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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 ofmatrix valued functions, which map matrices to matrices by applying avector-to-vector function to all eigenvalues/singular values of the underlyingmatrices. Spectral operators play a crucial role in the study of variousapplications involving matrices such as matrix optimization problems (MOPs){that include semidefinite programming as one of the most important exampleclasses}. In this paper, we will study more fundamental first- and second-orderproperties of spectral operators, including the Lipschitz continuity,$\rho$-order B(ouligand)-differentiability ($0<\rho\le 1$), $\rho$-orderG-semismoothness ($0<\rho\le 1$), and characterization of generalizedJacobians.

Spectral operators of matrices: semismoothness and characterizations of the generalized Jacobian