Let 
be a closed operator on a complex Banach
space 
.
The operator spectrum of 
is the complement in the complex
plane of its resolvent set,
 |
Thus 
belongs to 
if 
does not map 
bijectively onto 
with a bounded inverse, where 
is the identity operator.
For a bounded operator defined on all of 
, this reduces to the condition that
is not invertible on 
.
The resolvent set is an open set, so 
is closed.
