Classes of operators on vector valued integration spaces
Copyright policy )Classes of operators on vector valued integration spaces
AbstractLet Lp(Ω, K) denote the Banach space of weakly measurable functions F defined on a finite measure space and taking values in a separable Hilbert space K for which ∥ F ∥p = ( ∫ | F(ω) |p)1/p < + ∞. The bounded Hermitian operators on Lp(Ω, K) (in the sense of Lumer) are shown to be of the form , where B(ω) is a uniformly bounded Hermitian operator valued function on K. This extends the result known for classical Lp spaces. Further, this characterization is utilized to obtain a new proof of Cambern's theorem describing the surjective isometries of Lp(Ω, K). In addition, it is shown that every adjoint abelian operator on Lp(Ω, K) is scalar.
- University of Memphis United States
Spaces of vector- and operator-valued functions, Special classes of linear operators, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
Spaces of vector- and operator-valued functions, Special classes of linear operators, Spaces of measurable functions (\(L^p\)-spaces, Orlicz spaces, Köthe function spaces, Lorentz spaces, rearrangement invariant spaces, ideal spaces, etc.)
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