Real and complex operator norms
Abstract: Any bounded linear operator between real (quasi-)Banach spaces T : X ® Y has a natural bounded complex linear extension TC : XC ® YC defined by the formula TC(x+iy)=Tx+iTy for x,yÎX, where XC and YC are so called reasonable complexifications of X and Y, respectively. We are interested in the exact relation between the norms of the operators TC and T. This relation can be expressed in terms of the constant gX,Y appearing in the inequality||TC|| £ gX,Y ||T|| considered for all bounded linear operators T : X®Y between (quasi-)Banach spaces. The work on the constant gLp,Lq for 0 < p,q £ ¥, or shortly gp,q, is traced back to M. Riesz, Thorin, Marcinkiewicz, Zygmund, Verbitckii, Krivine, Gasch, Maligranda, Defant and others. In this thesis we try to summarize the results of these authors. We also present some new estimates for gp,q in the case when at least one of the spaces is quasi-Banach as well as in the case when the spaces are supplied with discrete measures. For example, we get that gp,q £ 2 for all 0 < p,q £ ¥. Furthermore we obtain some new results concerning the relation between complex and real norms of the operators between spaces of functions of bounded p-variation and between mixed norm Lebesgue spaces. Looking for the criteria of the equality of real and complex norms of operators from a Banach lattice into the same Banach lattice we find a number of examples of two dimensional Orlicz spaces different from Lebesgue spaces and a simple operator between them with non-equal real and complex norms. We also consider in detail the Clarkson inequality which can be interpreted in terms of a certain operator norm inequality appearing as an example in many parts of the thesis. It turns out that complex norm of this operator can be easily obtained but to find the real one is not so trivial. With the help of the Clarkson inequality we construct an operator between Lebesgue spaces with non-atomic measures which has different real and complex norms. Finally, we consider both complex and real versions of the Riesz-Thorin interpolation theorem in the first quadrant and by using numbers gp,q find, for example, that the real Riesz constant is bounded by 2 for all 0 < p,q £ ¥.
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