A converse of Loewner–Heinz inequality and applications to operator means

In a recent paper, [l], Dixmier has proved Heinz' inequality by deducing it from a lemma due to Thorin. In this note it is proved directly from a convexity theorem.Let(M(k), ℳ(k), μ(k)), k = 0, …, n, be measure spaces and Lq(k) (M(k), ℳ(k), μ(k)) be all the functions on M(k) such that

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GEODESICS AND OPERATOR MEANS IN THE SPACE OF POSITIVE OPERATORS

International Journal of Mathematics ◽

10.1142/s0129167x9300011x ◽

1993 ◽

Vol 04 (02) ◽

pp. 193-202 ◽

Cited By ~ 31

Author(s):

GUSTAVO CORACH ◽

HORACIO PORTA ◽

LÁZARO RECHT

Keyword(s):

Geometric Mean ◽

Geodesic Segment ◽

Finsler Metric ◽

Natural Structure ◽

Operator Inequalities ◽

C Algebra ◽

Operator Means ◽

Natural Notion ◽

Invertible Elements ◽

Classical Convexity

The set A+ of positive invertible elements of a C*-algebra has a natural structure of reductive homogeneous manifold with a Finsler metric. Because pairs of points can be joined by uniquely determined geodesics and geodesics are "short" curves, there is a natural notion of convexity: C ⊂ A+ is convex if the geodesic segment joining a, b ∈ C is contained in C. We show that this notion is related to the classical convexity of real and operator valued functions. Several results about convexity are proved in this paper. The expressions of these results are closely related to the operator means of Kubo and Ando, in particular to the geometric mean of Pusz and Woronowicz, and they produce several norm estimations and operator inequalities.

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A new class of operator monotone functions via operator means

Linear and Multilinear Algebra ◽

10.1080/03081087.2016.1160999 ◽

2016 ◽

Vol 64 (12) ◽

pp. 2463-2473 ◽

Cited By ~ 2

Author(s):

Rajinder Pal ◽

Mandeep Singh ◽

Mohammad Sal Moslehian ◽

Jaspal Singh Aujla

Keyword(s):

Monotone Functions ◽

New Class ◽

Operator Monotone ◽

Operator Means ◽

Operator Monotone Functions

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Some operator inequalities involving operator means and positive linear maps

Linear and Multilinear Algebra ◽

10.1080/03081087.2017.1346059 ◽

2017 ◽

Vol 66 (6) ◽

pp. 1186-1198 ◽

Cited By ~ 3

Author(s):

Maryam Khosravi ◽

Mohammad Sal Moslehian ◽

Alemeh Sheikhhosseini

Keyword(s):

Operator Inequalities ◽

Linear Maps ◽

Operator Means ◽

Positive Linear Maps

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A new quantum operator for distance

International Journal of Modern Physics A ◽

10.1142/s0217751x19500337 ◽

2019 ◽

Vol 34 (06n07) ◽

pp. 1950033

Author(s):

Daniel Katz

Keyword(s):

Quantum Mechanics ◽

Equivalence Principle ◽

Classical Mechanics ◽

Proof Of Concept ◽

Logical Relationship ◽

Weak Equivalence Principle ◽

Quantum Operator ◽

Operator Means ◽

Definition Of ◽

Future Work

We introduce a new semirelativistic quantum operator for the length of the worldline a particle traces out as it moves. In this article the operator is constructed in a heuristic way and some of its elementary properties are explored. The operator ends up depending in a very complicated way on the potential of the system it is to act on so as a proof of concept we use it to analyze the expected distance traveled by a free Gaussian wave packet with some initial momentum. It is shown in this case that the distance such a particle travels becomes light-like as its mass vanishes and agrees with the classical result for macroscopic masses. This preliminary result has minor implications for the Weak Equivalence Principle (WEP) in quantum mechanics. In particular it shows that the logical relationship between two formulations of the WEP in classical mechanics extends to quantum mechanics. That our result is qualitatively consistent with the work of others emboldens us to start the task of evaluating the new operator in nonzero potentials. However, we readily acknowledge that the looseness in the definition of our operator means that all of our so-called results are highly speculative. Plans for future work with the new operator are discussed in the last section.

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Ando–Hiai-type inequalities for operator means and operator perspectives

International Journal of Mathematics ◽

10.1142/s0129167x2050007x ◽

2019 ◽

Vol 31 (01) ◽

pp. 2050007

Author(s):

Fumio Hiai ◽

Yuki Seo ◽

Shuhei Wada

Keyword(s):

Positive Operators ◽

Extension Problem ◽

Operator Means ◽

Trotter Formula

We improve the existing Ando–Hiai inequalities for operator means and present new ones for operator perspectives in several ways. We also provide the operator perspective version of the Lie–Trotter formula and consider the extension problem of operator perspectives to non-invertible positive operators.

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