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

The Spectrum of Operators on C(K) with the Grothendieck Property and Characterization of J-Class Operators which are adjoints