Joint Spectral Properties for Pairs of Permutable Selfadjoint Transformations

Abstract

Let X be a complex Banach space and let T_j: D(T_j) ⊂ X → X (j = 1, 2) be linear transformations in X. Then the composite operator T₁T₂ is defined on the linear space

$$ D\left( {{T_1}{T_2}} \right) = \left\{ {x \in {\kern 1pt} D\left( {{T_2}} \right);{T_2}x \in D\left( {{T_1}} \right)} \right\} $$

in an obvious manner and we have, in general, T₁T₂ ≠ T₂T₁ on their joint domain of definition.