Some inequalities in Pseudo-Hilbert spaces

Authors

  • Loredana Ciurdariu Department of Mathematics,"Politehnica" University of Timisoara, P-ta. Victoriei, No.2, 300006-Timisoara, Romania.

DOI:

https://doi.org/10.5556/j.tkjm.42.2011.1033

Keywords:

Pseudo-Hilbertspaces( Loynes spaces ), Seminorms, topology, Admissible spaces

Abstract

The aim of this paper is to obtain new versions of the reverse of the generalized triangle inequalities given in \cite{SSDNA}, %[4],and \cite{SSDPR} %[5] if the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ from Theorem 1 of \cite{SSDNA} %[4] belongs to ${\mathbb C}\times\mathcal H $, where $\mathcal H$ is a Loynes $Z$-space instead of ${\mathbb K}\times X$, $X$ being a normed linear space and ${\mathbb K}$ is the field of scalars. By comparison, in \cite{SSDNA} %[4] the pair $(a_i,x_i),\;i\in\{1,\ldots,n\}$ belongs to $A^2$, where $A$ is a normed algebra over the real or complex number field ${\mathbb K}.$ The results will be given in Theorem 1, Theorem 3, Remark 2 and Corollary 3 which represent other interesting variants of Theorem 2.1, Remark 2.2, Theorem 3.2 and Theorem 3.4., see \cite{SSDNA}. %[4].

References

A. H. Ansari and M. S. Moslehian, Refinements of reverse triangle inequalities in inner product spaces, J. Inequal. Pure Appl. Math.,6 (2005), art 64, 12 pp.

S. A. Chobanyan and A. Weron, Banach-space-valued stationary processes and their linear prediction, Dissertations Math.,125, (1975), 1--45.

L. Ciurdariu, Classes of linear operators on pseudo-Hilbert spaces and applications, Part I, Monografii matematice, Tipografia Universitatii de Vest din Timisoara, 2006.

S. S. Dragomir, On some inequalities in normed algebras, Journal of inequalities in pure and applied mathematics, 9 (2008), Issue 1, Art. 5, 10 pp.

S. S. Dragomir, A generalization of the Pecaric-Rajic inequality in normed linear spaces, Preprint, RGMIA Res. Rep. Coll.,10 (2007), Art.3, ONLINE:http://rmgia.vu.edu.au/v10n3.html].

C. F. Dunkl and K. S. Willams, A simple norm inequality, Amer. Math. Monthly,71 (1964), 53--54.

M. Khosravi, H. Mahyar and M. S. Moslehian, Reverse triangle inequality in Hilbert $C^*$-modules, J. Inequal. Pure Appl. Math.,10 (2009), art. 110, 11 pp.

R. M. Loynes, Linear operators in $VH$-spaces, Trans. American Math. Soc.,116 (1965), 167--180.

R. M. Loynes, On generalized positive definite functions, Proc. London Math. Soc,3 (1965),373--384.

L. Maligranda, Simple norm inequalities, Amer. Math. Monthly,113 (2006), 256--260.

P. R. Mercer, The Dunkl-Williams inequality in an inner-product space, Math. Inequal. Appl.,10(2007), 447--450.

J. Pecaric and R. Rajic, The Dunkl-Wiliams inequality with n elements in normed linear spaces, Math. Ineq. Appl.,10(2007), 461--470.

A. Weron and S. A. Chobanyan, Stochastic processes on pseudo-Hilbert spaces (russian), Bull. Acad. Polon., Ser. Math. Astr. Phys., tom XX1, 9 (1973), 847--854.

A. Weron, Prediction theory in Banach spaces, Proc. of Winter School on Probability, Karpacz, Springer Verlag, London, 1975, 207-228.

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Published

2011-12-31

How to Cite

Ciurdariu, L. (2011). Some inequalities in Pseudo-Hilbert spaces. Tamkang Journal of Mathematics, 42(4), 483–492. https://doi.org/10.5556/j.tkjm.42.2011.1033

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