A non-2-starcompact, pseudocompact Tychonoff space whose hyperspace is 2-starcompact

For a topological space X, let H(X) be the collection of all nonempty closed subsets of X with the Vietoris topology and let C(X) be the collection of all nonempty compact subsets of X equipped with the subspace topology.We prove the following:(1)if C(X) is 1-starcompact, then X is K-starcompact;(2)if C(X) is -starcompact, then X is -starcompact; and(3)if X is a regular space which has a closed discrete subset E such that |E|=w(X)⩾ω, then C(X) is not -starcompact, where w(X) is the weight of X.And we show that there exists a non-2-starcompact, pseudocompact Tychonoff space X whose hyperspace H(X) is 2-starcompact under the assumption p=c.