Complete form of Furuta inequality

Abstract

Abstract. Let A and B be bounded linear operators on a Hilbert space satisfying A ≥ B ≥ 0. The well-known Furuta inequality is given as follows: Let. In order to give a self-contained proof of it, Furuta (1989) p+r p+r . This paper aims to show a sharpening of Furuta (1989)s+r p+r . We call it the complete form of Furuta inequality because the case p 0 = 1 of it implies the essential part (p > 1) of Furuta inequality for 1+r s+r ∈ (0, 1] by the famous Löwner-Heinz inequality. Afterwards, the optimality of the outer exponent of the complete form is considered. Lastly, we give some applications of the complete form to Aluthge transformation.

Some Properties of Furuta Type Inequalities and Applications

“…Tanahashi proved that the outer exponent min{1, } + above is optimal; see [3] for related topics. In order to establish the order structure on Aluthge transform of nonnormal operators, the complete form of Furuta inequality was showed in [4]. We call the theorem above the complete form of Furuta inequality because the case 0 = = 1 of it implies the essential part ( > 1) of Furuta inequality by the Loewner-Heinz inequality for (1+ )/( (1)+ ) ∈ (0,1].…”

An application of grand Furuta inequality to a type of operator equation

“…Afterwards, the studies of the theory of operator inequalities have been developed quickly and some results related to Furuta inequality have been obtained in recent twenty-five years, such as [1,2,9,17,23,24,25]. It is well known that Furuta inequality has many applications.…”

Operator inequalities dealing with operator equations

“…In 1996, Tanahashi proved the conditions for p and q in Furuta Inequality are the best possible if r 0 in [7]; In 2000, he proved the outer exponent value of Grand Furuta Inequality is the best possible in [8]. in [11]. There were many related studies on Complete form of Furuta Inequality, such as [12], [13].…”

“…(3-11) ⇒ (3-10). Because of (3)(4)(5)(6)(7)(8)(9)(10)(11) and the fact that S > O with S 1 ⇒ S −1 I, we have (A…”