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This paper is a continuation of the previous one . Here we show that it is still possible to have several characterizations of the Furuta inequality in terms of different types of the Heinz-Kato inequality. Some application are given. Mathematics subject classification (1991): 47A30, 47A62, 47A63.doi:10.7153/mia-02-25 fatcat:nmp2bpdfrza3xfwef65gy7qyxm