  This article deals with properties of spectra of operators on C(K)-spaces
with the Grothendieck property (e.g. l^{\infty}) and application to so called
J-class operators introduced by A. Manoussos and G. Costakis. We will show that
C(K) has the Grothendieck property if and only if the boundary of the spectrum
of every operator on C(K) consists entirely of eigenvalues of its adjoint. As a
consequence we will see that there does not exist invertible J-class operators
on C(K) with the Grothendieck property. In the third section we will give a
quantitative and qualitative characterization of all J-class operators on
l^{\infty} which are adjoints from operators on l^1.
