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sp.spectral concept – Relating basic spectral decomposition with Euclidean Jordan algebras retort

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sp.spectral concept – Relating basic spectral decomposition with Euclidean Jordan algebras

I’m presently entering into learning optimization issues over symmetric cones (NSCP) and I’m having some wretchedness to win one thing.

Let me first give some context, sorry whether it is repetitive to you: An Euclidean Jordan algebra $mathbb{J}$ is a vector
house outfitted with a Jordan product $circ$ and an internal product $langlecdot ,cdotrangle$. moreover,
if we assume the linear operator $L_x(y) = xcirc y$ for every $x,yinmathbb{J}$, we maintain that $$langle L_x(y),zrangle = langle y, L_x(z)rangle textual content{ for every } x,y,zinmathbb{J}.$$
That is, the operator $L_x(cdot)$ is self-adjoint witih respect to $langlecdot,cdotrangle$.

My query is how assassinate we relate the spectral decomposition of an part $xinmathbb{J}$ with the spectral decomposition of the operator $L_x(cdot)$. I’ve been investigating the primary examples ($mathbb{R}^n$, $mathbb{S}^n$, and $mathbb{R}^ntimesmathbb{R}$) and could not discover any frequent rule relating to this theme.

More typically, what’s the relation between Jordan frames and foundation from the underlying vector house, if there may be any?

Has anybody puzzled about these questions earlier than and will mission it out? Any help, pointers and references are welcome.

Thanks on your consideration!

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