transpose operator

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

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

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

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

It is clear that $T^{\prime }$ is well defined, i.e. $\varphi \circ T\in X^{\prime }$, 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 -

• When the vector spaces are finite dimensional, the transpose operator corresponds to transposing (http://planetmath.org/Transpose) the matrix associated to it.

• For Hilbert spaces, a somewhat similar definition is that of adjoint operator. But this two notions do not coincide: while the transpose operator corresponds to the transpose of a matrix, the adjoint operator corresponds to the conjugate transpose of a matrix.