Definition (resolvent operator, regular value, resolvent set, spectrum, spectral value)
Let be an operator in a complex normed space . We write
where is a complex number and is the identical operator on the domain of . If the operator is injective, that is, it has an inverse (with domain ), then we call
the resolvent operator of for . A regular value of is a complex number for which the resolvent exists, has dense domain and is bounded. The set of all regular values of is called the resolvent set of and denoted . The complement of the resolvent set in the complex plane is called the spectrum of and denoted . The elements of the spectrum of are called the spectral values of .
Definition (point spectrum, residual spectrum, continuous spectrum)
Let be an operator in a complex normed space . The point spectrum is the set of all for which the resolvent does not exist. The residual spectrum is the set of all for which the resolvent exists, but is not densely defined. The continuous spectrum is the set of all for which the resolvent exists and is densely defined, but unbounded.
|If exists,||is densely defined||and is bounded …||… then belongs to the|
By definition, the sets , , and form a partition of the complex plane.