Full-derivable points of $\mathcal {J}$-subspace lattice algebras

Abstract

Let $\mathcal{L}$ be a $\mathcal{J}$-subspace lattice on a complex Banach space $X$ and $\mbox{\rm Alg\,}{\mathcal L}$ the associated $\mathcal{J}$-subspace lattice algebra. We say that an operator $Z\in {\rm Alg\,}{\mathcal L}$ is a full-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a derivation and is a full-generalized-derivable point of $\mbox{\rm Alg\,}{\mathcal L}$ if every linear map $\delta$ from $\mbox{\rm Alg\,}{\mathcal L}$ into itself generalized derivable at $Z$ (i.e., $\delta(A)B+A\delta(B)-A\delta(I)B=\delta(Z)$ for any $A,B \in {\rm Alg\,}{\mathcal L}$ with $AB=Z$) is a generalized derivation. In this paper, we prove that if $Z\in\mbox{\rm Alg\,}{\mathcal L}$ is an injective operator or an operator with dense range, then $Z$ is a full-derivable point as well as a full-generalized-derivable point of ${\rm Alg\,}{\mathcal L}$.