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45
votes
0answers
2k 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 ...
26
votes
0answers
2k views

Example of a compact set that isn't the spectrum of an operator

This question is somewhat ill-posed (due to the word easy) and is triggered by idle curiosity: Is there an easy example of a (separable, infinite-dimensional) Banach space $X$ and a nonempty ...
15
votes
0answers
711 views

Decomposable Banach Spaces

An infinite dimensional Banach space $X$ is decomposable provided $X$ is the direct sum of two closed infinite dimensional subspaces; equivalently, if there is a bounded linear idempotent operator on ...
15
votes
0answers
525 views

Lipschitz Homeomorphisms Between Spheres of N-dimensional Spaces

Let $B_p^N$ be the unit ball of $\mathbb{R}^N$ under the $\ell_p^N$ norm. Question: Let $C_N$ be the infimum of all $C$ for which there is a homeomorphism $f_N$ from $B_\infty^N$ onto $B_2^N$ so ...
14
votes
0answers
443 views

Dual of the Ultraproduct of a Banach Space

Suppose we have a Banach space ultraproduct $(E_i)_U$. I can think of three natural objects which are "dual-like": $(E_i^*)_U$, the ultraproduct of the duals of the ground spaces. The space made up ...
13
votes
0answers
380 views

An open problem in convex geometry

Is it possible to find four norms $\| \cdot\|_k$ $( 1 \leq k \leq 4)$ on the plane such that a three-dimensional normed space containing four subspaces isometric to these normed planes does not exist? ...
13
votes
0answers
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 ...
12
votes
0answers
1k views

Infinite convex combinations in a Banach space

Let's say that a subset $C$ of a Banach space $X$ is $\sigma$-convex if the following property holds: For any sequence $(x_k)_{k\ge0}$ in $C$, and for any sequence of non-negative real numbers ...
10
votes
0answers
268 views

Can we find arbitrarily many elements of SU(2) generating a good copy of MAX($\ell_1^n$) inside VN(SU(2))?

In trying to prove that the answer to the title is "no", I was led to the following problem (which I think is equivalent to the question asked in the title, but can be stated independently). If ...
9
votes
0answers
448 views

Asymptotic non-distortion of the separable Hilbert space

By the work of E. Odell and Th. Schlumprecht, we know that the separable Hilbert space $\ell_2$ is arbitrarily distortable. But I don't know if an "asymptotic" version of their result is true. To ...
8
votes
0answers
224 views

Ultrapowers of Banach spaces without the continuum hypothesis

Let $\mathcal{U}$ be a non-trivial ultrafilter on the set of integers $\mathbb{N}$, and let $C(K)$ denote the Banach space of continuous functions on a compact $K$. Under the continuum hypothesis CH, ...
8
votes
0answers
225 views

Preduals of $\ell_1$

The space $\ell_1$ has loads of (isomorphic) predulas. They can be as weird as possible but I am interested in Banach lattices. Question: Let $X$ be a Banach lattice with dual isomorphic to $\ell_1$. ...
7
votes
0answers
241 views

Uniform closure of subspaces of Baire class 1

Describe a uniformly closed linear subspace $A \subset C([0,1])$ such that the space $B_1(A)$ is not uniformly complete. Here $B_1(A)$ is the set of all bounded functions $f$ which are pointwise ...
6
votes
0answers
184 views

Existence of injective operators with dense range

Given two separable (infinite dimensional) Banach spaces $X$ and $Y$, it is not difficult to show that there exists an injective (bounded linear) operator $T:X\to Y$ with range dense in $Y$. See S. ...
6
votes
0answers
119 views

Complex interpolation of a Banach space and its antidual when the space has a basis

Given a complex Banach space $X$ and its antidual $\hat X^*$, it is possible in some cases to apply the complex interpolation method, and get as $(X,\hat X^*)_{1/2}$ a Hilbert space. See [F. Watbled, ...
6
votes
0answers
214 views

Is there Ultracoproduct-like construction for topological spaces in general?

In http://arxiv.org/pdf/math/9704205.pdf they define the ultracoproduct of a sequence of compact Hausdorff spaces, $\sum_\mathcal{U}X_i$ along an ultrafilter $\mathcal{U}$ as the Wallman-Frink ...
6
votes
0answers
272 views

Minkowski's Inequality for Integrals in Orlicz spaces

EDIT: I have changed the question to have less parameters, fitting it into the context of Orlicz spaces. Suppose $f:[0,\infty)\to[0,\infty)$ is convex and increasing, ...
6
votes
0answers
351 views

Continuous projections in $\ell_1$ with norm $>1$

I was trying to find papers and articles about non-contractive continuous projections in $\ell_1(S)$ where $S$ is an arbitrary set. If it is not studied yet, I would like to know results for the case ...
6
votes
0answers
222 views

What is the intersection of the closures of left invertible operators and right invertible operators?

From Douglas Zare's answer (see Does $X$ embed in $Y$, and $Y$ embed in $X$, always imply that $X$ isomorphic onto $Y$?), one know that $$ \overline{G_{l}(X,Y)} \bigcap \overline{G_{r}(X,Y) } = ...
6
votes
0answers
435 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 ...
6
votes
0answers
1k views

Weak lower semi-continuity

Which conditions assure the weak lower semicontinuity of, say, an integral functional of the type $F(u):=\int_\Omega f(u(x),Du(x))dx$ on $W^{1,2}(\Omega,\mathbb{R}^N)$ for a bounded, if you will ...
6
votes
0answers
556 views

Hilbert subspaces of indefinite inner product spaces

Let $E$ be a real linear space, endowed with a non-degenerate symmetric bilinear form $(.,.)$. Suppose that the [indefinite] inner product space $(E,(.,.))$ satisfies the following [sequential] ...
5
votes
0answers
139 views

Products of spaces containing no copies of $\ell_2(\Gamma)$

Given an infinite set $\Gamma$, I would like to know if the class of Banach spaces containing no copies of $\ell_2(\Gamma)$ is stable under finite products. When $\Gamma$ is countable the answer is ...
5
votes
0answers
175 views

Second duals of Grothendieck spaces

The classical example of a Grothendieck space is $\ell_\infty$. It is also known that its even duals $\ell_\infty^{**}$, $\ell_\infty^{(4)}$, $\dots$ are Grothendieck spaces. (See, e.g., ...
5
votes
0answers
141 views

almost projective Banach space, complex scalars

It is well-known that if a real Banach space $E$ is "almost metrically projective" then $E$ is isometrically isomorphic to some $\ell^1(\Gamma)$. We say $E$ is "almost metrically projective" if ...
5
votes
0answers
149 views

Given that a conditional measure is Gaussian, how bad can the original measure be?

Let $X$ and $Y$ be Banach spaces, and let $\varphi : X \to Y$ be a continuous linear map. Suppose that $\mathbb P$ is a probability measure on $X$ which satisfies the continuous disintegration ...
4
votes
0answers
155 views

On the classification of injective Banach spaces

A Banach space $E$ is injective when it is complemented in each Banach space $X$ that contains it as a closed subspace. The space $E$ is $1$-injective if the copy of $E$ in $X$ is the range of a ...
4
votes
0answers
129 views

How to check a function belongs to a Banach space

My question comes from my confusion in studying the Fefferman-Stein inequality which says that for any $f\in L^p$ it holds $\|f\|_p\leq c \|f^\#\|_p$ with $f^\#$ the maximal function. This is an ...
4
votes
0answers
107 views

Hilbert metric of a sum of cones

Suppose that $K_{1}$ and $K_{2}$ are pointed closed cones in a finite-dimensional space $V$ whose Hilbert metrics $d_{1},d_{2}$ are known. Is there a way to express the Hilbert metric of $K_{1}+K_{2}$ ...
4
votes
0answers
433 views

Do the banded operators check the invariant subspace problem?

Let $H$ be an infinite dimensional separable Hilbert space and $B(H)$ the algebra of bounded operators. Invariant subspace problem: Let $T \in B(H)$. Is there a non-trivial closed $T$-invariant ...
4
votes
0answers
94 views

Categorical notions involving $\ell_p$ spaces.

First of all, apologies for a somewhat vague question but let me give a try. We know what the projective objects in the category of Banach spaces are: these are precisely $\ell_1(\Gamma)$-spaces. (One ...
4
votes
0answers
254 views

Self-dual finite-dimensional complex normed spaces

Suppose $X$ is a complex normed space of dimension 2 or 3 and $X$ is isometrically isomorphic to its dual. Is $X$ a Hilbert space? Remarks: There are easy counterexamples in the real case, and in ...
4
votes
0answers
326 views

Reflexive-saturated Banach spaces

Say that a Banach space $X$ is strongly saturated by reflexive subspaces if every closed subspace $Y\subset X$ contains a further reflexive subspace $Z\subset Y$ with $\mbox{dens }Y=\mbox{dens Z}$. If ...
4
votes
0answers
365 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 $X\oplus X$. ...
3
votes
0answers
116 views

Ultrapowers and complemented subspaces

Let $Y$ be a closed subspace of a Banach space $X$, and let $\mathcal{U}$ be a nontrivial ultrafilter on the set $\mathbb{N}$ of all integer numbers. It is not difficult to see that if $Y$ is ...
3
votes
0answers
120 views

Is the space $U_{0}$ constructed by J.Lindenstrauss Lipschitz complemented in $l^{1}$?

In $l^{1}$, we let $x_{n}=e_{n}-\frac{e_{2n+1}+e_{2n+2}}{2}(n=1,2,...)$, where $(e_{n})_{n=1}^{\infty}$ is the unit vector basis in $l^{1}$. We let $U_{0}$ be the subspace generated by the monotone ...
3
votes
0answers
58 views

$x\in Ext(B_X)$ has the Kadec property, implies that the slices form a neighborhood base of the norm topology

This is question 3.87 from Fabian's Functional Analysis and Infinite-Dimensional Geometry. The result is credited to Lin and Troyanski. Where on the net can I read a proof of this lemma? Any help ...
3
votes
0answers
369 views

question about bidual normed space

Consider a Banach $\mathbb{R}$-space E and an element $u\in E''$ such that : for all sequence $\phi_n\in E'$ which $\sigma(E',E)$-converges to $\phi \in E'$, one has $\lim u(\phi_n)=u(\phi)$. Is it ...
3
votes
0answers
126 views

Isometric automorphism of $c_0$ different than coordinate permutation

Does there exist an isometric automorphism of $c_0$ which is not a permutation of coordinates?
3
votes
0answers
1k views

Projective and injective tensor product

It is well known that for arbitrary Banach spaces $X$ and $Y$ we have that the dual space $(X \hat{\otimes}_{\pi} Y)^* = \mathcal{L}(X, Y^*)$. If we take $\ell^p$ and $\ell^q$ such that $p < ...
3
votes
0answers
569 views

Sigma-algebras on Banach Spaces.

I am very interested in counterexamples when cylindrical sigma-algebra is not equal to the borel. I read link text and link text. Especially interessted in l-infinity space. I've read Talgat and ...
3
votes
0answers
390 views

Banach lattice with AP but without BAP?

Is there an example of a Banach lattice with the approximation property of Grothendieck, but without the MAP (metric approximation property)?
2
votes
0answers
77 views

Birkhoff orthogonal of a Banach space in its bidual

Let $X$ be a Banach space embedded in $X^{**}$ in the usual way. We consider the set $$ O_X := \{ x^{**}\in X^{**} : \|x^{**}-x\|\geq \|x\| \textrm{ for all }x\in X\}. $$ I think this is the ...
2
votes
0answers
157 views

Centralizers and containment of $c_0$

I have this question also in MSE (see: http://math.stackexchange.com/questions/666053/centralizers-and-containment-of-c-0), but I have not got an answer there. So I thought I try my luck here. Let ...
2
votes
0answers
98 views

Projective tensor powers of Banach spaces over a normed field

Let $E$ be a Banach space over a complete normed field $\mathbb K$. Is it possible to classify all proper $E$ for which the projective seminorm $p_n$ defined on the $n$-th tensor power of $E$ is a ...
2
votes
0answers
100 views

Is reflexive Banach space valued scalarwise Lebesgue space isomorphic to the Bochner space?

I first specify the setting and then formulate the question precisely. (A very long post follows.) Definitions 1. For $E$ a (real Hausdorff) locally convex space, say that $E$ is suitable iff there ...
2
votes
0answers
68 views

Maximum Principle with Banach Control Space

This is a problem that seems very natural to me, but I couldn't find any formal statement in the literature for some time now. I am basically considering an autonomous optimal control problem in ...
2
votes
0answers
164 views

Cameron Martin space

I have seen two definitions of Cameron Martin space of a Gaussian measure $\nu$ on a Banach space (say $W$) and am unable to establish their equivalence. Any help would be appreciated. 1) It is the ...
2
votes
0answers
229 views

Finite codimensional subspaces of L(X,Y)

Let $X$ and $Y$ be separable Banach spaces and $L(X,Y) $ be the Banach space of bounded linear operators from $X$ to $Y$. Suppose $A$ is a norm closed finite codimensional subspace of $L(X,Y)$. My ...
2
votes
0answers
78 views

Form of finite dimensional contractive projection in $L_p$

Let $P$ be a finite dimensional contractive (norm 1) projection in $L_p$, $1 < p < \infty$. Then $P$ is of the following form: $Pf = \sum_{k=1}^n g_k \int h_kf$ Where $\|g_k\|_p = \|h_k\|_q = ...