The inverse to a c-continuous operator (Q1814631)

Let \(X\) be a locally compact set representable as a union of a countable number of compact subsets, \(\lambda\) be a positive measure on \(X\), \(E\) be a Banach space. Denote by \(L_ p\) the space of vector-valued functions \(L_ p(X,E)\), \(1\leq p\leq\infty\). A linear operator \(T: L_ p\to L_ p\) is said to be \(c\)-continuous provided (i) for any \(\varepsilon>0\) and any compact subset \(N\subset X\) there exists a compact subset \(M\subset X\) such that \(\forall y\in L_ p\), \(y\mid_ M=0\Rightarrow\|(Ty)\mid_ N\|\leq\varepsilon\| y\|\). The author gives an example of a \(c\)-continuous operator \(T\) such that \(T^{-1}\) exists but not \(c\)-continuous. Motivated by this, he introduces a class of \(o\)-continuous operators with some additive hypothesis for which \(T^{-1}\) is \(o\)-continuous provided that \(T\) is \(o\)-continuous.