Let 
be a linear operator on a separable Hilbert space. The spectrum 
of 
is the set of 
such that 
is not invertible on all of the Hilbert
space, where the 
s
are complex numbers and 
is the identity operator.
The definition can also be stated in terms of the resolvent of an operator
 |
and then the spectrum is defined to be the complement of 
in the complex plane.
It is easy to demonstrate that 
is an open set, which shows
that the spectrum 
is closed.
If 
is a domain in 
(i.e., a Lebesgue measurable subset of 
with finite nonzero Lebesgue
measure), then a set 
is a spectrum of 
if 
is an orthogonal
basis of 
(Iosevich et al. 1999).
