properties of the transpose operator
In this entry, whenever $V,W$ are normed vector spaces^{}, $\mathcal{B}(V,W)$ denotes the algebra of bounded linear operators $V\u27f6W$.
Let $X,Y,Z$ be normed vector spaces and ${X}^{\prime},{Y}^{\prime},{Z}^{\prime}$ be their continuous dual spaces. Let $T,S\in \mathcal{B}(X,Y)$, $R\in \mathcal{B}(Y,Z)$ and $\lambda \in \u2102$.
0.0.1 Basic properties

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${T}^{\prime}\in \mathcal{B}({Y}^{\prime},{X}^{\prime})$ and $\parallel T\parallel =\parallel {T}^{\prime}\parallel $.

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${(\lambda T)}^{\prime}=\lambda {T}^{\prime}$.

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${(S+T)}^{\prime}={S}^{\prime}+{T}^{\prime}$.

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${(RT)}^{\prime}={T}^{\prime}{R}^{\prime}$.

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If ${T}^{1}$ exists and ${T}^{1}\in \mathcal{B}(Y,X)$ then ${({T}^{\prime})}^{1}\in \mathcal{B}({X}^{\prime},{Y}^{\prime})$ and ${({T}^{\prime})}^{1}={({T}^{1})}^{\prime}$.
0.0.2 Miscellaneous properties

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If we endow ${X}^{\prime}$ and ${Y}^{\prime}$ with the weak* topology^{} then ${T}^{\prime}:{Y}^{\prime}\u27f6{X}^{\prime}$ is continuous.

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$T$ is an isometric isomorphism if and only if ${T}^{\prime}$ is an isometric isomorphism.

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If $T$ is compact^{} (http://planetmath.org/CompactOperator) then ${T}^{\prime}$ is also compact.

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If ${T}^{\prime}$ is compact and $Y$ is a Banach space^{}, then $T$ is also compact.
Title  properties of the transpose operator 

Canonical name  PropertiesOfTheTransposeOperator 
Date of creation  20130322 17:36:02 
Last modified on  20130322 17:36:02 
Owner  asteroid (17536) 
Last modified by  asteroid (17536) 
Numerical id  7 
Author  asteroid (17536) 
Entry type  Theorem 
Classification  msc 4600 
Classification  msc 47A05 
Synonym  transpose operator properties 