Editing Unbounded operator (section)

== Transpose ==
{{See also|Transpose of a linear map}}

Let <math>T : B_1 \to B_2</math> be an operator between Banach spaces. Then the ''[[transpose]]'' (or ''dual'') <math>{}^t T: {B_2}^* \to {B_1}^*</math> of <math>T</math> is the linear operator satisfying:
<math display=block>\langle T x, y' \rangle = \langle x, \left({}^t T\right) y' \rangle</math>
for all <math>x \in B_1</math> and <math>y \in B_2^*.</math> Here, we used the notation: <math>\langle x, x' \rangle = x'(x).</math><ref>{{harvnb | Yoshida|1980 | p= 193}}</ref>

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

For any Hilbert space <math>H,</math> there is the anti-linear isomorphism:
<math display=block>J: H^* \to H</math>
given by <math>J f = y</math> where <math>f(x) = \langle x \mid y \rangle_H, (x \in H).</math> 
Through this isomorphism, the transpose <math>{}^t T</math> relates to the adjoint <math>T^*</math> in the following way:<ref>{{harvnb | Yoshida | 1980 | p = 196}}</ref>
<math display=block>T^* = J_1 \left({}^t T\right) J_2^{-1},</math>
where <math>J_j: H_j^* \to H_j</math>. (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.