Continuity properties of the spectrum of operators on Lebesgue spaces

Abstract

Fix 1 ≤ p ≤ s ≤ ∞ 1 \leq p \leq s \leq \infty . Let T x , x ∈ [ p , s ] {T_x},x \in \left [ {p,s} \right ] , be the collection of bounded linear operators on the Lebesgue spaces L x {L^x} determined by some fixed operator T T . This paper concerns continuity properties of the map x → σ ( T x ) x \to \sigma \left ( {{T_x}} \right ) .