Local Spectral Formula for Integral Operators on L p ( T )

Let 1 ≤ p ≤ ∞ , f ∈ L p ( T ) and 0 ∉ supp f ̂ . Then, in this paper, we obtain the following local spectral formula for the integral operator I on L p ( T ) , the space of 2π-periodic functions belonging to Lp(−π,π): lim n → ∞ ∥ I n f ∥ p , T 1 / n = σ − 1 , where σ = min { | k | : k ∈ supp f ̂ } , If ( x ) = ∫ 0 x f ( t ) dt − c f , x ∈ ℝ and the constant cf is chosen such that ∫ 0 2 π If ( x ) dx = 0 . The local spectral formula for polynomial integral operators on L p ( T ) is also given.