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Note on finite dimensional \(weak^*\) upper semi-continuous mappings. (English) Zbl 1222.19002

First, the authors introduce the following concepts:
Let \(E\) be a Banach space and \(E^*\) the dual space of \(E\), and we do not distinguish between the zero in \(E\) and \(E^*\). An operator \(T:D(T)\subseteq E\to 2^{E^*}\) is said to be finite-dimensional weak* upper semi-continuous if for any finite-dimensional subspace \(F\) of \(E\) with \(F\cap D(T)\neq \emptyset\), the restriction \(T:D(T)\cap F\to 2^{E^*}\) is upper-semicontinuous in the weak* topology; \(T\) is said to be finite-dimensional bounded if \(T\) is bounded on any finite-dimensional subspaces.
A generalized index for a class of densely defined finite-dimensional weak* upper semi-continuous mappings is constructed. This index is applied to study the range problem of such type of mappings. The basic result is the following:
Let \(T:D(T)\subseteq E\to 2^{E^*}\) be a bounded and finite-dimensional weak\(^*\) upper semi-continuous operator with closed convex values, and \(D(T)\) a dense subspace of \(E\). Suppose that
\[ \liminf_{x\in D(T),\|x\|\to\infty,f\in Tx}\frac{(f,x)}{\|x\|^2}=+\infty. \]
Then \(\overline{T(D(T))}^*=E^*\).

MSC:

19K99 \(K\)-theory and operator algebras
55M10 Dimension theory in algebraic topology
47H11 Degree theory for nonlinear operators
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