1980
DOI: 10.1007/bf01371042
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Abstract: I. IntroductionMotivated by a study of electrical network connections, Anderson and Duffin [1] introduced a binary operation A:B, called parallel addition, for pairs of positive (semi-definite) matrices. Subsequently Anderson and Trapp [3] have extended this notion to positive (linear) operators on a Hilbert space and demonstrated its importance in operator theory. On the other hand, Pusz and Woronowicz [12] considered a binary operation A # B, called geometric mean, for pairs of positive operators.In this pap… Show more

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Cited by 709 publications
(509 citation statements)
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References 7 publications
(9 reference statements)
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“…As in the seminal paper [10], a binary operation σ on B(H) + is called a connection if it satisfies the following conditions. For any operators A, B, C, D ∈ B(H) + and sequences…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
See 1 more Smart Citation
“…As in the seminal paper [10], a binary operation σ on B(H) + is called a connection if it satisfies the following conditions. For any operators A, B, C, D ∈ B(H) + and sequences…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…One of the most important results in the Kubo-Ando theory says that there is an affine order-isomorphism between the class of all connections σ on B(H) + and the class of all nonnegative scalar valued operator monotone functions f on ]0, ∞[, see Theorem 3.2 in [10]. In fact, as seen in the proof of that theorem, if σ is a connection then the operator monotone function f associated with it is defined by the formula f (t) = Iσ(tI), t > 0.…”
Section: Introduction and Statement Of The Resultsmentioning
confidence: 99%
“…2, correspond to 18) so the inverse, 1/c, is equal to the harmonic, logarithmic and arithmetic mean, respectively. It is known that among all normalised, self-transposed, operator monotone functions f there is a minimal and a maximal one [36]. They are distinguished above by the appropriate labels.…”
Section: Bures Metricmentioning
confidence: 99%
“…Some of the underlying ideas can be found in [43] and [48]. We continue as in [28] to prove the following: …”
Section: The Problem Of Optimalitymentioning
confidence: 90%
“…Fujii [22] gave a one line proof of (2.6), having heard the result on May 8, 1979 in the Nakamura seminar in Osaka, based on the Kubo Ando theory [43] for means of operators. Fujii's proof, albeit short, is highly sophisticated as it fully relies on Löwner's theorem.…”
Section: Inequalities In One Variablementioning
confidence: 99%