transpose operator
Let X,Y be normed vector spaces and X′,Y′ be their continuous dual spaces.
- Let T:X⟶Y be a bounded linear operator. The operator T′:Y′⟶X′ given by
T′ϕ=ϕ∘T,ϕ∈Y′ |
is called the transpose operator of T or the conjugate operator of T.
It is clear that T′ is well defined, i.e. ϕ∘T∈X′, since the composition of two continuous linear operators is again a continuous linear operator.
Moreover, it can be easily checked that T′ 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 |