A Banach space operator T ∈ has a left m-inverse (resp., an essential left m-inverse) for some integer m ≥ 1 if there exists an operator S ∈ (resp., an operator S ∈ and a compact operator K ∈ ) such that ${\sum }_{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)S^{m-i}{T}^{m-i}=0$ (resp., ${\sum }_{i=0}^m{(-1)}^i\left(\genfrac{}{}{0pt}{}{m}{i}\right)T^{m-i}{S}^{m-i}=K$). If $T_i$ is left $m_i$-invertible (resp., essentially left $m_i$-invertible), then the tensor product T₁ ⊗ T₂ is left (m₁ + m₂-1)-invertible (resp., essentially left (m₁ + m₂-1)-invertible). Furthermore, if T₁ is strictly left m-invertible (resp., strictly essentially left m-invertible), then...

The boundedness properties of commutators for operators are of central importance in Mathematical Analysis, and some of these commutators arise in a natural way from interpolation theory. Our aim is to present a general abstract method to prove the boundedness of the commutator $[T,\Omega ]$ for linear operators $T$ and certain unbounded operators $\Omega$ that appear in interpolation theory, previously known and a priori unrelated for both real and complex interpolation methods, and also to show how the abstract result...

We prove that the separating space of a Lie homomorphism from a Banach algebra onto a Banach algebra is contained in the centre modulo the radical.

The questions when a derivation on a Jordan-Banach algebra has quasi-nilpotent values, and when it has the range in the radical, are discussed.

We prove that the spaces of (α,β)-derivations on certain operator algebras are topologically reflexive in the weak operator topology. Characterizations of automorphisms and (α,β)-derivations on reflexive algebras are also given.