  This paper is concerned with the design and analysis of a fully adaptive
eigenvalue solver for linear symmetric operators. After transforming the
original problem into an equivalent one formulated on $\ell_2$, the space of
square summable sequences, the problem becomes sufficiently well conditioned so
that a gradient type iteration can be shown to reduce the error by some fixed
factor per step. It then remains to realize these (ideal) iterations within
suitable dynamically updated error tolerances. It is shown under which
circumstances the adaptive scheme exhibits in some sense asymptotically optimal
complexity.
