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Inequalities between means of positive operators

Published online by Cambridge University Press:  24 October 2008

K. V. Bhagwat  and
R. Subramanian
K. V. Bhagwat
Affiliation:
Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India
R. Subramanian
Affiliation:
Bhabha Atomic Research Centre, Trombay, Bombay 400 085, India
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Extract

One of the most fruitful – and natural – ways of introducing a partial order in the set of bounded self-adjoint operators in a Hilbert space is through the concept of a positive operator. A bounded self-adjoint operator A denned on is called positive – and one writes A ≥ 0 - if the inner product (ψ, Aψ) ≥ 0 for every ψ ∈ . If, in addition, (ψ, Aψ) = 0 only if ψ = 0, then A is called positive-definite and one writes A > 0. Further, if there exists a real number γ > 0 such that A — γI ≥ 0, I being the unit operator, then A is called strictly positive (in symbols, A ≫ 0). In a finite dimensional space, a positive-definite operator is also strictly positive.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 83 , Issue 3 , May 1978 , pp. 393 - 401
Copyright
Copyright © Cambridge Philosophical Society 1978

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References

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