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Convex Spectral Functions of Compact Operators

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J. M. Borwein, J. Read, A. S. Lewis and Q. J. Zhu

**Abstract**:
In this paper we consider
convex spectral functions. These are functions which are defined
on the ``spectral sequence'' of compact
\underline{self-ad}j\underline{oint} operators
on the complex Hilbert space of square summable sequences.
The spectral sequence of a compact self-adjoint operator is the
$c_0$ sequence of
(real!)
eigenvalues listed with multiplicity.
We see that the non-uniqueness of this spectral sequence is not
important given the proper conditions on the convex spectral function
under consideration.
We relate the subdifferentail and differential of
convex spectral functions on the space of
such operators to this simpler class of functions which act only on the
spectral sequence of the operator.
We also develop a number of conjugate formulas.
Extensions to nonconvex Lipschitz functions are considered.
A number of motivating examples are discussed:
the differentiability of the norms of
the Schatten $p$-spaces, duality formulae for the Calder\'on norms,
self-concordant barrieres and the $k$th largest eigenvalue of a selfadjoint
operator.

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