Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Uniqueness of unconditional basis of $H_p(\mathbb{T})\oplus\ell_{2}$ and
  $H_p(\mathbb{T})\oplus\mathcal{T}^{(2)}$ for $0<p<1$   Our goal in this paper is to advance the state of the art of the topic of
uniqueness of unconditional basis. To that end we establish general conditions
on a pair $(\mathbb{X}, \mathbb{Y})$ formed by a quasi-Banach space
$\mathbb{X}$ and a Banach space $\mathbb{Y}$ which guarantee that every
unconditional basis of their direct sum $\mathbb{X}\oplus\mathbb{Y}$ splits
into unconditional bases of each summand. As application of our methods we
obtain that, among others, the spaces $H_p(\mathbb{T}^d)
\oplus\mathcal{T}^{(2)}$ and $H_p(\mathbb{T}^d)\oplus\ell_2$, for $p\in(0,1)$
and $d\in\mathbb{N}$, have a unique unconditional basis (up to equivalence and
permutation).

30
The categories of this paper are: q-alg
27
The categories of this paper are: comp-gas
6
The categories of this paper are: astro-ph
34
The categories of this paper are: chem-ph
22
The categories of this paper are: chao-dyn
8
The categories of this paper are: hep-ex
1
The categories of this paper are: cs
32
The categories of this paper are: atom-ph
13
The categories of this paper are: quant-ph
11
The categories of this paper are: hep-lat
25
The categories of this paper are: patt-sol
3
The categories of this paper are: physics
35
The categories of this paper are: mtrl-th
28
The categories of this paper are: alg-geom
16
The categories of this paper are: stat
33
The categories of this paper are: plasm-ph