General Adjoint on a Banach Space

Abstract

In this paper, we show that the continuous dense embedding of a separable Banach space $\mathcal B$ into a Hilbert space $\mathcal H$ offers a new tool for studying the structure of operators on a Banach space. We use this embedding to demonstrate that the dual of a Banach space is not unique. As a application, we consider this non-uniqueness within the $\mathbb C[0,1] \subset L^2[0,1]$ setting. We then extend our theory every separable Banach space $\mathcal B$. In particular, we show that every closed densely defined linear operator $A$ on $\mathcal B$ has a unique adjoint $A^*$ defined on $\mathcal B$ and that $\mathcal L[\mathcal B]$, the bounded linear operators on $\mathcal B$, are continuously embedded in $\mathcal L[\mathcal H]$. This allows us to define the Schatten classes for $\mathcal L[\mathcal B]$ as the restriction of a subset of $\mathcal L[\mathcal H]$. Thus, the structure of $\mathcal L[\mathcal B]$, particularly the structure of the compact operators $\mathbb K[\mathcal B]$, is unrelated to the basis or approximation problems for compact operators. We conclude that for the Enflo space $\mathcal B_e$, we can provide a representation for compact operators that is very close to the same representation for a Hilbert space, but the norm limit of the partial sums may not converge, which is the only missing property.