**2**

votes

**1**answer

37 views

### A question on unconditionally $p$-summable sequences

We say that a sequence $(x_{n})_{n}$ in a Banach space $X$ is unconditionally $p$-summable ($1\leq p<\infty$) if
$$\sup_{x^{*}\in B_{X^{*}}}(\sum_{n=m}^{\infty}|\langle ...

**2**

votes

**0**answers

43 views

### On the relationship between the factorizations of an operator $T$ and its second adjoint $T^{**}$

Let $T$ be an operator from a space $X$ into a space $Y$ and let $1\leq p<\infty$. If $T^{**}$ has a factorization $T^{**}=RS:X^{**}\xrightarrow{S} l_{p}\xrightarrow{R}Y^{**}$, where $S$ is compact ...

**2**

votes

**1**answer

52 views

### Post composition of integral

Setup:
If $\langle \Omega, \mathfrak{F},\mu \rangle$ is a measure space, $f:\Omega \rightarrow E$ is a weakly-measurable function to a Banach space $E$, $g: E \rightarrow E'$ is a diffeomorphism and ...

**1**

vote

**0**answers

36 views

### On compact operators with domain $c_{0}$

Let $X$ be a Banach space and $T$ be a compact operator from $c_{0}$ into $X^{*}$. Let $\epsilon>0$. Is there a compact operator $S$ from $X$ into $l_{1}$ such that $S^{*}|_{c_{0}}=T$ and ...

**4**

votes

**0**answers

23 views

### How to characterize an operator $T$ that factors through a special space?

Let $T\in \mathcal{L}(X,Y)$ and $1<p<\infty$. My question is: Is there a convienent and useful characterization of the operator $T$ factoring through a space $Z$ satisfying ...

**6**

votes

**1**answer

264 views

### Two vector spaces with homeomorphic open subsets are isomorphic?

Is it true that if $ E,F$ are two topological vector spaces (or say Banach spaces) over $\mathbb{R}$ such that they have nonempty open subsets $U\subset E, V\subset F$ which are homeomorphic, then the ...

**1**

vote

**0**answers

58 views

### An operator factoring through a Banach space containing no copy of $l_{1}$

Is there an operator $T:X\rightarrow Y$ that factors through a Banach space $Z$ containing no complemented copy of $l_{1}$, but does not factor through any Banach space $W$ containg no copy of ...

**5**

votes

**1**answer

60 views

### A question on characterizing a Banach space containing no copy of $l_{1}$

Let $X$ be a Banach space. My question is: $X$ contains no copy of $l_{1}$ if and only if any operator from $X$ to $l_{1}$ is compact? I guess that the necessary part may be true. But is the ...

**5**

votes

**1**answer

60 views

### A question on compact operators with domain $l_{p}$

Suppose that $T$ is an operator from a Banach space $X$ to a Banach space $Y$. Let $1<p<q<\infty$. If $TS$ is compact for any operator $S:l_{p}\rightarrow X$, is $TR$ compact for any operator ...

**11**

votes

**2**answers

195 views

### Existence of closed operators with arbitrary dense domain of a given banach space

Consider any Banach space $X$, and let $Y$ be any dense subspace, then does it necessarily exist a closed linear operator $T$ defined on $X$, such that the domain of $T$ is exactly $Y$, i.e., ...

**7**

votes

**0**answers

190 views

### What is the name for a Banach space property closed under ultraproducts?

In Banach space theory, a super-property is a property of a Banach space that is preserved under ultrapowers. (Update (2015-09-28): The property must also be closed under isometric embeddings.) ...

**9**

votes

**1**answer

89 views

### Approximation via finite rank Cameron-Martin projections

Let $(W, \|\cdot\|_W)$ be a real separable Banach space equipped with
a non-degenerate Gaussian Borel measure $\mu$. Let $H \subset W$ be
the corresponding Cameron-Martin Hilbert space (also known as ...

**0**

votes

**0**answers

87 views

### does every compact convex set in c0 have but countably many extreme points

This seems plausible, given the properties of the unit ball of $c_0$.
I have a compact set in a complex Banach space $X$ whose closed convex hull has uncountably many extreme points. It would be ...

**1**

vote

**0**answers

86 views

### The dual of the space of smooth functions that vanish at infinity

Let $U \subset \Bbb R ^n$ be an open subset and let $\mathcal C$ be the space of the smooth functions on $U$ that vanish at infinity, endowed with the seminorms $p_\alpha (f) = \sup \limits _{x \in U} ...

**4**

votes

**1**answer

105 views

### How to characterize a Banach space $X$ such that any operator from $X$ to $l_{p}$ is compact?

Let $X$ be a Banach space and $1<p<\infty$. How to characterize $X$ such that any operator from $X$ to $l_{p}$ is compact? Are there any known or new results?

**6**

votes

**1**answer

149 views

### Is $T^{**}$ unconditionally $p$-summing whenever $T$ is unconditionally $p$-summing?

A sequence $(x_{n})_{n}$ in a Banach space $X$ is said to be unconditionally $p$-summable if $$\sup_{x^{*}\in B_{X^{*}}}\Bigl(\sum_{n=m}^{\infty}\lvert\langle ...

**2**

votes

**2**answers

248 views

### “Generalisation” of one-parameter semigroups

Let $(Y,\left\|\cdot\right\|_Y)$ be a Banach space and $A:D(A)\subset Y \to Y$ a closed operator. Studying dynamical systems of the form
\begin{equation}
u'=Au
\end{equation}
quickly leads to the ...

**11**

votes

**1**answer

252 views

### Is there a Gelfand-Naimark-like characterization of group algebras $L_1(G)$?

The Gelfand-Naimark theorem establishes that a complex commutative Banach algebra $A$ with an identity and an involution $x\to x^*$ satisfying $\|x x^*\|=\|x\|^2$ is (isometrically isomorphic to) a ...

**3**

votes

**2**answers

191 views

### Is there an easier proof to show that the closed convex hull of a normalized weakly null sequence is weakly compact?

In a paper that I am reading there is a following step:
Let $X$ be a Banach space and let $(x_k) \subset X$ be a normalized sequence that converges weakly to $0$.
Then $\overline{co}(x_k)$ is a ...

**3**

votes

**1**answer

245 views

### Proof of Pinelis (1992) - Banach space inequalities

I am reading Pinelis "An approach to inequalities for the distributions of infinite -dimensional martingales" and cannot follow his proof of Theorem 3:
Let $(f_n)$ be a martingale in a separable ...

**0**

votes

**1**answer

87 views

### Solutions of an nonlinear evolution problem

We consider the following continuous-time nonlinear evolution problem
\begin{equation}
\begin{cases} \dot{y}(t)=Ay(t)+F(y(t),u(t)),\quad t\geq0\\y(0)=f\in\mathcal{X}\end{cases}
\end{equation}
where ...

**0**

votes

**0**answers

83 views

### When do block sequences yield disjoint subspaces?

Let $X$ be a Banach space having a (unconditional, normalized) Schauder basis $(e_n)_n$. Suppose that $Y$ and $Z$ are (closed) block subspaces of $X$ having normalized block bases (with respect to ...

**2**

votes

**1**answer

131 views

### Completion of $C_0^{\infty}(\mathbb{R}^N)$ with norm $\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, \mathrm{d}x \Bigg)^{\frac{1}{2}}. $

I have a question that I could not find it any where.
Is the completion of $C_0^{\infty}(\mathbb{R}^N)$ with the respect to norm
$$\|u\|= \Bigg(\int_{{\mathbb{R}}^N} |\Delta u |^2 \, ...

**4**

votes

**2**answers

211 views

### Well-complemented copies of $\ell_p^n$

This must be surely known but I couldn't locate this problem in the literature. It popped out in a priori unrelated approximation problem but if true, would help me greatly.
Let $p\in (1,\infty)$.
...

**5**

votes

**1**answer

174 views

### Is the space of vectorial functions that are Dunford integrable complete?

Let $X$ a Banach Space and $(\Omega, \Sigma, \mu)$ a measure space. A function $F:\Omega\rightarrow X$ is Dunford integrable if $x^\ast\circ F$ is $\mu$-integrable for every $x^\ast\in X^\ast$. The ...

**3**

votes

**2**answers

137 views

### On the Lorentz sequence space $d(w,1)$

I am interested in examples of dual Banach spaces $X$ with the Schur property (weakly convergent sequences in $X$ are norm convergent) like $\ell_1$.
The Lorentz spaces $d(w,1)$ [Lindenstrauss and ...

**5**

votes

**0**answers

118 views

### Is there a quotient of $c_0$ without the approximation property?

The famous example of Enflo of a Banach space without the approximation property is actually a subspace of $c_0$. Is there a quotient of $c_0$ without the approximation property?
This would follow if ...

**6**

votes

**2**answers

279 views

### Non strictly-singular operators and complemented subspaces

If $T$ is a bounded operator which is not strictly singular, acting on a separable Banach space $X$, can one always find an infinite dimensional, closed and complemented, subspace $Y$ such that $T$ ...

**5**

votes

**1**answer

200 views

### Hahn Banach type extension of a Lipschitz map

The problem that I posted was a much generalized form of what I had in my mind. All I want to know the literature of Hahn-Banach type extension of Lipschitz map. I know only about the result by ...

**1**

vote

**1**answer

166 views

### Noncommutative analogs of classical Banach geometric properties

The scale of Schatten-von Neumann classes is noncommutatitve analog of classical $\ell_p$-spaces. A lot of researchers devoted their lives to study Banach geometric structure of these spaces. ...

**4**

votes

**0**answers

112 views

### $L_\infty(\mu)$ spaces non-isomorphic to a dual space

Given a measurable space $(\Omega,\mu)$ such that $L_\infty(\mu)$ is isomorphic to a dual space, $L_\infty(\mu)$ is an injective Banach space. Indeed, given a subspace $Y$ of $X$ and a norm-one ...

**0**

votes

**0**answers

102 views

### Unitarizability of group representations

Let $G$ be a Lie (or more general) group. Consider its continuous representation in a Banach space by isometries, i.e. preserving the Banach norm. Under what conditions this representation is ...

**4**

votes

**2**answers

339 views

### Is a specific sequentially closed subset of $M([0,1])$ closed?

Let $M([0,1])$ be the set of finite signed measures on $[0,1]$
(with the topology generated by the sets $\left\{ \mu \in M([0,1]) : \left| \int f(x) \mu(dx)- a\right| \leq \delta\right\}$ for all ...

**3**

votes

**1**answer

126 views

### A differential equation with continuous coefficient and no solution in a reflexive Banach space?

Is there a reflexive Banach space $B$ and a continuous map $f:B\to B$ such that the differential equation
$$ \frac{d x (t)}{dt} = f(x(t)) $$
with some initial condition $x(0)=x_0$ has no solution?

**-1**

votes

**1**answer

69 views

### Infinitesimal generator is bounded [closed]

Consider a strongly continuous semigroup of bounded linear operators $S(t):X\to X$. The infinitesimal generator of $S(t)$ is the linear operator $A:D(A)\subseteq X \to X$ defined by
...

**2**

votes

**1**answer

114 views

### Besov and Triebel-Lizorkin spaces

Let me start with a couple of notational reminders. For $\xi\in \mathbb R^n$,
$$
1=\varphi_{0}(\xi)+\sum_{\nu \ge 1}\varphi_{\nu}(\xi),\quad \varphi_{0}\in C^\infty_c(\mathbb R^{n}),\quad ...

**3**

votes

**0**answers

86 views

### Approximating the norm of a finite dimensional representation on a Banach space by irreducible representations

Let $G$ be a compact group, let $X$ be a Banach space and let $\pi$ be a linear and isometric representation of $G$ on $X$ that is continuous with respect to the strong operator norm. For $v \in X$, ...

**4**

votes

**2**answers

196 views

### Almost isometric embeddability implies isometric embeddability

Consider the following situation: Suppose $X$ is a Banach space such that for each finite metric space $M$ and each $\epsilon > 0$ for which $M$ bi-lipschitz embeds into $X$ with Lipschitz constant ...

**1**

vote

**0**answers

59 views

### Quadratic Variation of a Martingale in Hlibert Spaces

I'm looking at a Martingale (actually a Martingale difference sequence),
$$
M_n = \sum \delta M_n,
$$
and I'd like to prove something about convergence. If Martingale is Hilbert space valued ...

**4**

votes

**1**answer

125 views

### Operator on a Banach space

Let $T$ be a continuous operator on a Banach space $V$. Assume there exist $T$-stable finite-dimensional subspaces $V_i$ such that $\bigoplus_{i=1}^\infty V_i$ is dense in $V$, on $V_i$ the operator ...

**9**

votes

**0**answers

169 views

### Subspaces and quotients in Banach space theory

In Banach space theory (closed) subspaces and quotient seem to play a symmetric role. However, since the behavior of subspaces is more intuitive, subspaces appear more frequently. E.g., the theory of ...

**6**

votes

**0**answers

83 views

### Real interpolation space between the Wiener algebra and $L^2$

The Wiener algebra $W_n$ is the image by the Fourier transform of $L^1(\mathbb R^n)$. What is the (complex) interpolation space between $W_n$ and $L^2(\mathbb R^n)$? It is probably not true that for ...

**2**

votes

**0**answers

43 views

### Dual cone of 'positive' Bochner integrable functions

If we consider the space of integrable functions $L^1([0,1];\mathbb{R})$, it can be ordered by the convex cone of positive integrable functions $L^1([0,1];\mathbb{R}_+)$. It is known that the ...

**1**

vote

**1**answer

88 views

### A nullspace identity for operator exponentials

Let $X$ be a complex Banach space. Does validity of
$$
\mbox{ker}\left(e^{2\pi \imath \, T} - 1\right) = \overline{\sum\nolimits_{k\in \mathbb{Z}} \mbox{ker} (T-k) }\quad \forall \, T \in B(X,X)
$$
...

**0**

votes

**0**answers

53 views

### interpolation between Bochner spaces

Is there a reference for the interpolation result stating the existence of an embedding
\begin{equation}
L^2(I;W^{2,p}(\Omega)) \cap H^1(I;L^p(\Omega)) \hookrightarrow ...

**3**

votes

**0**answers

85 views

### independent symmetric 3-valued random variables in Lp

Consider the following excerpt from this paper:
Given $1<p<2$, $0<w\leq 1$ and $n\in\mathbb{N}$, we fix once and for all a sequence $f_j^{(n)}=f_{p,w,j}^{(n)}$, $1\leq j\leq n$, of ...

**1**

vote

**0**answers

110 views

### What is the trace of this operator in $L^\infty$ (if this question make sense)?

Let $\Omega$ be a closed domain in $\mathbb{R}^N$, and $\lambda$ the corresponding Lebesgue measure. We define in $(L^\infty(\Omega), \Vert \cdot \Vert_\infty)$ the following operator :
...

**2**

votes

**1**answer

106 views

### Rearrangments of Fourier series

Suppose one has a schauder basis $\{f_n\}_{n\in\mathbb{N}}$ for $L^p([0,1])$ and we wish to expand a function $f \in L^p([0,1])$ in our basis to get the expression
$$f(y)=\sum_{n=0}^{\infty} a_n ...

**2**

votes

**1**answer

111 views

### A question on p-summing operators

Let $(x_{n})_{n}$ be a $p$-summable sequence in a Banach space $X$. Define an operator $T$ from $l_{q}$(where $q=p/(p-1)$) to $X$ by $Te_{n}=x_{n}(n=1,2...)$. Is $T$ a $p$-summing operator?

**2**

votes

**1**answer

132 views

### When do Borel $\sigma$-algebras generated by the total variation norm and the weak* topology coincide?

I am almost certain that I read somewhere that the following is true, but I cannot seem to locate the reference. I would be most appreciative if someone could point me to a reference. The result was ...