Various Continuity Properties of the Deconvolution (or Epigraphical Difference)

Abstract

It is known that the inf-convolution (or epigraphical sum)

$$ (f\mathop + \limits_e g)(x) = \inf \{ f(u) + g(x - u):u \in X\} $$

of convex proper lower semicontinuous functions defined on a normed linear space is continuous, under some assumptions, with respect to the usual variational convergences (Mosco convergence in the reflexive case, bounded Hausdorff topology in the general case). The deconvolution (or epigraphical difference) defined by

$$ (k\mathop {{\text{ }} - }\limits_e g)(x) = \sup \{ k(x + u) - g(u):g(u) < + \infty \} $$

is the smallest solution, when it exists, of the equation $\xi \mathop {{\text{ }} + }\limits_e g = k$. In this paper we present continuity results on the mappings $k \mapsto k\mathop {{\text{ }} - }\limits_e g$ and $g \mapsto k\mathop {{\text{ }} - }\limits_e g$ with respect to the quoted topologies and to some intermediate topologies lying between Mosco and bounded Hausdorff topologies. Applications are given to continuity of the parallel subtraction of operators in Hilbert spaces.