The Adjoint of an Operator

The task of finding it is simplified enormously by virtue of the fact that a Hilbert space has an inner product. Its availability permits us, among others, to introduce the adjoint of the operator $A-\lambda B$ , an indispensible tool in integrating linear differential equations in one, two, and more dimensions. Because of its mathematical (as well as its physical) significance, let us remind ourselves about the adjoint of a linear operator. Its definition is as follows:

Definition. In an inner product space, say $\cal H$ , where

$$\langle f,g \rangle =\overline{\langle g,f \rangle}~~,$$

the Hermitian adjoint $T^H$ of a given operator $T$ is defined by the requirement that

$$\langle f,Tg \rangle =\langle T^H f,g \rangle$$   for all$$~f,g\in \cal H ~~.$$

What can one say about $(T^H)^H$ ? It follows from the this definition that

$$(T^H)^H=T ~~.$$

(Show this!) Operators which coincide with their adjoints are of particular importance and hence warrant their own

Definition. An operator T is said to be self-adjoint or Hermitian if $T^H=T$ , which is to say that they are defined by the requirement

$$\langle f,Tg \rangle =\langle T f,g \rangle$$   for all$$~f,g\in \cal H ~~.$$

Thus, if the hermitian adjoint of an operator equals the operator itself, then the operator is said to be Hermitian.