Acharya, Cheban P.

Spectral theory, mathematical system theory, evolution equations, differential and difference equations [electronic resource] : 21st International Workshop on Operator Theory and Applications, Berlin, July 2010.It is known that lattice homomorphisms and G-solvable positive operators on Banach lattices have cyclic peripheral spectrum (See [17]). In my thesis I prove that positive contractions whose spectral radius is 1 on Banach lattices with increasing norm have cyclic peripheral point spectrum. I also prove that if the Banach lattice is a K B space satisfying the growth conditon and º is an eigenvalue of a positive contraction T such that [º] = 1, then 1 is also an eigenvalue of T as well as an eigenvalue of T¨, the dual of T. I also investigate the conditions on contraction operators on Hilbert lattices and AL-spaces which guanantee that 1 is an eigenvalue. As we know from [17], if T : E-E is a positive ideal irreducible operator on E such the r (T) = 1 is a pole of the resolvent R(º, T), then r (T) is simple pole with dimN (T -r(T)I) and ºper(T) is cyclic. Also all points of ºper(T) are simple poles of the resolvent R(º,T). SInce band irreducibility and º-order continuity do not imply ideal irreducibility [2], we prove the analogous results for band irreducible, º-order continuous operators.