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On the intrinsic and the spatial numerical range | | Miguel Martin
; Javier Meri
; Rafael Paya
; | | Date: |
4 Mar 2005 | | Subject: | Functional Analysis MSC-class: 46B20; 47A12 | math.FA | | Affiliation: | Granada, Spain | | Abstract: | For a bounded function $f$ from the unit sphere of a closed subspace $X$ of a Banach space $Y$, we study when the closed convex hull of its spatial numerical range $W(f)$ is equal to its intrinsic numerical range $V(f)$. We show that for every infinite-dimensional Banach space $X$ there is a superspace $Y$ and a bounded linear operator $T:Xlongrightarrow Y$ such that $ar{co} W(T)
eq V(T)$. We also show that, up to renormig, for every non-reflexive Banach space $Y$, one can find a closed subspace $X$ and a bounded linear operator $Tin L(X,Y)$ such that $ar{co} W(T)
eq V(T)$. Finally, we introduce a sufficient condition for the closed convex hull of the spatial numerical range to be equal to the intrinsic numerical range, which we call the Bishop-Phelps-Bollobas property, and which is weaker than the uniform smoothness and the finite-dimensionality. We characterize strong subdifferentiability and uniform smoothness in terms of this property. | | Source: | arXiv, math.FA/0503076 | | Services: | Forum | Review | PDF | Favorites | |
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