Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Second derivatives of norms and contractive complementation in
  vector-valued spaces   We consider 1-complemented subspaces (ranges of contractive projections) of
vector-valued spaces $\ell_p(X)$, where $X$ is a Banach space with a
1-unconditional basis and $p \in (1,2)\cup (2,\infty)$. If the norm of $X$ is
twice continuously differentiable and satisfies certain conditions connecting
the norm and the notion of disjointness with respect to the basis, then we
prove that every 1-complemented subspace of $\ell_p(X)$ admits a basis of
mutually disjoint elements. Moreover, we show that every contractive projection
is then an averaging operator. We apply our results to the space
$\ell_p(\ell_q)$ with $p,q\in (1,2)\cup (2,\infty)$ and obtain a complete
characterization of its 1-complemented subspaces.

25
The categories of this paper are: patt-sol
22
The categories of this paper are: chao-dyn
11
The categories of this paper are: hep-lat
8
The categories of this paper are: hep-ex
0
The categories of this paper are: hep-ph
4
The categories of this paper are: cond-mat
17
The categories of this paper are: q-fin
23
The categories of this paper are: solv-int
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
29
The categories of this paper are: funct-an
14
The categories of this paper are: math-ph
24
The categories of this paper are: dg-ga
16
The categories of this paper are: stat
36
The categories of this paper are: cmp-lg
1
The categories of this paper are: cs