  Spectral operators of matrices proposed recently in [C. Ding, D.F. Sun, J.
Sun, and K.C. Toh, Math. Program. {\bf 168}, 509--531 (2018)] are a class of
matrix valued functions, which map matrices to matrices by applying a
vector-to-vector function to all eigenvalues/singular values of the underlying
matrices. Spectral operators play a crucial role in the study of various
applications involving matrices such as matrix optimization problems (MOPs)
{that include semidefinite programming as one of the most important example
classes}. In this paper, we will study more fundamental first- and second-order
properties of spectral operators, including the Lipschitz continuity,
$\rho$-order B(ouligand)-differentiability ($0<\rho\le 1$), $\rho$-order
G-semismoothness ($0<\rho\le 1$), and characterization of generalized
Jacobians.
