Abstract The notion of capacity of a subspace which was introduced in  is used to prove new estimates on the shift of the eigenvalues which arises if the form domain of a self-adjoint and semibounded operator is restricted to a smaller subspace. The upper bound on the shift of the spectral bound given in  is improved and another lower bound is proved which leads to a generalization of Thirring's inequality if the underlying Hilbert space is an L 2-space. Moreover we prove a similar capacitary upper bound for the second eigenvalue. The results are applied to elliptic constant coefficient differential operators of arbitrary order. Finally it is given a capacitary characterization for the shift of the spectral bound being positive which works for operators with spectral bound of arbitrary type.