Closed Operator - Transpose

Let T : B₁ → B₂ be an operator between Banach spaces. Then the transpose (or dual) of T is an operator satisfying:

for all x in B₁ and y in B₂*. Here, we used the notation: .

The necessary and sufficient condition for the transpose of T to exist is that T is densely defined (for essentially the same reason as to adjoints, as discussed above.)

For any Hilbert space H, there is the anti-linear isomorphism:

given by where . Through this isomorphism, the transpose T' relates to the adjoint T∗ in the following way:

,

where . (For the finite-dimensional case, this corresponds to the fact that the adjoint of a matrix is its conjugate transpose.) Note that this gives the definition of adjoint in terms of a transpose.