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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 infinite-dimensional 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, idempotentsCategories:46B20, 47B49

2. CJM 2007 (vol 59 pp. 63)

Ferenczi, Valentin; Galego, Elói Medina
 Some Results on the Schroeder--Bernstein Property for Separable Banach Spaces We construct a continuum of mutually non-isomorphic separable Banach spaces which are complemented in each other. Consequently, the Schroeder--Bernstein 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 Schroeder--Bernstein Property for spaces with an unconditional finite-dimensional Schauder decomposition. Keywords:complemented subspaces,, Schroeder-Bernstein propertyCategories:46B03, 46B20
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