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You got the definition of $L^+$ wrong. It is not $L^* \setminus \{\epsilon\}$. Rather, it is $$L^+ = \bigcup_{n=1}^* L^n.$$ You can check that $\epsilon \in L^+$ iff $\epsilon \in L$. Therefore: If $\epsilon \notin L$ then $L^+ = L^*\setminus\{\epsilon\}$. If $\epsilon \in L$ then $L^+ = L^*$.
Linear-bounded automata accept the class of context-sensitive languages. In contrast, Turing machine (deterministic as well as nondeterministic) accept the class of recursive languages. Every context-sensitive language is recursive, but the converse doesn't hold. For example, the halting problem for nondeterministic Turing machines running in space $n^2$ is ...