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 -

•

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.

Title	transpose operator
Canonical name	TransposeOperator
Date of creation	2013-03-22 17:34:19
Last modified on	2013-03-22 17:34:19
Owner	asteroid (17536)
Last modified by	asteroid (17536)
Numerical id	5
Author	asteroid (17536)
Entry type	Definition
Classification	msc 47A05
Classification	msc 46-00
Synonym	conjugate operator
Related topic	Transpose
Related topic	Adjoint5