Tepper L Gill and Woodford W Zachary
Adjoint for Operators in Banach Spaces II
(17K, amsart)
ABSTRACT. In a previous paper [GBZS] it was shown that each bounded linear operator $A$, defined on a separable Banach space $\mathcal{B}$, has a natural adjoint $A^*$. In this paper we prove that, for each closed linear operator $C$ defined on $ \mathcal{B}$, there exists a pair of contractions $A,\;B$ such that $C=AB^{-1}$. We also prove that, if $C$ is densely defined, then $B= (I-A^*A)^{-1/2}$. This result allows us to amend an oversight of [GBZS] by showing that every closed densely defined linear operator on $\mathcal{B}$ has a natural adjoint.