We show that every Lorentz sequence space $d(\textbf {w},p)$ admits a 1-complemented subspace Y distinct from $\ell _p$ and containing no isomorph of $d(\textbf {w},p)$ . In the general case, this is only the second nontrivial complemented subspace in $d(\textbf {w},p)$ yet known. We also give an explicit representation of Y in the special case $\textbf {w}=(n^{-\theta })_{n=1}^\infty $ ( $0<\theta <1$ ) as the $\ell _p$ -sum of finite-dimensional copies of $d(\textbf {w},p)$ . As an application, we find a sixth distinct element in the lattice of closed ideals of $\mathcal {L}(d(\textbf {w},p))$ , of which only five were previously known in the general case.

Let $\mathcal{H}$ and $\mathcal{K}$ be infinite-dimensional separable Hilbert spaces and $\text{Lat}\,\mathcal{H}$ the lattice of all closed subspaces oh $\mathcal{H}$. We describe the general form of pairs of bijective maps $\phi ,\,\psi :\,\text{Lat}\,\mathcal{H}\,\to \,\text{Lat}\,\mathcal{K}$ having the property that for every pair $U,\,V\,\in \,\text{Lat}\,\mathcal{H}$ we have $\mathcal{H}\,=\,U\,\oplus \,V\,\Leftrightarrow \,\mathcal{K}\,=\,\phi \left( U \right)\,\oplus \,\psi \,\left( V \right)$. 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.

We study the complementation of the space $W\left( X,Y \right)$ of weakly compact operators, the space $K\left( X,Y \right)$ of compact operators, the space $U\left( X,Y \right)$ of unconditionally converging operators, and the space $CC\left( X,Y \right)$ of completely continuous operators in the space $L\left( X,Y \right)$ of bounded linear operators from $X$ to $Y$. Feder proved that if $X$ is infinite-dimensional and ${{c}_{0}}\,\to \,Y$, then $K\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$. Emmanuele and John showed that if ${{c}_{0}}\,\to \,K(X,\,Y)$, then $K\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$. Bator and Lewis showed that if $X$ is not a Grothendieck space and ${{c}_{0}}\,\to \,Y$, then $W\left( X,Y \right)$ is uncomplemented in $L\left( X,Y \right)$. In this paper, classical results of Kalton and separably determined operator ideals with property $\left( * \right)$ are used to obtain complementation results that yield these theorems as corollaries.

The Banach spaces $L(X,Y),K(X,Y),{{L}_{{{w}^{*}}}}({{X}^{*}},Y)$, and ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ are studied to determine when they contain the classical Banach spaces ${{c}_{o}}$ or ${{l}_{\infty }}$. The complementation of the Banach space $K(X,Y)$ in $L(X,Y)$ is discussed as well as what impact this complementation has on the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in $K(X,Y)$ or $L(X,Y)$. Results of Kalton, Feder, and Emmanuele concerning the complementation of $K(X,Y)$ in $L(X,Y)$ are generalized. Results concerning the complementation of the Banach space ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ in ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$ are also explored as well as how that complementation affects the embedding of ${{c}_{o}}$ or ${{l}_{\infty }}$ in ${{K}_{{{w}^{*}}}}({{X}^{*}},Y)$ or ${{L}_{{{w}^{*}}}}({{X}^{*}},Y)$. The ${{l}_{p}}$ spaces for $1\,=\,p\,<\,\infty $ are studied to determine when the space of compact operators from one ${{l}_{p}}$ space to another contains ${{c}_{o}}$. The paper contains a new result which classifies these spaces of operators. A new result using vector measures is given to provide more efficient proofs of theorems by Kalton, Feder, Emmanuele, Emmanuele and John, and Bator and Lewis.

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.