Questions tagged [fa.functional-analysis]
Banach spaces, function spaces, real functions, integral transforms, theory of distributions, measure theory.
9,773 questions
7
votes
1
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a.e. convergence of the powers of an operator built from rotations
Consider two numbers $a,b\in R/Z$ and some integer $p\geq 1$. Let $T:L^p(R/Z)\rightarrow L^p(R/Z)$ be the operator given by
$$T(f)(x)=1/2(f(x+a)+f(x+b))$$
For which values of $a,b$ do we have almost ...
- 18.7k
4
votes
0
answers
487
views
Convolutions and Toeplitz Operators
Let be $d>0$ an integer number and consider the Cartesian product $\mathbb Z^d$ as metric space, with the distance between $x,y\in\mathbb Z^d$ given by $\|x-y\|_1=\sum_{j=0}^d|x_j-y_j|$.
Let be $...
- 2,044
8
votes
0
answers
605
views
convergence rate in Wiener's approximation theorem
Wiener has the following fantastic results about approximations using translation families:
Given a function $h: \mathbb{R} \to \mathbb{R}$, the set $\{\sum a_i h(\cdot - x_i): a_i, x_i \in \mathbb{...
- 1,839
0
votes
0
answers
301
views
Lifting of product of a Banach algebra
Let $A$ be a non unital Banach algebra. The product induces a bounded linear map $T:A \otimes_{\gamma} A\to A$ where $\otimes_\gamma$ denotes the Banach projective tensor product.
A lifting of $T$ is ...
- 1,222
3
votes
0
answers
84
views
Application and relevance of Sobolev gradients
The Sobolev gradient concept has been developed in the 1970s, with a first publication in 1985, and an introduction can be found at: Ranka
I would like to learn how strong the impact of Sobolev ...
- 5,327
4
votes
0
answers
167
views
Relationship between sequential compactness of a convex set and its extremal points
Suppose that $X$ is a compact convex subset of a topological vector space. Suppose also that the extremal points of $X$ have the additional property that any sequence $x_n$ of extremal points has a ...
- 93
0
votes
1
answer
297
views
Continuity of cylindrical functions.
Let $C_c^\infty(\mathbb R^n)$ be the functions from $\mathbb R^n$ to $\mathbb R$ with compact support, further let $X$ be a separable Hilbert space with a fixed orthonormal basis $(e_n)_n$. Define the ...
- 455
4
votes
1
answer
221
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existence of charaterization of amenable groups by complementation?
Recall that we say that a closed space $F$ of a Banach space $E$ is complemented if there exists a contractive projection $P$ from $E$ onto $F$.
Do you know a charaterization of discrete amenable ...
- 1,222
11
votes
0
answers
310
views
Combinatorial Hilbert spaces
Any closed subspace $V\subset {\ell}^2(\omega)$ has associated to it a subset ${\cal S}_V$ of ${\cal P}(\omega)$, call it a combinatorial Hilbert space, namely the set of all supports of all vectors ...
- 17.6k
1
vote
0
answers
76
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h-oscillating function
I need help understanding the following condition:
$u_h\in L^2(\mathbb{T}^d)$, $\|u_h\|_{L^2(\mathbb{T}^d)}=1$, where $h$ is the semiclassical parameter and $\mathbb{T}^d$ is the flat torus, is ...
- 11
2
votes
1
answer
265
views
Multiple ergodic averages with varying number of terms
Hi. I've been stuck on the following question for some time.
Consider a sequence of functions $\left( f_n \right)$ from an ergodic space $\left( \mathsf{X}, \mathsf{S}, \mu \right)$ to $\left[ 0,1 \...
1
vote
0
answers
125
views
base change for distributions
For distributions on smooth manifolds one can consider the push-forward which is defined for proper maps, and the pull-back which is defined under certain condition on the wave front set see ...
- 2,649
6
votes
0
answers
354
views
Ordering of completely bounded maps
Let A be a C*-algebra, let H be a Hilbert space, and let $T:A\rightarrow B(H)$ be a completely bounded (cb) map (that is, the dilations to maps $M_n(A)\rightarrow M_n(B(H))$ are uniformly bounded). ...
- 18.7k
2
votes
0
answers
146
views
Subspace where an operator is positive
Given a self-adjoint operator $\hat{T}$ on a Hilbert space $\mathcal{H}$, and assuming it has a basis of eigenvectors $\{\phi_n\}$ such that $\hat{T}\phi_n=\lambda_n\phi_n$, one can consider the ...
- 2,358
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
1
vote
0
answers
125
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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
2
votes
0
answers
114
views
non-closed weak graph limit of symmetric operators
Hi Everyone,
I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence of symmetric ...
- 21
1
vote
0
answers
149
views
Banach spaces with simple best approximate solutions
Let $\langle V,||.||\rangle$ be a Banach space such that:
$\;\;$ for all continuous linear maps $\: L : V\to V \:$ and members $v$ of $V$, there exists a unqiue member $u$ of $V$
$\;\;$ that ...
user5810
1
vote
1
answer
263
views
Need help with references on the status of a "Littlewood Problem"
The "Littlewood Problem" in the title asks for a characterization of finite sequences
n1< ...< nk of integers such that zn1+zn2+...+znk≠0
for any complex number z of unit modulus.
Does ...
1
vote
0
answers
136
views
Boundedness of Integral
Suppose we operate on the unit simplex $\Delta \subset \mathbb{R}^d$ with $0$ as a corner point.
Define the integral
$$
Iu(x):=\int_0^1 t^{|\beta|-1}x^\beta u(tx)\mathrm{d}t,\quad x\in \Delta
$$
and ...
- 233
2
votes
1
answer
194
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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 $...
- 23
5
votes
0
answers
104
views
Regularity of simplices, part deux
This question is directly inspired by Pietro Majer's question and my answer to it.
One can define a simplex, and the dihedral angles thereof in an infinite dimensional Hilbert space (one has to take ...
- 96.4k
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
3
votes
1
answer
280
views
An analogue of an old proposition
For the absolute value $|C|=(C^*C)^\frac{1}{2}$ and the
Hilbert-Schmidt norm
$\parallel C\parallel_{HS}=(trC^*C)^\frac{1}{2}$ of the operator $C$. The
following inequality is shown by Araki et al in ...
- 223
0
votes
0
answers
320
views
A result about Fredholm operator
When I read the article "Index Theory" in Handbook of global analysis, I meet a result as below(Corollary 2.13):
If every $F_0\in \mathcal {F}(H_1,H_2)$, there is an open neighborhood $U_0\subseteq \...
- 381
1
vote
0
answers
135
views
Inequality involving BV norm and a regularizing kernel
In the same article by Benoit Perthame: http://www.mendeley.com/research/uniqueness-error-estimates-first-order-quasilinear-conservation-laws-via-kinetic-entropy-defect-measure/# (related to this ...
- 2,222
1
vote
0
answers
159
views
variational problem under convexity constraints
I wonder if there is any method to compute variational problems subject to certain shape constraints (e.g., convexity, monotonicity, etc.). The literature I found on this topic (which I am no expert ...
- 1,839
4
votes
1
answer
228
views
When can closedness of the range of an operator be checked on a positive cone?
Let $T:X\to Y$ be an operator between Banach spaces $X$ and $Y$. Assume that $X$ has a positive cone $C\subset X$, which generates $X$: every element of $X$ can be written as a difference of elements ...
user2412
6
votes
0
answers
299
views
Spectrum of an operator arising in a dynamical problem
(Question edited according to Denis Serre comment).
While studying the action of dilating map of the circle on probability measures, I ran across the following operator:
$$\mathcal{K}^* : L^2_0(\mu)\...
- 14.4k
3
votes
1
answer
181
views
Reference request - spectral radius formula for linear transformations in char p
I am finishing up a paper and I would like to be able to quote a theorem that does what
is said in the title. To be specific let me introduce some notations:
${\bf F}$ is a local field of ...
1
vote
0
answers
237
views
Variation of a function
There are probably some of you guys who already know some of the terms that I am going to use so in order to be not so boring I will put the definition to the end.
Let $f$ be a piecewise expanding ...
- 11
4
votes
0
answers
189
views
Boundedness criterion for operators on mixed Lebesgue spaces
Define the mixed Lebesgue space $l_{p,q}$ as the space of all doubly indexed sequences
${\bf a}= (a(i,j))_{i,j\in\mathbb{Z}}$ such that
...
- 979
0
votes
0
answers
138
views
Notion of simplicity of a function(al)
Given a function (functional actually) $f(x,g(x))$, can a notion of simplicity be attached with respect to the function $g(x)$? (all functions and args are real).
Specifically, intuitively one could ...
- 59
-1
votes
1
answer
311
views
A differential equation
let $g(s)$ be real-valued function defined on $[0,T]$ such that $g(T)=0$ and suppose that $g$ is a "nice function"
Assume that $0<\gamma<1$, $v$ is a positive number, and
$$\frac{dg}{ds}+(v\...
- 1
4
votes
1
answer
321
views
What functorial topologies are there on the space of linear maps between LCTVS?
Setup: we consider the category of locally convex topological vector spaces with morphisms as continuous linear maps. This time, I'm explicitly allowing the axiom of choice (or at least the Hahn-...
- 26.8k
4
votes
0
answers
257
views
A matrix minimisation problem
Feel free to edit the title!
Suppose A is a C*-algebra and $a,b\in A$ are self-adjoint. I'd be very happy with A being just $n\times n$ matrices.
Question: If there are $t\in\mathbb R$ and $\...
- 18.7k
2
votes
1
answer
493
views
Convergence of Affine Transformations
Hi,
I was wondering if anyone could point me to any sources regarding the convergence of iterated affine transformation, i.e. sequences where {a_n} is a set of affine transforms and the sequence:
...
- 690
2
votes
0
answers
137
views
Invariant linear manifolds for multiplication by the independent variable in L^2 (R)
In general I am trying to determine when the self-adjoint operator $M$ of multiplication by the independent variable in $L^2 (\mathbb{R})$ has a symmetric restriction to a dense linear manifold (non-...
- 31
5
votes
0
answers
420
views
Direct integrals and fields of operators
Suppose we have a measure space $(X,\mu)$ and a measurable field of Hilbert spaces $H_x$ on it. We can form the direct integral ${\cal{H}} = \int H_x \ d \mu$, which is a Hilbert space.
Suppose now ...
- 3,897
0
votes
0
answers
45
views
compactness related to some distance defined on the space of increasing functions2
Let $I=[0,1]$ and denote by $C^{+}(I)$ the space of continuous increasing functions. Can we find a distance $d$ for $C^+(I)$ such that the set of the form
$$d(f,g)\rightarrow 0\Longrightarrow f(1)\...
- 1,835
1
vote
0
answers
309
views
Loynes spaces, also called pseudo-Hilbert spaces
Let me first define my object:
First, a locally convex space $Z$ is called admissible in the sense of Loynes if
$Z$ is complete
There is a closed convex cone in $Z$, called $Z_+$, satisfying (for $x\...
- 2,600
2
votes
0
answers
82
views
Description of the norm of certain interpolation space
Dear all,
I suspect that there should be some detailed description of the norm (or of the unit ball) of the following complex interpolation space (for any $0< \theta < 1$): $$\Big(B(\ell_1^n, \...
- 769
7
votes
0
answers
161
views
Seeking reference - criterion for the existence of a positive linear functional on an ordered vector space below a given function
The following surely appears somewhere, I would greatly appreciate a reference. (The aim is to get a measure via Riesz representation, but that has nothing to do with the statement.)
Let $X$ be an ...
- 577
1
vote
0
answers
283
views
Density of Dolean exponentials in L2 and Wiener Measure
Assume that W is the classical Wiener space C([0,1],R) note $\mu$ the Wiener measure, and denote by $\mu_s$ the image of $\mu$ under the maping $T: W ->W$ such that$ T(w)= \sqrt(s) w$ . Denote by $...
- 19
1
vote
0
answers
114
views
Mappings preserving convex compactness
Let $H$ be a Hilbert space.
How can one describe continuous mappings $F:H \to H$
that satisfy the following condition:
There exist two elements $c$, $F(c) \neq c$
and a convex compact $M$ containing ...
- 11
0
votes
0
answers
155
views
General form of a symplectic map
A symplectic automorphism of a Hilbert space has the form $T=U(\cosh S+J\sinh S)$ for a unitary $U$, an antilinear involution $J$ and a positive operator $S$. In fact a version of this goes through in ...
- 1,411
3
votes
0
answers
223
views
Extension of positive operators and Bauer-Namioka
When $X$ is a vector subspace of an ordered vector space $A$, any positive linear functional $f: X \to R$ extends to all of $A$ as a positive linear functional provided one can find a nonvoid, ...
- 31
2
votes
0
answers
200
views
Fredholmness and invertibility in a C* algebra generated convolution-type operators
Let $PC$ be the algebra of complex-valued, piecewise-continuous functions from $[-\infty,+\infty]$, $SO$ be the algebra of bounded, continuous, complex-valued functions on $\mathbb R$ which are slowly ...
- 21
0
votes
1
answer
130
views
Maximal length vector under constraints
Consider a criculant symmetric $M$ an $n \times n$ matrix with $0$ and $1$ entries and $r$ entries of $1$ in each row with the diagonal values taken as $1$. I am looking for a $0-1$ vector $v$ with ...
- 800
1
vote
0
answers
52
views
Extension of $S_+$ type operators
Let $X$ be a reflexive Banach space and $G\subset X$ a open bounded set. Let $F:\overline{G}\rightarrow X^\star$ be a $S_+$ operator, i.e., if for any sequence $x_n$ in $G$ for which $x_n\...
- 71