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In present paper, we give some new reverses of the Young type inequalities which were established by X. Hu and J. Xue [7] via Kantorovich constant. Then we apply these inequalities to establish corresponding inequalities for the Hilbert-Schmidt norm and the trace norm.

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Abstract:In this article, we first present some improved Young type inequalities for scalars, then according to these inequalities we give the Hilbert-Schmidt norm and the trace norm versions.

A Banach space X has the alternative Daugavet property (ADP in short)if max |ω|=1 Id + ωT = 1 + T (aDE) holds for all rank-1 operators T : X → X. We prove that if every weakly compact operator on X satisfies (aDE) , then X has the ADP. We present some examples that shows generally the ADP from space into subspace is not transmitted and vice-versa. Also we shows that ADP is a isometric property of a Banach space.