A quasinilpotent operator is a bounded operator 
on a nonzero complex Banach
space whose spectral radius is zero. The spectral radius formula expresses this condition
as
 |
(1)
|
It does not require a power of 
to vanish. For example, a nilpotent
matrix is quasinilpotent: 
has 
.
An example whose powers never vanish is obtained by integration. On the complex Banach space ![C[0,1]](/images/equations/QuasinilpotentOperator/Inline5.svg)
of continuous functions
with the supremum norm, it is defined for 
by
 |
(2)
|
For every positive integer 
, repeated integration
gives 
, where 1 is the constant
function with value 1. The bound 
,
together with this test function, gives the operator
norm
 |
(3)
|
Thus 
, while 
for every 
(Schenker 2008, pp. 29-2-29-3).
Quasinilpotence is a hypothesis in one formulation of the hyperinvariant subspace problem.
