This expository paper is mainly dedicated to structural properties of the elementary operators \(\mathcal{E}_{A, B}: s\mapsto\sum_{j=1}^nA_jsB_j\), where \(A=(A_1,\dots, A_n)\), \(B=(B_1,\dots, B_n)\in L(X)^n\) are fixed \(n\)-tuples of bounded operators on \(X\) and \(X\) is a (classical) Banach space. It concentrates on aspects of the theory of elementary operators that, roughly speaking, involves “Banach space techniques”, which mean, e.g., basic sequence techniques applied in \(X\) or in the compact operators \(K(X)\) on \(X\), facts about the structure of complemented subspaces of classical Banach spaces \(X\), as well as useful special properties of the space \(X\) (such as approximation properties or the Dunford–Pettis property). It turns out that Banach space techniques are helpful also when \(X\) is a Hilbert space.
In Section 2, the authors discuss various qualitative properties such as (weak) compactness or strict singularity of the basic two-sided multiplications \(S\mapsto ASB\) for \(A, B\in L(X)\). In Section 3, the authors concentrate on questions related to the norms and spectra of elementary operators in various settings. The authors include a quite detailed proof, using only elementary concepts, of the known formula \(\sigma(\mathcal E_{A, B})=\sigma_T(A)\circ\sigma_T(B)\) for the spectrum of \(\mathcal E_{A,B}\) in terms of the Taylor joint spectra of the \(n\)-tuples \(A\) and \(B\). The authors also describe the state of the art in computing the norm of elementary operators. Section 4 discusses properties of the included elementary operators on the Calkin algebra \(L(X)/K(X)\), such as a solution to the Fong–Sourour conjecture in the case where the Banach space \(X\) has an unconditional basis, and various rigidity properties of these operators. The results included here demonstrate that elementary operators have nicer properties on the Calkin algebra. There is some parallel research about tensor product operators \(A\hat\otimes_\alpha B\) for various tensor norms \(\alpha\) and fixed operators \(A,B\), which may be more familiar to readers with a background in Banach space theory.
For the entire collection see [Zbl 1106.46007].