A dissipative operator on a Banach space 
is a linear operator 
satisfying
 |
for every 
in the domain of 
and every 
, where 
is the identity operator.
On a complex Hilbert space, this is equivalent to
![R[<Ax,x>]<=0](/images/equations/DissipativeOperator/NumberedEquation2.svg) |
for every 
in the domain of 
,
where 
is the inner product of 
and 
, and 
denotes its real part. A dissipative
operator is maximal dissipative if it has no dissipative operator
extension 
for which 
is a proper subset of 
. It is 
-dissipative if 
has range 
for some 
. This range condition implies maximal dissipativity,
but the converse need not hold on a general Banach space. The Lumer-Phillips
theorem identifies the densely defined 
-dissipative operators as the infinitesimal
generators of contraction semigroups.
