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. 638)
 MacDonald, Gordon W.

Distance from Idempotents to Nilpotents
We give bounds on the distance from a nonzero idempotent to the
set of nilpotents in the set of $n\times n$ matrices in terms of
the norm of the idempotent. We construct explicit idempotents and
nilpotents which achieve these distances, and determine exact
distances in some special cases.
Keywords:operator, matrix, nilpotent, idempotent, projection Categories:47A15, 47D03, 15A30 
