Communications in Mathematical Analysis

The Adjoint Problem on Banach Spaces

Tepper L. Gill and Woodford W. Zachary

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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.

Article information

Commun. Math. Anal., Volume 8, Number 1 (2010), 1-11.

First available in Project Euclid: 7 April 2010

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 45xx
Secondary: 46xx

adjoints Banach space embeddings Hilbert spaces


Gill, Tepper L.; Zachary, Woodford W. The Adjoint Problem on Banach Spaces. Commun. Math. Anal. 8 (2010), no. 1, 1--11.

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