Spectral Conjugate

A spectral function is one expressible as the composition of a symmetric function and the ordered eigenvalues. In a Euclidean Jordan algebra X of degreen, the theme is that the calculation of the conjugate and the subdifferential of a spectral function effectively reduces to a calculation in Rn. The trace inner product is less than or equal to the inner product of the ordered eigenvalues: $$\left \langle \boldsymbol {x}^{\ast },\boldsymbol {x}\right \rangle \leq \left \langle \boldsymbol {\lambda }\left ( \boldsymbol {x}^{\ast }\right ) ,\boldsymbol {\lambda }\left ( \boldsymbol {x}\right ) \right \rangle \!.$$ Equality holds if and only if the two vectors have a simultaneous ordered spectral decomposition. A spectral function is one expressible as the composition of a symmetric function and the ordered eigenvalues: $$f\left [ \boldsymbol {\lambda }\left ( \boldsymbol {x}\right ) \right ]$$ . The conjugate (f∘λ)*=f*∘λ. Spectral conjugacy is widespread in applications.