Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-10T02:20:02.935Z Has data issue: false hasContentIssue false
  • English
  • Français

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
Get access Rights & Permissions [Opens in a new window]

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1Landau, E.. Einige Ungleichungen für zweimal differentiierbare Funktionen. Proc. London Math. Soc. 13 (1913), 43–49.Google Scholar
2Hadamard, J.. Sur le module maximum d'une fonction et de ses dérivées. Bull. Soc. Math. France 42 (1914), 68–72.Google Scholar
3Hardy, G. H., Littlewood, J. E. and Polya, G.. Inequalities (Cambridge Univ. Press, 1938).Google Scholar
4Shilov, G. E.. On inequalities between derivatives. Sb. Rab. Stud. Naučn. Kruzkov. Moskov. Gos. Univ. (1937), 17–27.Google Scholar
5Kolmogorov, A. N.. On inequalities between supremums of derivatives of functions on infinite interval. Učen. Zap. Moskov. Gos. Univ. 30 (1939), 3–16.Google Scholar
English translation: Amer. Math. Soc. Transl. (1) 2, 233–243).Google Scholar
6Gorny, A.. Contribution à l'étude des fonctions dérivables d'une variable réelle. Acta Math. 71 (1939), 317–358.CrossRefGoogle Scholar
7Cartan, H.. Sur les classes de fonction définies par des inégalitiés portant sur leurs dérivées successive. Actualitiés Sci. Indust. 867 (1940).Google Scholar
8Rodov, A.. Dependence between supremums of derivatives of real variable functions. Izv. Akad. Nauk SSSR Ser. Mat. 10 (1946), 257–260.Google Scholar
English translation: Amer. Math. Soc. Transl. (1) 2, 245–274).Google Scholar
9Ljubic, Ju. J.. The problem whether powers of an operator on a vector belong to a certain class. Dokl. Acad. Nauk SSSR 102 (1955), 881–884.Google Scholar
10Ljubic, Ju. J.. On inequalities between powers of linear operators. Izv. Acad. Nauk SSSR Ser. Mat. 24 6 (1960), 825–864.Google Scholar
English translation: Amer. Math. Soc. Transl. (2) 40, 39–84).Google Scholar
11Kuptsov, N. P.. Kolmogorov's estimates for norm of derivatives in L2 (0, ∞). Trudy Mat. Inst. Steklov 138 (1975), 94–117. (English translation: Amer. Math. Soc.)Google Scholar
12Hardy, G. H. and Littlewood, J. E.. Some integral inequalities connected with the calculus of variation. Quart. J. Math. Oxford 3 (1932), 241–252.CrossRefGoogle Scholar
13Bradley, J. S. and Everitt, W. N.. On the inequality Quart. J. Math. Oxford 182 (1973), 303–321.Google Scholar
14Everitt, W. N.. On an extension to an integro-differential inequality of Hardy, Littlewood and Polya. Proc. Roy. Soc. Edinburgh Sect. A 69 (1971/1972), 295–333.Google Scholar
15Brodlie, K. W. and Everitt, W. N.. On an inequality of Hardy and Littlewood. Proc. Roy. Soc. Edinburgh Sect. A 72 (1974), 179–186.CrossRefGoogle Scholar
16Dawson, E. R. and Dicker, R. M.. Generalizations of the Hardy-Littlewood inequality (preprint).Google Scholar
17Kato, T.. On an inequality of Hardy, Littlewood and Polya. Advances in Math. 7 (1971). 217–218.CrossRefGoogle Scholar
18Akhiezer, N. J. and Glazman, J. M.. Theory of linear operators in Hilbert space. (Moscow: Nauka, 1966; New York: Ungar, 1961).Google Scholar
19Nagy, S. and Foias, C.. Analyse harmonique des operateurs de l'espace de Hilbert. (Budapest: Akademai Kiado, 1967).Google Scholar
20Sobolev, S. L.. Some applications of functional analysis to mathematical physics. (Leningrad: 1950).Google Scholar
21Phong, Vu Quoc. On inequalities for powers of linear operators and for quadratic forms. Dopovidi Acad. Nauk. Ukrain. S.S.R. (A) 11 (1977),Google Scholar
22Phong, Vu Quoc. On universality of operator i(d/dx) in L2 (0, ∞) and inequalities for dissipative operators. J. Functional Anal. Appl. Moscow 4 (1979).Google Scholar