Convex Spectral Functions of Compact Operators

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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