properties of the transpose operator

In this entry, whenever $V,W$ are normed vector spaces, $ℬ(V,W)$ denotes the algebra of bounded linear operators $V⟶W$.

Let $X,Y,Z$ be normed vector spaces and $X^{\prime },Y^{\prime },Z^{\prime }$ be their continuous dual spaces. Let $T,S\in ℬ(X,Y)$,$R\in ℬ(Y,Z)$ and $\lambda \in ℂ$.

0.0.1 Basic properties

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

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

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

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

• If $T^{-1}$ exists and $T^{-1}\in ℬ(Y,X)$ then ${(T^{\prime })}^{-1}\in ℬ(X^{\prime },Y^{\prime })$ and ${(T^{\prime })}^{-1}={(T^{-1})}^{\prime }$.

0.0.2 Miscellaneous properties

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

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

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

• If $T^{\prime }$ is compact and $Y$ is a Banach space, then $T$ is also compact.