  Recently a novel family of eigensolvers, called spectral indicator methods
(SIMs), was proposed. Given a region on the complex plane, SIMs first compute
an indicator by the spectral projection. The indicator is used to test if the
region contains eigenvalue(s). Then the region containing eigenvalues(s) is
subdivided and tested. The procedure is repeated until the eigenvalues are
identified within a specified precision. In this paper, using Cayley
transformation and Krylov subspaces, a memory efficient multilevel eigensolver
is proposed. The method uses less memory compared with the early versions of
SIMs and is particularly suitable to compute many eigenvalues of large sparse
(non-Hermitian) matrices. Several examples are presented for demonstration.
