Approximation property and nuclearity on mixed-norm $L^p$, modulation and Wiener amalgam spaces

Abstract

In this paper we first prove the metric approximation property for weighted mixed-norm $L_w^{(p_1,\dots ,p_n)}$ spaces. Using Gabor frame representation this implies that the same property holds in weighted modulation and Wiener amalgam spaces. As a consequence, Grothendieck's theory becomes applicable, and we give criteria for nuclearity and $r$-nuclearity for operators acting on these space as well as derive the corresponding trace formulae. Finally, we apply the notion of nuclearity to functions of the harmonic oscillator on modulation spaces.

22 pages; slightly updated final version, to appear in J. Lond. Math. Soc

References in corpus (3)

• Schatten classes on compact manifolds: Kernel conditions
• Bounded Approximation Property for Sobolev spaces on simply-connected planar domains
• Continuity and Schatten properties for pseudo-differential operators with symbols in quasi-Banach modulation spaces or Hörmander classes

Cited by in corpus (3)

• Modulation spaces associated to tensor products of amalgam spaces
• Grothendieck-Lidskii trace formula for mixed-norm and variable Lebesgue spaces
• Nuclear pseudo-differential operators in Besov spaces on compact Lie groups