For $f\in L^2(0,\infty),$ define $(Tf)(x)=x^{-1}\int_0^x f(s)ds,$ for $x\in(0,\infty),$ then from hardy's inequality, $T\in B(L^2),$ my question is how to show that $T$ is not comp …

It seems to be a folk result that l_infinity has the approximation property, even the bounded approximation property, and also, I think, even the so-called propery pi (approximatio …

Let $X$ be a separable Banach space with its Borel $\sigma$-algebra $\mathcal F$. Let $x_n \to x$ in $X$. Fix a Gaussian covariance operator $K$, and let $\mathbb P_n$ and $\math …

Hello everyone, could anyone please tell me where can I find information about the Elton–Odell theorem? It states: For any infinite dimensional Banach space $X$ there is a $q &gt …

This is a sequel to another question I have asked. The notion of disintegration is a refinement of conditional probability to spaces which have more structure than abstract probab …

Let be $d$ a positive integer, $\Omega=\mathbb{R}^{\mathbb{Z}^d}$ and fix $R\geq 2$. We define weighted Banach spaces $$ \Omega_p:=\left\{ x\in \Omega\left| \left[\sum_{i\in\math …

Is there a "right" definition of the nuclear operator in the nonlinear framework ? Of course, such an operator must be compact, while a linear operator should be "nonlinearly" nucl …

This seems like a really simple question, but I'm struggling with it. Let $X$ be a separable Banach space, $H$ be a separable Hilbert space, and suppose $i : H \hookrightarrow X$ …

I'm wondering: What is the tensor product of $L^p({\bf R})$ with $L^q({\bf R})$? (For p=q=2, the answer clearly should be $L^2({\bf R}^2)$; for other values of $p$ and $q$, it is …

Lemma 1 from Anderson & Trapp's Shorted Operators, II isLet $A$ and $B$ be bounded operators on the Hilbert space $\mathcal H$. The following statements are equivalent: (1) r …

Consider the metric space on, say, ℝ2 induced by the various $L^p$ norms, and the group of isometries from that space into itself that preserve the origin. When $p=2$ I get t …

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 …

Let $K$ be the closed unit ball of $C[0,1]$, and let $f$ in $C(K,\mathbb{\, R})$. Is it true that there exists an infinite dimensional reflexive subspace $E$ of $C[0,1]$ s.t. $f(K\ …

Let $E$ be an infinite dimensional Banach space, let $E^{*}$ denote its continuous (i.e., Banach space) dual, and let $E^'$ be its algebraic dual. Clearly, $E^{*}$ is a proper vect …

I would like to elaborate a little bit on my previous question which can be found here. Firstly, let me recall that a separable Banach space $(X, \| \cdot \|)$ is said to be arbi …