ISSN:
1420-8989
Source:
Springer Online Journal Archives 1860-2000
Topics:
Mathematics
Notes:
Abstract The essence of an interpolation theorem of Peetre can be re-formulated as follows: for P an injective, positive bounded operator, T any bounded operator, and Φ a continuous, non-negative, non-decreasing, concave function on [0, ‖P‖2], if the operator PTP−1 is bounded then so is the operator Φ(P2)1/2 T Φ(P2)−1/2. We strengthen this theorem to show that $$\parallel \Phi (P^2 )^{{\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} T \Phi (P^2 )^{ - {\raise0.5ex\hbox{$\scriptstyle 1$}\kern-0.1em/\kern-0.15em\lower0.25ex\hbox{$\scriptstyle 2$}}} \parallel \leqslant \sqrt 2 max \left\{ {\parallel T\parallel , \parallel PTP^{ - 1} \parallel } \right\}$$ for such Φ, and also obtain similar results for ‖Φ(P) T Φ(P)−1‖. We present two different approaches, one of which is based on the study of invariant operator ranges of bounded operators.
Type of Medium:
Electronic Resource
URL:
http://dx.doi.org/10.1007/BF01196120
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