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Krein–Rutman type property and exponential separation of a noncompact operator

We investigate exponentially separated property for a noncompact linear operator 

 
 T
 T
 

 on a Banach space. We obtain the relationship between exponentially separated property and the well-known Krein–Rutman type property for a noncompact operator. Under the assumption of an essential spectral gap, we prove that any 

 
 u
 u
 

-bounded operator 

 
 T
 T
 

 with a reproducing cone admits the exponentially separated property and, hence, is of Krein–Rutman type automatically. We also establish an amenable sufficient condition for the exponentially separated property of some degenerate linear parabolic systems.