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Norms of Operators from Lp into Lq in the Real and the Complex Case
2004 (English)Independent thesis Advanced level (professional degree), 20 credits / 30 HE creditsStudent thesis
Abstract [en]

For 1 <= p,q <= ¥ the natural complexification of any bounded linear operator between real Banach spaces T:Lp(m) -> Lq(n) denoted by TC:LCp(m) -> LCq(n) is also bounded and we are interested in the exact relation between the norm of the operator T in the real case and in the complex case. This relation can be expressed in terms of the constants gp,q appearing in the inequality ||TC||p,q <= gp,q||T||p,q for all operators T as above and all positive s-finite measures m and n. In analysis the considered relation has, for instance, its applications in the interpolation theory and connection with the constants from the fundamental Grothendieck inequality. The work on the constants gp,q is traced back to Marcinkiewicz, Zygmund, Verbickii-Sereda, Figiel-Iwaniec-Pelczynski, Krivine, Gasch-Maligranda, Defant and others. In this paper we try to summarize the results of these authors and show how to obtain the estimates for gp,q as well as present a number of these estimates. We begin with the case when gp,q=1. In addition, we show that the norms of the operators acting between spaces of functions of bounded p-variation and the norms of the positive linear operators between Lp spaces coincide with the norms of their complexifications. When calculating the biggest constant g¥,1 Krivine described it in terms of certain tensor product, that is why this necessary complication appears in this work. Moreover, this description helps to derive some basic properties of the constants gp,q. Finally, we specify the measures to be counting and consider the operators acting between finite-dimensional lp spaces. We present a number of the estimates of gp,q (lnp,lmq) in this particular case. However, the exact value of gp,q for 1 <= q < 2 < p <= ¥ and (p,q) ¹ (¥,1) has not been found for arbitrary finite-dimensional lp spaces as well as in more general settings.

Place, publisher, year, edition, pages
Keyword [en]
Technology, matematik
Keyword [sv]
URN: urn:nbn:se:ltu:diva-50464ISRN: LTU-EX--04/170--SELocal ID: 7b84a117-ef1d-43c3-9d89-3b80d2605225OAI: diva2:1023823
Subject / course
Student thesis, at least 30 credits
Educational program
Computer Science and Engineering, master's level
Validerat; 20101217 (root)Available from: 2016-10-04 Created: 2016-10-04Bibliographically approved

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