# transpose operator

Let $X,Y$ be normed vector spaces and $X^{\prime},Y^{\prime}$ be their continuous dual spaces.

- Let $T:X\longrightarrow Y$ be a bounded linear operator. The operator $T^{\prime}:Y^{\prime}\longrightarrow X^{\prime}$ given by

 $T^{\prime}\phi=\phi\circ T,\;\;\;\phi\in Y^{\prime}$

is called the transpose operator of $T$ or the conjugate operator of $T$.

It is clear that $T^{\prime}$ is well defined, i.e. $\phi\circ T\in X^{\prime}$, since the composition of two continuous linear operators is again a continuous linear operator.

Moreover, it can be easily checked that $T^{\prime}$ is a bounded linear operator.

Remarks -

Title transpose operator TransposeOperator 2013-03-22 17:34:19 2013-03-22 17:34:19 asteroid (17536) asteroid (17536) 5 asteroid (17536) Definition msc 47A05 msc 46-00 conjugate operator Transpose Adjoint5