The Adjoint Problem on Banach Spaces

Abstract

In this paper we survey recent work on the existence of an adjoint for operators on Banach spaces and applications. In [GBZS] it was shown that each bounded linear operator $A$, defined on a separable Banach space $\mathcal{B}$, has a natural adjoint $A^*$ defined on the space. Here, we show 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 show that, if $C$ is densely defined, then $B= (I-A^*A)^{-1/2}$. This result allows us to extend the results of [GBZS] (in a domain independent way) by showing that every closed densely defined linear operator on $\mathcal{B}$ has a natural adjoint. As an application, we show that our theory allows us to provide a natural definition for the Schatten class of operators in separable Banach spaces. In the process, we extend an important theorem due to Professor Lax.