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General Adjoint on a Banach Space

Commun. Math. Anal.
Volume 20, Number 2 (2017), 31 - 47

General Adjoint on a Banach Space

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Abstract

In this paper, we show that the continuous dense embedding of a separable Banach space $\mcB$ into a Hilbert space $\mcH$ 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 $\C[0,1] \subset L^2[0,1]$ setting. We then extend our theory every separable Banach space $\mcB$. In particular, we show that every closed densely defined linear operator $A$ on $\mcB$ has a unique adjoint $A^*$ defined on $\mcB$ and that $\mcL[\mcB]$, the bounded linear operators on $\mcB$, are continuously embedded in $\mcL[\mcH]$. This allows us to define the Schatten classes for $\mcL[\mcB]$ as the restriction of a subset of $\mcL[\mcH]$. Thus, the structure of $\mcL[\mcB]$, particularly the structure of the compact operators $\K[\mcB]$, is unrelated to the basis or approximation problems for compact operators. We conclude that for the Enflo space $\mcB_e$, 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.