All Questions
10,199 questions
8
votes
3
answers
2k
views
Measures on general topological groups
I am interested in the group algebras of non-locally compact groups. What references can you advise?
This is a wide question, so I list more concretely what I would like to see:
Here X can be even ...
- 1,152
11
votes
1
answer
1k
views
Quasi-nilpotent trace class operators as limits of nilpotents
In as yet unwritten work with T. Figiel and A. Szankowski we make an observation in a Banach space context that for Hilbert spaces reduces to:
If $T$ is a quasi-nilpotent (i.e., has only $0$ in its ...
- 31.5k
3
votes
1
answer
571
views
Subspaces of a Sobolev space
For $a \in \mathbb{R}^N\setminus\{0\}, N \ge 2$, and $\lambda \in \mathbb{R}$ let
$$
X_{\lambda,a}=\{u(\cdot+\lambda a):\, u(x)=u(|x|) \in W^{1,2}(\mathbb{R}^N)\}.
$$
Denote by $X_a$ the closure of ...
- 397
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
240
views
Conditions for differentiability of minima and minimizers of linear functionals?
Let $B$ be a (real) Banach space and $C$ be a compact convex subset of $B$.
For every continuous linear functional $F$ on $B$, define
$V(F)=min_{c\epsilon C} F(c)$ and
$S(F)= { \lbrace c \epsilon C :...
- 13
4
votes
1
answer
230
views
A convergence problem about integral operator in the space of representations
This would be a basic problem in representation theory.
Let $G$ be a unimodular real Lie group, $(\pi,V)$ a smooth representation of $G$ in a Frechet space $V$. Let $f$ be a smooth function on $G$. ...
- 2,709
7
votes
0
answers
624
views
"Liftings" of L^\infty functions
This is motivated by this question: Is there an inclusion of $L_\infty(G)$ into $C_0(G)^{**}$? and Bill Johnson's comments there.
Let $X$ be a locally compact Hausdorff space and $\mu$ a Radon ...
- 18.7k
8
votes
1
answer
656
views
When is the adjoint of a hypoelliptic operator also hypoelliptic?
Suppose that $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$.
Recall that $L$ is a hypoelliptic differential operator if for ...
- 329
15
votes
1
answer
1k
views
Intersection of complemented subspaces of a Banach space
The following seems a very basic question in the theory of complemented subspaces of Banach spaces, but I was not able to find a reference, so I wish to ask it here.
Question. Let $X$ be a Banach ...
- 60.6k
2
votes
0
answers
242
views
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
vote
0
answers
2k
views
Derivative total variation function
Hello everyone,
I would like some help on proving the following statement:
Let $f\in\mathrm{BV}[a,b]$, i.e. $f$ is of bounded variation and let $T(x)$ be the total variation of $f$ on $[a,x]$ for $x\...
- 11
2
votes
1
answer
544
views
Approximation of the radon-derivative
I am looking for the following statement.
Let $X$ be a topological space and let $\mu$, $\nu$ be Radon-Borel measures in the same measure class i.e. the the zero-sets are identical. Denote then by $...
- 21
12
votes
1
answer
1k
views
Decomposition of positive definite matrices.
It is known that a $n^2 \times n^2$ positive semidefinite matrix $A$ cannot always be written as a finite sum
$$
A=\sum_{j} B_j \otimes C_j
$$
with $B_j$ and $C_j$ positive semidefinite matrices (of ...
12
votes
2
answers
1k
views
Matrix inequality $(A-B)^2 \leq c (A+B)^2$ ?
Let A and B be positive semidefinite matrices. It is not hard to see that $(A-B)^2 \leq 2A^2 + 2B^2$. In fact, $2A^2 + 2B^2 - (A-B)^2 = (A+B)^2$ is positive semidefinite.
My question is: Is there a ...
- 123
0
votes
0
answers
104
views
Differential equation with switched parameters and boundary conditions in integral form
Sorry for the title, I didn't find a better description (showing that I have no idea for the solution). Feel free to put in a better title and change the tags if you can grasp a view on the problem.
...
- 43
5
votes
0
answers
1k
views
Generalized Stone Weierstrass theorem
Given a smooth function $f$ on some compact $K$ in the euclidean space $\mathbb{R}^d$, does exist a sequence of polynomial functions $p_n$ such that $p_n$ and all of its derivatives converge uniformly ...
- 367
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 ...
2
votes
1
answer
362
views
Quasinilpotent example [duplicate]
Possible Duplicate:
Quasinilpotent operator
Do you know any example of a quasinilpotent operator such that every its power is non-compact?
Of course direct sum of nilpotent operators(or Volterra ...
- 21
3
votes
2
answers
384
views
ED compact $K$ such that $C(K)$ is not a dual Banach space
Almost every mathematician working in von Neumann algebras says that there are certainly extremely disconnected compact spaces $K$ such that $C(K)$ is not a dual space (they are not hyperstonean). ...
- 55
0
votes
1
answer
437
views
Möbius Transform of a Continuous Possibility Function
In order to be able to use a basic possibility function as a Body of Evidence in the Dempster-Shafer Theory of Evidence, it is needed to transform the function to its Möbius representation.
There is ...
- 103
5
votes
2
answers
412
views
Wiener Tauberian Theorem for nonunimodular group
Is there a nonunimodular group for which Wiener's Tauberian theorem is true?
Is a locally compact topological group whose volume grows polynomially with radius always unimodular?
- 415
3
votes
1
answer
740
views
Centralizers in C*-algebra
Let $a,b\in A$ be self-adjoint elements in $C^*$-algebra $A$ with equal centralizers, $\{x\in A; [a,x]=0\}=\{x\in A; [b,x]=0\}$. Can we say anything about the correspondence between $a$ and $b$?
For ...
- 179
2
votes
2
answers
551
views
L^2 basis of class functions on a compact Lie group that are point-wise small
Consider first the torus group $\mathbb{T}^k$. A natural $L^2$ basis is given by the 1-dimensional complex representations: $(\theta_1, \ldots, \theta_k) \mapsto e^{i \sum_j c_j \theta_j}$ for ...
- 4,466
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
26
votes
4
answers
5k
views
Can $L^{2}$ be represented as a space of functions (not equivalence classes)?
Let $X$ be the vector space of all Lebesgue-measurable functions $f:\left[a,b\right]\rightarrowℝ$ such that $\int^{b}_{a}\left|f\left(x\right)\right|^{2}dx<\infty$ (Lebesgue integral). Then we can ...
- 4,597
34
votes
1
answer
4k
views
Theme of Isbell duality
Let $C$ be a small category. Isbell duality provides an adjunction $\widehat{C} {{\mathcal{O} \atop \longrightarrow} \atop {\longleftarrow \atop \mathrm{Spec}}}\widehat{C^{\mathrm{op}}}^{\mathrm{op}}$....
- 63.1k
4
votes
0
answers
820
views
Calderón's complex interpolation: what is the corresponding classical theorem?
This question is closely related to my answer to Dan's question, which contains the definitions of some terms I use here. In addition, the notion of exact interpolation functor of exponent $\theta$ is ...
- 1,202
21
votes
2
answers
2k
views
Uncertainty principle and Cramer-Rao bound - is there relation?
Just out of curiosity.
The two things sounds a little bit similar - 1) Uncertainty principle 2) Cramer-Rao bound.
Saying that we cannot measure something with certain accuracy.
However looking closer ...
- 24.9k
4
votes
1
answer
287
views
Second conjugate operators to operators on $c_0$
I posted my question at MS but unfortunately it is still without a response, so let me ask it here.
We can think about a bounded operator $T\colon c_0\to c_0$ as a double-infinite matrix $[T_{mn}]_{m,...
- 51
7
votes
3
answers
1k
views
Gamma-function analogues for Gauss sums
I have a Gauss sum, which I have to calculate. I have heard that it has an analogues form with the Gamma function, but couldn't find its formula shape. It would be so nice of you to help me and write ...
- 71
6
votes
0
answers
8k
views
Dual space of continuous functions
Let $C_b(\Omega,V )=$ { $ f:\Omega\rightarrow V $ } is the Banach space of all bounded continuous functions in Banach space $V$ with a norm $\|\cdot\|$ defined as $\|f\|_\infty=\sup _{x\in\Omega}\|f(x)...
- 385
1
vote
1
answer
931
views
Finding a counter example for Minkowski's integral inequality for $p=\infty$ [closed]
Dear All,
As we know that this following Miskowski's integral inequality is true for $1\leq p<\infty$
$
[\int_{S_1}|\int_{S_2}F(x,y)d\mu_1(x)|^pd\mu_2(y)dy]^{1/p} \leq \int_{S_2}[\int_{S_1}|F(x,y)...
- 51
1
vote
1
answer
423
views
Variation of function
Let $f(t,v)$, $t\in[0,T]$, $v\in \mathbb R$ be absolutely continuous in $t$ for every fixed $v$ and absolutely continuous in $v$ for every fixed $t$. Let $X(t)$, $t\in [0,T]$ be a continuous function ...
- 115
1
vote
1
answer
233
views
Structure of Measurable Subsets of the Unit Square
If A is a (Lebesgue-)measurable subset of the unit square that has positive measure, does there exist a subset B contained in A that has a product structure (is the product of two subsets of the real ...
- 99
3
votes
1
answer
248
views
relation between SOT-convergence of T and T'
I am trying to prove or to break the following statement (I assume that the statment is correct):
Assumptions: Let $H$ be a Hilbert-space (or more generally a reflexive space) and $T\in \mathcal{L}(H)...
- 65
3
votes
2
answers
403
views
Cesaro bounded Operator which is not power bounded
Good evening!
Let X be a banachspace and T a bounded linear operator on X.
The cesaro avearges of T are defined as:
$A_n:=\frac{1}{n} \sum\limits_{j=0}^{n-1}T^j $
We call T cesaro bounded if: $\...
- 65
71
votes
16
answers
21k
views
Is there a nice application of category theory to functional/complex/harmonic analysis?
[Title changed, and wording of question tweaked, by YC, because the original title asked a question which seems different from the one people want to answer.]
I've read looked at the examples in most ...
6
votes
0
answers
484
views
Square and cube?
Gowers and Maurey proved in their remarkable paper(s), that there is a Banach space $X$ such that $X$ is isomorphic to its cube $X\oplus X\oplus X$ but not to isomorphic to its square $X\oplus X$. ...
- 113
11
votes
3
answers
1k
views
Is there a Plancherel Theorem for Gowers norms?
In the process of counting arithmetic sequences in sets, the Gowers norms
$$ ||f||\_{U^s[N]}^{2^s} = \frac{1}{N^s} \sum_{\vec{h} ,\\, n } \Delta_{h_1}\dots\Delta_{h_s}f(n) $$
where the sum is $ \...
- 22.8k
4
votes
1
answer
254
views
M-bases for $C(K)$-spaces, $K$ -scattered
Recall that a biorthogonal system $\{(x_i, x^*_i)\colon i\in I\}$ in a Banach space $E$ is a M-basis if $\{x_i\}_{i\in I}$ is linearly dense in $E$ and $\{x_i^*\}_{i\in I}$ separates points. Let me ...
- 11.3k
7
votes
1
answer
2k
views
On the Paley-Wiener theorem
Is there any even Schwartz function whose restriction to $[0,\infty)$ is monotone and whose Fourier transform is compactly supported? In other words, is there any entire analytic function satisfying ...
- 329
3
votes
2
answers
428
views
Approximating smooth functions with polynomials subject to constraints.
Suppose that we are given a smooth function $h:\mathbb{R}^n \to \mathbb{R}$ which satisfies $h \circ F= h \circ G$ for two polynomial functions $F,G:\mathbb{R}^m \to \mathbb{R}^n$ (i.e. each component ...
- 15.5k
4
votes
1
answer
626
views
Reasonable crossnorm on Banach algebra tensor product constructed from isometric non-degenerate representations
The following is an abstraction of a more specific problem I've been grappling with, so I might give some extra unnecessary information. (The `bounded approximate
left identities' assumption is ...
33
votes
3
answers
3k
views
Reference request for translating from Top to C*-alg
Some recent questions on MO (for example, Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?) have been about Gelfand duality — namely, that the categories of ...
- 18.7k
0
votes
1
answer
396
views
Characterization of Measureable Sets [closed]
Every countable union of rectangles in R2 is a Lebesgue measurable set. Is the converse true, too?
Specifically, I wonder whether the following statement is true:
Let A be a set in the unit square ...
- 99
7
votes
1
answer
682
views
$c_0$-direct sum of $\mathcal{K}(\mathcal{H})$
Let $\mathcal{K}(\mathcal{H})$ be the C*-algebra of compact operators on a Hilbert space $\mathcal{H}$. I am interested in the ($c_0$-)sum
$A=\sum \mathcal{K}(\mathcal{H})$
of countably many ...
- 113
17
votes
4
answers
2k
views
Banach-Mazur applied to a Hilbert space
The Banach-Mazur theorem says that every separable Banach space is isometric to a subspace of $C^0([0;1],R)$, the space of continuous real valued functions on the interval $[0;1]$, with the sup norm.
...
- 6,924
10
votes
5
answers
2k
views
Do subalgebras of C(X) admit a description in terms of the compact Hausdorff space X?
In light of the well-known theorem of Gelfand that, bluntly put, ends up saying that unital abelian C*-algebras are the 'same' as compact Hausdorff topological spaces, I tried to compile a dictionary ...
- 1,105
14
votes
4
answers
3k
views
Representing a product of matrix exponentials as the exponential of a sum
In Proof of a conjectured exponential formula, R. C. Thompson (1986) [edit: apparently, assuming Horn's conjecture] proved that if $A$ and $B$ are Hermitian matrices, then there exist unitary matrices ...
- 28.6k
4
votes
1
answer
775
views
Algebraically simple Banach algebras
There are plenty of semi-simple Banach algebras - this broad class includes C*-algebras and algebras of bounded operators on a given Banach space. On the other hand, it seems unlikely to me that there ...
- 113