Inequalities of Reid type and Furuta

Two of the most useful inequality formulas for bounded linear operators on a Hilbert space are the Ldwner-Heinz and Reid's inequalities. The first inequality was generalized by Furuta (so called the Furuta inequality in the literature). We shall generalize the second one and obtain its related results. It is shown that these two generalized fundamental inequalities are all equivalent to one another. 1. NOTATIONS AND INTRODUCTION Throughout the paper we use capital letters to denote bounded linear operators acting on a Hilbert space H. T is positive (written T > 0 ) in case (Tx, x) > 0 for all x EH. If S and T are Hermitian, we write T > S in case T-S > O. T = U I T I is the polar decomposition of T with U the partial isometry, whereas I T I is the positive square root of the operator T*T, and U*U is the initial projection. Let I denote the identity operator. Under the polar decomposition of T we recall a well-known relation: I T* Iq= U I T Iq U* for q > 0 [1, p. 752]. The aim of this article is to define and characterize a generalized Reid inequality. Consequently, related inequalities and improved inequalities are given. Recall that if S > 0 and SK is Hermitian, then for x EH, I (SKx, x) I? 11 K 11 (Sx, x) is Reid's inequality [8], and the inequality was sharpened by Halmos [4, pp. 51, 244], where II K II is replaced by the spectral radius of K. Let a, ,3 E [0, 1] with a + ,3 > 1. For x,y EH let us call I (SK I SK Ic+,-1 x,y) I'll K IIc+/31 Sax SI y I an extended Reid's inequality. We shall call, for an obvious reason, the inequality (2) in Theorem 1 below a generalized Reid inequality, whereas (1) is the well-known altered Furuta inequality [2]. Our main result is characterizations of this inequality.