arXiv subjects
Publications and source records attributed to Ricardo García.
We study those operators on a Hilbert space that can be lifted / extended to any twisted Hilbert space. We prove that these form an ideal of operators which contains all the Schatten classes. We characterize those multiplication operators on $\ell_p$ that are liftable / extensible through centralizers.
The purpose of this paper is to lay the foundations for the study of the problem of when $\Ext^n(X, Y)=0$ in Banach/quasi-Banach spaces. We provide a number of examples of couples $X,Y$ so that $\Ext^n(X,Y)$ is (or is not ) $0$, including the first example of a separable Banach space $\mathscr K$ so that $\Ext^n(\mathscr K, \mathscr K)\neq 0$ for all $n\in \N$. Such space moreover provides the first example of Banach spaces with infinite homological dimension/codimension. We also show that the homological dimension/codimension of Hilbert spaces is infinite. The final section is devoted to compare $\Ext^2(\cdot, \cdot)$ in Banach and Quasi-Banach spaces.
We show that $\Ext^2(\ell_2, \ell_2)\neq 0$ in the category of Banach spaces. This solves a sharpened version of Palamodov's problem and provides a solution to the second order version of Palais problem. We also show that $\Ext^2(\ell_1, \K)\neq 0$ in the category of quasi Banach spaces which solves the four-space problem for local convexity.
Let $X$ be a Banach space and let $κ(X)$ denote the kernel of a quotient map $\ell_1(Γ)\to X$. We show that $\Ext^2(X,X^*)=0$ if and only if bilinear forms on $κ(X)$ extend to $\ell_1(Γ)$. From that we obtain i) If $κ(X)$ is a $\mathcal L_1$-space then $\Ext^2(X,X^*)=0$; ii) If $X$ is separable, $κ(X)$ is not an $\mathcal L_1$ space and $\Ext^2(X,X^*)=0$ then $κ(X)$ has an unconditional basis. This provides new insight into a question of Palamodov in the category of Banach spaces.
We use the Aron-Berner extension to prove that the set of extreme points of the unit ball of the space of integral polynomials over a real Banach space $X$ is $\{\pm ϕ^k: ϕ\in X^*, \| ϕ\|=1\}$. With this description we show that, for real Banach spaces $X$ and $Y$, if $X$ is a non trivial $M$-ideal in $Y$, then $\hat\bigotimes^{k,s}_{ε_{k,s}} X$ (the $k$-th symmetric tensor product of $X$ endowed with the injective symmetric tensor norm) is \emph{never} an $M$-ideal in $\hat\bigotimes^{k,s}_{ε_{k,s}} Y$. This result marks up a difference with the behavior of non-symmetric tensors since, when $X$ is an $M$-ideal in $Y$, it is known that $\hat\bigotimes^k_{ε_k} X$ (the $k$-th tensor product of $X$ endowed with the injective tensor norm) is an $M$-ideal in $\hat\bigotimes^k_{ε_k} Y$. Nevertheless, if $X$ is Asplund, we prove that every integral $k$-homogeneous polynomial in $X$ has a unique extension to $Y$ that preserves the integral norm. We explicitly describe this extension. We also give necessary and sufficient conditions (related with the continuity of the Aron-Berner extension morphism) for a fixed $k$-homogeneous polynomial $P$ belonging to a maximal polynomial ideal $\Q(^kX)$ to have a unique norm preserving extension to $\Q(^kX^{**})$. To this end, we study the relationship between the bidual of the symmetric tensor product of a Banach space and the symmetric tensor product of its bidual and show (in the presence of the BAP) that both spaces have `the same local structure'. Other applications to the metric and isomorphic theory of symmetric tensor products and polynomial ideals are also given.
We study the problem of extension and lifting of operators belonging to certain operator ideals, as well as that of their associated polynomials and holomorphic functions. Our results provide a characterization of $\mathcal{L}_1$ and $\mathcal{L}_{\infty}$-spaces that includes and extends those of Lindenstrauss-Rosenthal \cite{LR} using compact operators and González-Gutiérrez \cite{GG} using compact polynomials. We display several examples to show the difference between extending and lifting compact (resp. weakly compact, unconditionally convergent, separable and Rosenthal) operators to operators of the same type. Finally, we show the previous results in a homological perspective, which helps the interested reader to understand the motivations and nature of the results presented.