Spectrum (functional Analysis) - Spectrum of An Unbounded Operator

Spectrum of An Unbounded Operator

One can extend the definition of spectrum for unbounded operators on a Banach space X, operators which are no longer elements in the Banach algebra B(X). One proceeds in a manner similar to the bounded case. A complex number λ is said to be in the resolvent set, that is, the complement of the spectrum of a linear operator

if the operator

has a bounded inverse, i.e. if there exists a bounded operator

such that

A complex number λ is then in the spectrum if this property fails to hold. One can classify the spectrum in exactly the same way as in the bounded case.

The spectrum of an unbounded operator is in general a closed, possibly empty, subset of the complex plane.

For λ to be in the resolvent (i.e. not in the spectrum), as in the bounded case λI − T must be bijective, since it must have a two-sided inverse. As before if an inverse exists then its linearity is immediate, but in general it may not be bounded, so this condition must be checked separately.

However, boundedness of the inverse does follow directly from its existence if one introduces the additional assumption that T is closed; this follows from the closed graph theorem. Therefore, as in the bounded case, a complex number λ lies in the spectrum of a closed operator T if and only if λI − T is not bijective. Note that the class of closed operators includes all bounded operators.