On operator order and chaotic operator order for two operators

Operator inequalities dealing with operator equations

“…Theorem 1.4 [5]. For r 0 and a nonnegative integer n 0 such that (1 + r)(n + 1) = p + r (so, p 1), the following assertions are equivalent.…”

“…Recently, some kinds of operator equations have been shown and given deep discussion via Furuta inequality and grand Furuta inequality (see [5], [9]). In [5], C.-S.…”

“…In this section , we will show further development of characterizations of operator order A B > 0 and chaotic operator order log A log B for positive definite operators A, B in terms of operator equations due to Lin's results in [5].…”

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In this paper, we study the existence of solutions of some kinds of operator equations via operator inequalities. First, we investigate characterizations of operator order A B > 0 and chaotic operator order log A log B for positive definite operators A, B in terms of operator equations, and generalize the results in [5]. Then, we introduce applications of complete form of Furuta inequality in operator equations.

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Characterizations of operator order for k strictly positive operators

“…In [3], C. -S. Lin proved several characterizations of operator order A 2 A 1 in terms of Furuta inequality [1] and Pedersen-Takesaki type operator equation [6]; Afterwards, C.…”