Spectral Theory for Compact and Self‐Adjoint Fractional Resolvent Families

ABSTRACT This paper concerns with self‐adjoint fractional resolvent families. We show that a self‐adjoint operator generating a fractional resolvent family if and only if is bounded above. And a spectral decomposition form for compact and self‐adjoint fractional resolvent families is provided. We apply such decomposition to study the ergodic limits for the solutions of some inhomogeneous fractional differential equations.