  The class of absolutely norming operators on complex Hilbert spaces of
arbitrary dimensions was introduced in [6] and a spectral characterization
theorem for these operators was established in [11]. In this paper we extend
the concept of absolutely norming operators to various symmetric norms. We
establish a few spectral characterization theorems for operators on complex
Hilbert spaces that are absolutely norming with respect to various symmetric
norms. It is also shown that for many symmetric norms the absolutely norming
operators have the same spectral characterization as proven earlier for the
class of operators that are absolutely norming with respect to the usual
operator norm.
  Finally, we prove the existence of a symmetric norm on the algebra $\mathcal
B (\mathcal H)$ with respect to which even the identity operator does not
attain its norm.
