An interpolation theorem and operator ranges

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.