to see the meaning of " $\|\...
- 11
-1
votes
2
answers
407
views
Almost isometric subspaces of $\ell_p$
1) Given $p\in (1,\infty)$.
2) Let us fix two, non-isometric subspaces $X,Y\subseteq \ell_p$ isomorphic to $\ell_p$.
3) Are there an $\varepsilon\in (0,1)$ and an isomorphism $S\colon X\to Y$ such ...
- 537
4
votes
1
answer
882
views
What is the domain of the "average operator"?
I can try to define an averaging operator for functions, namely let
$$A: D \subset L^\infty([0,\infty]) \to \mathbb{R}$$
by
$$Af = \lim_{N\to\infty} \frac{1}{N}\int_0^N f(x)dx$$
whenever the limit ...
- 225
7
votes
3
answers
498
views
Sums of unitaries with small norm in full group $C^*$-algebras
Suppose $G$ is a finitely generated group, with given generating set $S={g_1, \dots, g_n}$. (Assume that if $g\in S$, then $g^{-1}\notin S$. (EDIT: Also assume that $S$ is minimal in the sense that ...
- 2,361
3
votes
1
answer
503
views
When an AW*-algebra is a W*-algebra
In a very old book of Kaplansky "Rings of operators", on p. 123 one can find the following sentence:
It is a standing conjecture that an AW${}^\ast$-algebra is W${}^\ast$ if its center is W${}^\ast$.
...
- 537
4
votes
1
answer
293
views
Deforming Fredholm sections
Suppose we have a Fredholm section S, (the differential is Fredholm at the 0-set) of some
Banach vector bundle over X, transverse to the 0-section, with Fredholm index 1 and such that the 0-set of ...
- 187
2
votes
1
answer
624
views
The perturbed KdV Equation
I'm now studying KdV Equation$$u_t-6uu_x+u_{xxx}=0$$To solve the initial-value problem,we can use method of Lax pair,so we can alter the original problem to the problem of solving out $u$ in the ...
- 61
3
votes
1
answer
254
views
gluing along a real analytic manifold
hi,
I have a general question. Assume we have a real analytic $n-$dim. manifold $X$ and $M$ a real analytic compact submanifold of $X$ (of dimension less that the dimension of $X$, say $k < n$). ...
- 89