Communications in Mathematical Analysis

General Adjoint on a Banach Space

Tepper L. Gill

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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.

Article information

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

Dates
First available in Project Euclid: 5 December 2017

Permanent link to this document
https://projecteuclid.org/euclid.cma/1512442821

Mathematical Reviews number (MathSciNet)
MR3722961

Zentralblatt MATH identifier
06841184

Subjects
Primary: 47B37: Operators on special spaces (weighted shifts, operators on sequence spaces, etc.)
Secondary: 28C20: Set functions and measures and integrals in infinite-dimensional spaces (Wiener measure, Gaussian measure, etc.) [See also 46G12, 58C35, 58D20, 60B11] 46B10: Duality and reflexivity [See also 46A25] 46C99: None of the above, but in this section

Keywords
dual space adjoint of operator basis Schatten classes

Citation

Gill, Tepper L. General Adjoint on a Banach Space. Commun. Math. Anal. 20 (2017), no. 2, 31--47. https://projecteuclid.org/euclid.cma/1512442821


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