Hermitian Operator

linear-algebra

Definition

Hermitian Operator

Let be a linear operator on a complex inner product space . The operator is hermitian if it is equal to its adjoint, that is,

Equivalently:

Spectral Representation

If is self-adjoined, then has a spectral representation of form:

where are the (real-valued) eigenvalues of of and each is an orthogonal projection.

Note that the convergence of this series is in the sense of the Hilbert-space norm, i.e., it holds that:

Operator Function

On the basis of the spectral representation, operator functions for self-adjoined operators can be defined as follows:
For a complex-valued function and a self-adjoined operator , define

In particular, for , can be represented as series:

It holds that is unitary and

where denotes the adjoint operator.