Communications in Mathematical Analysis

The Adjoint Problem on Banach Spaces

Tepper L. Gill and Woodford W. Zachary

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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.

Article information

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

Dates
First available in Project Euclid: 7 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.cma/1270646490

Mathematical Reviews number (MathSciNet)
MR2551489

Zentralblatt MATH identifier
1172.47001

Subjects
Primary: 45xx
Secondary: 46xx

Keywords
adjoints Banach space embeddings Hilbert spaces

Citation

Gill, Tepper L.; Zachary, Woodford W. The Adjoint Problem on Banach Spaces. Commun. Math. Anal. 8 (2010), no. 1, 1--11. https://projecteuclid.org/euclid.cma/1270646490


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