All Questions
9,959 questions
3
votes
1
answer
598
views
is a non-invertible operator a boundary point of the group of invertible operators?
Good evening,
I have a question concerning non-invertible operators.
Let $H$ be a Hilbert space and $T$ a non-invertible bounded operator on $H.$ Is it true that $T$ is the limit of some sequence ...
- 25.8k
0
votes
1
answer
901
views
Schwartz space inequality
Let $g$ be a function in the Schwartz space $\mathscr S (\mathbb R)$. Show that for any $l \ge 0$, we have $\sup_x |x|^l |g(x-y)|\le A_l (1+|y|)^l$ by considering separately the cases $|x|\le 2|y|$ ...
CommunityBot
- 1
1
vote
1
answer
565
views
Cesaro means for $\alpha<1$ and Banach limits
I am interested in conditions in terms of standard scales of summation methods that guarantee the existence of an averaged limit for all almost convergent sequences. For the Cesaro summation method $(...
CommunityBot
- 1
8
votes
1
answer
2k
views
Fastest decay of Fourier Transform for Generalized Functions of compact support
What is the fastest decay possible for the Fourier transform of a generalized function with compact support and finite value at the origin? I know that regular functions cannot attain exponential ...
10
votes
1
answer
1k
views
Separating vectors for C$^*$-algebras
(I asked this on math.stackexchange, without response).
Let $A$ be a C$^*$-algebra, concretely acting on a Hilbert space $H$. Suppose that $\xi_0\in H$ is cyclic and separating for $A$ (that is, the ...
CommunityBot
- 1
3
votes
2
answers
221
views
Do infinite products commute with functor of smooth sections?
Similarly to my previous question about direct limits, I have now basically the same question about inverse limits. It seems in fact, that I only need the result for products.
Question: Is there a ...
CommunityBot
- 1
10
votes
4
answers
1k
views
References: Infinite dimensional Lie algebras
What I really want are properties (if it is abelian, nilpotent, solvable, simple, or semisimple; Cartan subalgebras...) of the Lie algebra of smooth functions on a symplectic manifold $(M,\omega)$; ...
- 3,377
4
votes
1
answer
276
views
Abelian sub-W*-algebras
Let $M$ be a von Neumann algebra which acts faithfully on a Hilbert space of density character $\kappa$ but does not on a Hilbert space of density character $\lambda<\kappa$ (that is, the density ...
- 4,624
2
votes
1
answer
372
views
Complementable subspaces of $(c_{00}(S),\Vert\cdot\Vert_1)$
Let $\ell_{1,0}(S)=(c_{0,0}(S),\Vert\cdot\Vert_1)$ be a space of functions on a set $S$ with finite support, endowed with $\ell_1$ norm. Could you answer the at least one of the following questions
...
- 31.5k
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
0
votes
2
answers
424
views
Unbounded sequences in Banach spaces
Let $X$ be a Banach space and let $T$ be a bounded operator acting on $X$. Suppose for each linearly independent unbounded sequence $(x_n)$ in $E$, the sequence $(Tx_n)$ is unbounded. Must $T$ be ...
- 54.2k
2
votes
2
answers
816
views
Principle of Local Reflexivity
I'm having a hard time trying to understand a proof of the Principle of Local Reflexivity. I'm following the proofs from
1) Topics in Banach Space Theory (by Fernando Albiac, Nigel J. Kalton)
2) ...
- 228
12
votes
1
answer
838
views
A measure theory question
Here's an interesting problem one can formulate for a student. This problem arises when considering special ergodic theorems:
On a finite dimensional manifold $M$ with a Lebesgue measure $\mu$, does ...
- 17.6k
6
votes
0
answers
98
views
Do the translates of integrable function approximate its radial part?
For an integrable function $f$ on $\mathbb R^n$ we consider its ``radial'' part
$$R(f)(x)=\int_{\mathrm{SO}(n)} f(kx)dk.$$ What is the minimal condition on $f$ so that the span of translates of $f$ (...
- 415
3
votes
2
answers
340
views
Perturbing upper-semi Fredholm operators
Let $T\colon X\to X$ be an upper-semi Fredholm operator acting on a $B$-space $X$ (the range of $T$ is closed and kernel is finite-dimensional) with complemented range. Suppose $S\colon X\to X$ is ...
- 60.5k
1
vote
2
answers
139
views
Given f(t) = \sum_k C_k exp(2 pi i w_k t ) + noise. Need to estimate C_k and w_k .
Simpliefied setup.
Assume I am given some function f(t).
I know that it is constructed as $f(t) = \sum_{k=1...M} C_k exp(2 \pi~ i~ w_k t ) + noise(t)$.
where $noise(t)$ is some random set of numbers ...
- 41
2
votes
1
answer
2k
views
If any perfect set is uncountable in a metric space which is not complete?
We know that every ferfect set $E$ in a complete metric space $X$ is uncountable. My question is if there exists a metric space which is not complete, but every ferfect set in it is uncountable. The ...
- 1,872
1
vote
1
answer
287
views
General compactness criterion in functional spaces
What follows is a total boundness criterion in the space $L^1(X)$, where $X$ is arbitrary space with probabilistic continuous measure (Lebesgue space). Of course, all such spaces $X$ and hence $L^1(X)$...
- 60.5k
21
votes
2
answers
1k
views
Is there an L^p tauberian theorem?
From Wiener's tauberian theorem we know that linear combinations of translates of f \in L^1(R) are dense in L^1(R) if and only if the Fourier transform of f never vanishes. It is also known that ...
- 415
20
votes
1
answer
994
views
Which spaces are characterized by functions with compact support ?
It's well known that two locally compact Hausdorff spaces $X, Y$ are homeomorphic iff the rings $C_0(X), C_0(Y)$ (continuous functions vanishing at infinity) are isomorphic.
Is there a class $\...
- 607
3
votes
0
answers
409
views
Continuous function sort
If you have a real-valued function f(x), positive, continuous and bounded on some interval, then what kind of transform would convert this to a monotonic function g(x) on that interval analogously to ...
- 529
7
votes
3
answers
4k
views
Is a semicontinuous real function Borel measurable?
Let $f(x,u): [0,1]^2 \mapsto \mathbb{R}$ be a continuous
function.
[Q] Is $g(x) = \inf_{u\in [0,1]} f(x,u)$ always Borel measurable?
If not, can one find a counter-example?
Note that, for any $c$,
...
- 279
1
vote
0
answers
180
views
iterated traces for sobolev functions
It is well known that if $M$ is a smooth $(n-1)$-dimensional surface in $\mathbb R^n$ (e.g. a subspace) then there is a continuous trace operator $W^{s,p}(\mathbb R^n)\to W^{s-1/p,p}(M)$. Now suppose ...
- 2,041
1
vote
2
answers
295
views
The operator preseving two disjoint dense operator ranges invariant
Let $\mathcal{H}$ be a separable infinite dimensional Hilbert space. Suppose that $\mathcal{H}_0\subset\mathcal{H}$ which is a dense proper subspace is the range of some bounded linear operator $T$. ...
1
vote
1
answer
686
views
analytic continuation of a Laplace transform from a countably infinite set of points?
Let $f(\lambda)=\int_0^\infty e^{-\lambda s} F(ds)$, where $F$ is the distribution of a positive random variable. Suppose I know the value of $f(n)$ for $n=0,1,2,\cdots$. Is this enough to uniquely ...
- 4,921
2
votes
0
answers
564
views
Young inequality in weighted spaces
Let $U$ be a bounded open set in $\mathbb{R}^2$, $g\in L^1_{\mathrm{loc}}(\mathbb{R}^2)$.
Let moreover $w$ be a weight (i.e. a non vanishing locally integrable function) on $U$ and $p\geq2$.
Does ...
- 1,205
8
votes
2
answers
1k
views
Approximation by polynomials
Let $f:[a,b] \rightarrow \mathbb{R}$ be of class $C^n$. Let $ x_0, ..., x_m$ be different numbers from $[a,b]$.
Does for each $\varepsilon >0$ there exist a polynom $P$ such that $P^{(k)}(x_i)=f^{...
- 60.5k
2
votes
2
answers
2k
views
A point in the weak closure but not in the weak sequential closure
I'm trying to find a proof of this counterexample by von Neumann:
Let $x_{mn}\in \ell^2$ be defined by
$$x_{mn}(m)=n \quad,\quad x_{mn}(n)=m \quad,\quad x_{mn}(k)=0 \hbox{ otherwise,} $$
and let $S=\...
- 31.5k
7
votes
2
answers
484
views
Extension of weakly compact operators from $\ell_1$ into $c_0$
Is every weakly compact operator from $\ell_1$ into $c_0$ extendible
to any larger space? Equivalently, is every weakly compact operator from $\ell_1$ into $c_0$ extendible to $\ell_\infty$?
- 31.5k
-2
votes
1
answer
3k
views
Multiplying two Fourier series gives one Fourier series, but what are the new coefficients? [closed]
If I have $A(x)=B(x) C(x)$ (sine periodic from 0 to 1) rewritten as
$\sum_n A_n \sin(n \pi x)=\sum_m B_m \sin(m \pi x)\sum_p C_p \sin(p \pi x)$
is there any easier way to compute $A_n$ from $B_m,...
- 96.4k
5
votes
3
answers
821
views
are the smooth vectors of a Frechet space dense?
Given an action $\alpha$ of $V$ a Lie group on $B$ a Fréchet space with seminorms $ \{ \| \cdot \|_j \} $, let $B^\infty$ be the space of smooth vectors. Is this dense in $B$? Can I guarantee it is ...
- 6,924
4
votes
1
answer
266
views
Exotic uniform algebras
The first non-trivial example of a uniform algebra which comes to mind is the disc algebra $A(\mathbb{D})$. In a similar manner one can define its relatives $P(U)$ and $R(U)$, where $U$ is any region ...
- 25.8k
7
votes
0
answers
266
views
Problem with Shelah and Stern's paper on the Hanf number of the theory of Banach spaces
I have been trying to understand "The Hanf number of the first order theory of Banach spaces" by Shelah and Stern (Trans. AMS 244 (1978) 147-241). They construct a normed space $M$ from a Hilbert ...
- 879
5
votes
1
answer
3k
views
Inner product of linear bounded operators between Hilbert spaces
Let $X$ and $Y$ be Hilbert spaces, and let $L(X,Y)$ be the set of bounded linear operators between Hilbert spaces.
Can we equip $L(X,Y)$ with a natural inner product? I think it should look like
$\...
- 2,378
4
votes
2
answers
519
views
Factorization through $\ell_{1}$ and operator ideals
Recently, I bumped into the class of operators that factor through $\ell_{1}(X)$ for some set $X$. For now, $X$ is a set with arbitrary cardinality but if it leads to a more concrete answer to my ...
0
votes
1
answer
221
views
Sort-of extension of Young inequality to arbitrary measures
Hello folks,
Suppose we're working in ${\mathbb T}^q$, the $q$-times Cartesian product of $[-\pi,\pi]$ with the endpoints identified. So, $ L^p := L^p({\mathbb T}^q) $.
The Young inequality may be ...
5
votes
1
answer
540
views
Cosets of groups of functions
Let's consider an interval $I\subseteq\mathbb R$, and let $\mathcal F(I)$ be the set of bijective functions $f:I\to I$ so that the graph of $f$ is a analytic curve in $I\times I$.
The set $\mathcal ...
- 148k
1
vote
1
answer
138
views
Estimating norms of derivatives
Let $(M,g)$ be a closed, compact Riemannian manifold. Let $u$ be a smooth function. Let $H^{-k}(M)$,, $k$ is a positive integer, be the dual Hilbert space of $H^{k}(M)$. Does it follow that $|| |\...
- 34.7k
19
votes
3
answers
1k
views
Is there "Schur-Weyl duality" for infinite dimensional unitary group?
To what extent does the relation between the diagonal representation of $SU(n)$ in $(\mathbb{C}^n)^{\otimes k}$ and representations of the symmetric group $S_k$ remain valid when instead of the group $...
- 965
2
votes
3
answers
3k
views
Extension of pointwise convergence of a sequence of uniformly continuous functions that converges on a dense set
It is known that a sequence of continuous functions on a metric space that converges pointwise on a dense subset need not converge pointwise on the full space. But what about if one assumes uniform ...
- 2,481
1
vote
0
answers
178
views
Inequalities between self-adjoint operators
Let $T_s$ ($s\ge0$) be a smooth family of non-negative self-adjoint operators in a separable Hilbert space $H$. Suppose that, for some $C'>C>0$, we have $T_0+Cs^2\le T_s\le T_0+C's^2$ for all $s$...
- 329
10
votes
0
answers
509
views
Lacunary hyperbolic groups and weak amenability
In the paper called Lacunary Hyperbolic group, Y. Ol'shanskii, D. Osin and M. Sapir define and characterize the lacunary hyperbolic groups, which contains the hyperbolic groups but also Tarski's ...
- 195
8
votes
1
answer
487
views
Continuous selections from sums of compact sets
This question is somehow related to the last open problem from Grothendieck's thesis about completeness of locally convex inductive limit. However, a particular case of the problem boils down to a ...
- 16.4k
5
votes
2
answers
944
views
Realize a homomorphism $\mathcal{C}(X) \to \mathbb{R}$ as an evaluation
Let $X$ be a compact Hausdorff space. It is well-known that every homomorphism $F : \mathcal{C}(X) \to \mathbb{R}$ is the evaluation $f \mapsto f(x)$ at some point $x \in X$. The usual proof is not ...
CommunityBot
- 1
6
votes
2
answers
749
views
Transpose of unbounded operators between Banach spaces.
Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator
$L' : \operatorname{...
- 34.7k
7
votes
1
answer
1k
views
laplace equation on manifolds with boundary
in aubin's book on page 104 theorem 4.7 there is the theorem: Let $(M,g)$ be a compact $C^{\infty}$ Riemannian manifold. There exists a weak solution $\varphi \in H_{1}$ of $\Delta \varphi = f $ if ...
- 23.3k
11
votes
4
answers
1k
views
Orthogonality in non-inner product spaces
I have come across a notion of orthogonality of two vectors in a normed space not necessarily inner product space. Two vectors $x$ and $y$ in a normed space are said to be orthogonal (represented $x\...
- 2,545
1
vote
1
answer
295
views
A nice overview (and maybe derivation) of the Poincaré transformations of the Vector Spherical Harmonics
With $Y_{lm}(\vartheta,\varphi)$ being the Spherical Harmonics and $z_l^{(j)}(r)$ being the Spherical Bessel functions ($j=1$), Neumann functions ($j=2$) or Hankel functions ($j=3,4$) defining $$\psi_{...
- 21.5k
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 ...
CommunityBot
- 1
1
vote
1
answer
163
views
Maximum number of orthonormal vectors contained in an open cone
Let $H$ be a separable Hilbert space, $\Pi:H\to L$ the orthogonal projection to a linear subspace of finite dimension $p$, and $U$ the open cone of vectors $u\in H$ such that $\langle u,\Pi u\rangle&...
- 54.2k