| Research Abstract |
In what follows, a capital letter means a bounded linear operator on a Hilbert space. Furuta inequality (1987) asserts that if A 【greater than or equal】 B 【greater than or equal】 0, then for r 【greater than or equal】 0,
(*) (A^<r/2> A^p A^<r/2>)^<1/q> 【greater than or equal】 (A^<r/2> B^p A^<r/2>)^<1/q>
holds for p 【greater than or equal】 0 and q 【greater than or equal】 1 with (1 + r)q 【greater than or equal】 p + r. Furuta inequality yields the famous Lowner-Heinz one (1934), that is, A 【greater than or equal】 B 【greater than or equal】 0 ensures A^p 【greater than or equal】 B^p for 1 【greater than or equal】 p 【greater than or equal】 0 when we put r = 0 in (*). We obtained a lot of applications of Furuta inequality in the following three branches, (a) operator ibnequalities, (b) norm inequalities and (c) operator equations. We cite some of them as follows : (a_1) relative operator entropy, (a_2) Ando-Hiai log majorization, (a_3) Aluthge transformation, (b_1) Heinz-Kato inequality, (b_2) Kosaki trace inequality, (c_1) Pedersen-Takesaki operator equation. Recently we obtained a one page simplified proof of generalized Furuta inequality which interpolates Furuta inequality itself and an inequality equivalent to the main theorem on log majorizaton by Ando-Hiai. Further applications of Furuta inequality to some operator equatios and relative operator entoropy will be expected in near future.
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