In this note we obtain a reverse version of the Haagerup Theorem. In particular, if $ A \in \mathbb{M}_{n}$ has a $ 2\times2- $ principal submatrix as $ \left[ \begin{array}{cc}1& \alpha \\\beta & 1\\\end{array}\right]$ with $ \beta \neq \bar{\alpha}, $ then $ \Vert S_{A} \Vert > 1$ where the operator $ S_{A}:\mathbb{M}_{n}\longrightarrow \mathbb{M}_{n} $ is defined by $S_{A}(B) := A \circ B $ where $ "\circ " $ stands for Schur product.

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We study in a unitary way the Schur-convexity or concavity of the Stolarsky and Gini means Da,b (x, y) and Sa,b (x, y), for fixed x, y>0, x¹y.

By introducing the concept of limited completely continuous operators between two arbitrary Banach spaces X and Y, we give some properties of this concept related to some well-known classes of operators and specially, related to the Gelfand–Phillips property of the space X or Y. Then some necessary and sufficient conditions for the Gelfand–Phillips property of closed subspace M of some operator spaces, with respect to limited complete continuity of some operators on M, so-called, evaluation operators, are verified.