## Communications in Mathematical Analysis

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

### The Adjoint Problem on Banach Spaces

Tepper L. Gill and Woodford W. Zachary

#### 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