Definition
Given an operator
{% T:H \rightarrow K %}
if there is another operator
{% T^* : K \rightarrow H %}
such that
{% \langle Tx, y \rangle = \langle x, T^*y \rangle %}
see inner product
then {% T^* %} is called the adjoint of {% T %}.
Existence
Given two Hilbert spaces {% H %} and {% K %}, and a bounded operator {% T %}, then there exists a unique Hilbert adjoint {% T^* %} of {% T %}.
see Robinson chpt 13
Physics
In the physics literature, the definition of adjoint is reversed.
{% \langle x | Tt \rangle = \langle T^*x | t \rangle %}
this is due to the
Dirac notation
which labels a vector as a
key {% |\psi \rangle %}
Topics
- Self Adjoint - a self adjoint operator is an operator where {% T^* = T %}