1. CJM 2013 (vol 66 pp. 1143)
 Plevnik, Lucijan; Šemrl, Peter

Maps Preserving Complementarity of Closed Subspaces of a Hilbert Space
Let $\mathcal{H}$ and $\mathcal{K}$ be infinitedimensional separable
Hilbert spaces and ${\rm Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$.
We describe the general form of pairs of bijective maps $\phi , \psi :
{\rm Lat}\,\mathcal{H} \to {\rm Lat}\,\mathcal{K}$ having the property that for every pair
$U,V \in {\rm Lat}\,\mathcal{H}$ we have $\mathcal{H} = U \oplus V \iff \mathcal{K} = \phi (U) \oplus \psi (V)$. Then we reformulate this theorem as a description
of bijective image equality and kernel equality preserving maps acting on bounded linear idempotent operators. Several known
structural results for maps on idempotents are easy consequences.
Keywords:Hilbert space, lattice of closed subspaces, complemented subspaces, adjacent subspaces, idempotents Categories:46B20, 47B49 

2. CJM 2007 (vol 59 pp. 63)
 Ferenczi, Valentin; Galego, Elói Medina

Some Results on the SchroederBernstein Property for Separable Banach Spaces
We construct a continuum of mutually
nonisomorphic
separable Banach spaces which are complemented in each other.
Consequently, the SchroederBernstein Index of any of these spaces is
$2^{\aleph_0}$. Our
construction is based on a Banach space introduced by W. T. Gowers
and
B. Maurey in 1997.
We also use classical descriptive set theory methods, as in some
work of the first author and C. Rosendal, to improve some results
of P. G. Casazza and
of N. J. Kalton on the
SchroederBernstein Property for
spaces with an unconditional finitedimensional Schauder
decomposition.
Keywords:complemented subspaces,, SchroederBernstein property Categories:46B03, 46B20 
