# 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 $V\longrightarrow W$.

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

## 0.0.1 Basic properties

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

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

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

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

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

## 0.0.2 Miscellaneous properties

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

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