All Questions
4,339 questions with no upvoted or accepted answers
77
votes
0
answers
4k
views
2, 3, and 4 (a possible fixed point result ?)
The question below is related to the classical Browder-Goehde-Kirk fixed point theorem.
Let $K$ be the closed unit ball of $\ell^{2}$, and let $T:K\rightarrow K$
be a mapping such that
$$\Vert Tx-Ty\...
- 4,060
46
votes
0
answers
2k
views
Set-theoretic reformulation of the invariant subspace problem
The invariant subspace problem (ISP) for Hilbert spaces asks whether every bounded linear operator $A$ on $l^2$ (with complex scalars) must have a closed invariant subspace other than $\{0\}$ and $l^2$...
- 42.8k
33
votes
0
answers
1k
views
Subalgebras of von Neumann algebras
In the late 70s, Cuntz and Behncke had a paper
H. Behncke and J. Cuntz, Local Completeness of Operator Algebras, Proceedings of the American Mathematical Society, Vol. 62, No. 1 (Jan., 1977), pp. 95-...
- 25.5k
31
votes
0
answers
2k
views
Do there exist infinite-dimensional Banach spaces in which every bounded linear operator attains its norm?
Let $X$ be a Banach space, $L(X)$ the space of all bounded linear operators on $X$. We say that $A ∈ L(X)$ attains its norm if there exists $x ∈ X$ such that $\|x\| = 1$ and $\|Ax\| = \|A\|$. The ...
- 4,878
31
votes
0
answers
1k
views
"Three great cocycles" in Complex Analysis as cohomology generators
In his lecture notes, C. McMullen discusses "the three great cocycles" in Complex Analysis: the derivative $$f\mapsto\log f',$$ the non-linearity $$f\mapsto (\log f')'dz$$
and the Schwarzian ...
- 8,992
31
votes
0
answers
1k
views
When are two C*-algebras isomorphic as Banach spaces?
We may consider each $C^*$-algebra as a Banach space (by forgetting the multiplication and adjoint). I wonder how drastic this step is, i.e., which properties of the $C^*$-algebra are reflected by its ...
- 3,497
28
votes
0
answers
1k
views
Number of real roots of a polynomial
Let $P\in \mathbb{R}[x]$ be a polynomial such that $(P, P') = 1$. Suppose that we want to calculate the number of real roots of $P$ in the interval $[a, b]$ (to simplify, let us assume that $P(a), P(b)...
- 6,476
27
votes
1
answer
1k
views
The dual of $\mathrm{BV}$
$\DeclareMathOperator\BV{BV}\DeclareMathOperator\SBV{SBV}$I'm going to let $\BV := \BV(\mathbb{R}^d)$ denote the space of functions of bounded variation on $\mathbb{R}^d$. My question concerns the ...
- 683
27
votes
0
answers
1k
views
Unital $C^{*}$ algebras whose all elements have path connected spectrum
A unital $C^{*}$ algebra is called a "Path connected algebra" if the spectrum of all its elements is a path connected subset of $\mathbb{C}$.
What is an example of a non commutative ...
- 356
23
votes
0
answers
717
views
Which proofs of the fundamental theorem of algebra are "essentially the same" vs. "essentially different"?
The classic MO thread Ways to prove the fundamental theorem of algebra contains $60$ proofs of FTA, and I'm sure there are many more in the literature. It would be nice to have some way to organize ...
- 118k
23
votes
0
answers
1k
views
Laplace Transform in the context of Gelfand/Pontryagin
Questions:
Is there a class of objects (presumably related to locally compact abelian groups) for which the quasi-characters canonically generalize the Laplace transform?
If not, is there a ...
- 1,104
23
votes
0
answers
1k
views
Is analytic capacity inner regular?
For a compact set $K$ in the complex plane, define the analytic capacity of $K$ by
$$\gamma(K) := \sup |f'(\infty)|$$
where the supremum is taken over all functions $f$ holomorphic and bounded by $1$ ...
- 2,154
21
votes
0
answers
2k
views
Cartan–Oka vanishing in one variable without $\overline{\partial}$?
This is a literature question, about possible proofs of some very basic results in complex analysis.
Some key facts about holomorphic functions are proved via reduction to smooth functions, using $\...
- 21.3k
21
votes
0
answers
869
views
Noncommutative arithmetic mean geometric mean inequality and symmetric polynomials
While analyzing convergence speed of stochastic-gradient methods for convex optimization problems, Recht et al (2011) posed a tantalizing conjecture. It seems quite tricky, so after having struggled a ...
- 28.6k
21
votes
0
answers
732
views
Closed connected additive subgroups of the Hilbert space
It is a classical result that a closed and connected additive subgroup of $\mathbb{R}^n$ is necessarily a linear subspace. However, this is no longer true in infinite dimension: a very easy example is ...
- 60.5k
21
votes
0
answers
876
views
Are the eigenvalues of the Laplacian of a generic Kähler metric simple?
It is a theorem of Uhlenbeck that for a generic Riemannian metric, the Laplacian acting on functions has simple eigenvalues, i.e., all the eigenspaces are 1-dimensional. (Here "generic" means the set ...
- 6,247
20
votes
0
answers
666
views
Polynomials with roots in convex position
Let $\mathcal P_n$ denote the set of all monic polynomials of degree $n$ with real or complex coefficients such that $P\in\mathcal P_n$ if for all $k\in\lbrace 0,1,\dots,n-2\rbrace$ the $n-k$ roots of ...
- 17.9k
19
votes
0
answers
552
views
Talagrand's "Creating convexity" conjecture
We say a subset $A$ of $\mathbb{R}^N$ is balanced if
\begin{equation}
x \in A, \lambda \in [-1,1] \implies \lambda x \in A.
\end{equation}
Given a subset $A$ of $\mathbb{R}^N$, we write
\begin{...
- 484
17
votes
0
answers
677
views
Are dualizable topological vector spaces finite-dimensional?
Consider the symmetric monoidal category TVS of complete Hausdorff topological vector spaces equipped with the completed projective, injective, or inductive tensor product.
Every finite-dimensional ...
- 37.8k
17
votes
0
answers
488
views
Large almost equilateral sets in finite-dimensional Banach spaces
Question: Does there exist a function $C:~(0,1)\to
(0,\infty)$ such that for each $\varepsilon\in(0,1)$ every Banach space
$X$ of dimension $\ge C(\varepsilon)\log n$ contains an $n$-point
set $\{x_i\...
- 4,878
16
votes
0
answers
519
views
Gabriel's theorem for complex analytic spaces
Let $X,Y$ be noetherian schemes over $\mathbb{C}$.
Then, it is known that
$$
\text{Coh}(X) \simeq \text{Coh}(Y) \Rightarrow X \simeq Y,
$$
by P. Gabriel(1962).
Are there some results in the case of ...
- 889
16
votes
0
answers
531
views
Aligned roots of irreducible polynomials
It is well known from this famous question that the roots of a random polynomial tend to be close to the unit circle. So I was wondering in a somewhat converse sense: for an irreducible polynomial, is ...
- 13.4k
16
votes
0
answers
542
views
$C^*$-algebra generated by those operators that are bounded on every $\ell_p$
Suppose $T: c_{00} \to c_{00}$ is a linear map such that, when regarded as an infinite matrix, there is a uniform bound on the $\ell_1$-norms of its columns, and a uniform bound on the $\ell_1$-norms ...
- 25.8k
16
votes
0
answers
1k
views
What is the appropriate setting for Cauchy's Integral Formula?
For a $C^1$ function $f:U\to\mathbb{C}$, where $D\subset U\subseteq \mathbb{C}$ with piecewise $C^1$ boundary $\partial D$, we have the following generalized Cauchy integral formula:
$$
f(\zeta) = \...
- 1,104
16
votes
0
answers
1k
views
Finite Rank Commutators
My former student Detelin Dosev and I are interested in classifying the commutators in $L(X)$, the bounded linear operators on the Banach space $X$ (see our joint paper on my home page or the ArXiv ...
- 31.5k
15
votes
1
answer
601
views
Topological spaces in which countable intersections of dense open sets have dense interior
In certain topological spaces, known as Baire spaces (e.g., completely metrizable spaces), a countable intersection of dense open sets is dense.
Now consider the following strengthening of the Baire ...
- 32.5k
15
votes
0
answers
477
views
Quantitative Skorokhod embedding
The Skorokhod embedding theorem says that any random variable $X$ with $\mathbb E X=0$ and $\mathbb E[X^2]<\infty $ can be written as $X=B_{\tau }$ where $B$ is a Brownian motion and $\tau $ is a ...
- 723
15
votes
0
answers
365
views
Admissible relations in a Banach algebra
Suppose that $\mathbb{C}\left\langle x, y \right\rangle = R$ is a free (associative and unital) algebra and $f \in R$. I wonder whether there exists a (unital) Banach algebra $A$ and a non-zero pair $...
- 193
15
votes
0
answers
349
views
Is there support for the term "Gelfand algebra"?
In this question Yemon Choi asked whether there is a standard term for Banach algebras for which the submultiplicative law
($\|ab\| \leq \|a\| \|b\|$) is weakened to merely requiring the product to be ...
- 42.8k
15
votes
0
answers
1k
views
Homotopy equivalence vs weak homotopy equivalence in Gromov's h-principle
My question concerns Gromov's h-principle for open diffeomorphism-invariant partial differential relations on open manifolds; see e.g. Eliashberg/Mishachev: Introduction to the h-principle, §6.2.A and ...
- 648
14
votes
0
answers
860
views
strong topologies on $C_c^\infty$
UPDATE (27/08/2020): I realized after a comment from Jochen Wengenroth that there was at least one false premise behind my question, owing to the fact that analysts sometimes use the words "...
- 486
14
votes
0
answers
718
views
Lower bounds on analytic functions connected to Fox H
The question is related to the one I asked before and never got an answer to. Fourier transform of $f_a(x)= a^{-2}\exp(-|x|^a)$, $a \in (0,2)$, is decreasing in $a$ . I need to demonstrate that the ...
- 178
14
votes
0
answers
205
views
Have there been further developments on this scheme for polytope approximations to the unit ball of $\ell_p^n$?
A long time ago I happened to look at, and save (on a floppy disk!) for future reading, a copy of the following article:
W. T. Gowers, Polytope approximations of the unit ball of $l^n_p$.
In Convex ...
- 25.8k
14
votes
0
answers
633
views
Classes of (non-continuous) functions with the fixed point property
Let $K$ be a convex body in $ R^d$. (Say, a ball, say a cube...) For which classes $ \cal C$ of functions, every function $ f \in {\cal C}$ which takes $K$ into itself admits a fixed point in $K$.
...
- 24.7k
14
votes
0
answers
1k
views
Analytic continuation of the Dirichlet generating series of the multiplicative partition function
Apologies for the lengthy question, but it seems it's the only way i can convey my thoughts. Consider the Dirichlet series:
$$\kappa(s)=\prod_{m=2}^{\infty}\frac{1}{1-m^{-s}}=\sum_{n=1}^{\infty}\frac{\...
- 181
14
votes
0
answers
3k
views
Tanh version of a Fourier Transform?
I am trying to perform some computations in an environment where it is much easier to compute the hyperbolic tangent function (tanh) than cosines or sines. This prevents me from performing Fourier ...
- 3,979
14
votes
0
answers
3k
views
Meaning of Cauchy integral theorem - the (co)homology viewpoint
I'm not sure what follows is not just a complicated way to deduce a blatant triviality, or if is even correct. Let's try.
In the elementary theory of analytic functions of $1$ complex variable, one ...
- 23.3k
14
votes
0
answers
2k
views
Schwartz kernel theorem for A-linear operators
Let $X,Y \subset \mathbb{R}^n$ be open subsets. Denote by $C^\infty(X)$ the smooth functions on $X$, let $\mathcal{E}'(Y)$ be its dual space considered as a space of distributions. Let $L(C^\infty(X), ...
- 7,637
13
votes
0
answers
573
views
Classical (i.e. commutative) spaces with quantum symmetry but no classical symmetry
In a recent preprint (arXiv:2311.04889), my coauthors and I constructed a sequence of graphs with no classical symmetry which nevertheless have quantum symmetry.
For graphs this had been an open ...
- 2,339
13
votes
0
answers
174
views
Existence of more than two C*-norms on algebraic tensor product of C*-algebras
Let $A$ and $B$ be two C*-algebras. Then $(A,B)$ is called is a nuclear pair if there is a unique $C^*$-norm on the algebraic tensor product $A\odot B$.
If $A$ or $B$ is nuclear, then all pairs $(A,B)$...
- 536
13
votes
0
answers
818
views
Covering number estimates for Hölder balls
Let $\alpha \in (0,1]$, $r>0$ and $L>0$, and positive intwgers $n$ and $m$. The Arzela-Ascoli Theorem guarantees that the set $X(\alpha,L,r)$ of $f:[-1,1]^n\rightarrow [-r,r]^m$ with $\alpha$-...
- 5,405
13
votes
0
answers
491
views
Does Hahn-Banach for $\ell^\infty$ imply the existence of a non-measurable set?
Working over ZF but without the Axiom of Choice (AC), assume that the Hahn–Banach Theorem holds for $\ell^\infty$. Does it follow that there exists a set of real numbers that is not Lebesgue ...
- 82.7k
13
votes
0
answers
395
views
Converse to Riesz-Thorin Theorem
Let $T$ be an operator on simple functions on (say) $\mathbb{R}$.
The Riesz-Thorin interpolation theorem, in one form, says that the Riesz type diagram of $T$ is a convex subset of $[0,1]\times[0,1]$....
- 854
13
votes
0
answers
497
views
Is it possible that the following integral is $0$?
Given any integer $n\geqslant1$, let $E,F$ be two subsets of $\{\{i,j\}:1\leqslant i<j\leqslant n\}$ such that every two sets in $F$ are disjoint.
It is not difficult to see that
$$\int_{1<|z|&...
- 1,997
13
votes
0
answers
372
views
Finite dimensional approximation of Donaldson theory
In addition to the Seiberg-Witten invariant there has been further success with "finite dimensional approximations" of the Seiberg-Witten theory: Bauer-Furuta's stable (co)homotopy invariants, and ...
- 17.5k
13
votes
0
answers
324
views
Banach spaces with $d(X,Y) = 1$
We recall that the Banach-Mazur distance between two isomorphic Banach spaces is given by $d(X,Y) = \inf \{ \|T\| \|T^{-1}\| : T$ is an isomorphism from $X$ to $Y\}$.
It is a classical result that we ...
- 538
13
votes
0
answers
462
views
Is there a simple and reflexive Banach algebra?
There are many Banach algebras which, as Banach spaces, are reflexive. Of course, unitisation is just adding one dimension so this operation preserves reflexivity, hence there are many reflexive, ...
- 11.3k
13
votes
0
answers
323
views
Kolmogorov width for cartesian products
For an operator $T:X\to Y$ between Banach spaces with unit balls $B_X$ and $B_Y$ the sequence of Kolmogorov widths is
$$
\delta_n(T)=\inf\lbrace \delta>0: T(B_X)\subseteq \delta B_Y +L \text{ for ...
- 16.4k
13
votes
0
answers
474
views
Does anybody know if the Fourier algebra of SL(3,Z) has an approximate identity?
(Note to those who like to tidy LaTeX, or ${\rm \LaTeX}$: I kindly request that you don't put any LaTeX in the title of this question, nor change the bolds below to blackboard bold.)$\newcommand{\FA}{{...
- 25.8k
13
votes
0
answers
483
views
Where to use differential calculus on space of measures?
One great inside of Felix Otto is that the Wasserstein metric from optimal transportation gives the space of (finite second moment, probability) measures on $\mathbb{R}^n$ (or a manifold) a kind of ...
- 14.4k