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On inequalities for powers of linear operators and for quadratic forms

Published online by Cambridge University Press:  14 November 2011

Vũ Qúôc Phóng
Vũ Qúôc Phóng
Affiliation:
Institute of Mathematics, Hànôi, Viêtnam
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Synopsis

Let H be a Hilbert space in which a symmetric operator S with a dense domain Ds is given and let S have a finite deficiency index (r, s). This paper contains a necessary and sufficient condition for validity of the following inequalities of Kolmogorov type

and a method for calculating the best possible constants Cn,m(S).

Moreover, let φ be a symmetric bilinear functional with a dense domain Dφ such that Ds ⊂ Dφ and φ(f, g) = (Sf, g) for all f ∈ Ds, g ∈ Dφ. A necessary and sufficient condition for validity of the inequality

as well as a method for calculating the best possible constant K are obtained. Then an analogous approach is worked out in order to obtain the best possible additive inequalities of the form

The paper is concluded by establishing the best possible constants in the inequalities

where T is an arbitrary dissipative operator. The theorems are extensions of the results of Ju. I. Ljubič, W. N. Everitt, and T. Kato.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 89 , Issue 1-2 , 1981 , pp. 25 - 50
Copyright
Copyright © Royal Society of Edinburgh 1981

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