Local Spectral Formula for Integral Operators on L p ( T ) $L_{p}({\mathbb T})$

Let \(1\leq p\leq \infty \), \(f\in L_{p}({\mathbb T})\) and \(0 \notin \text {supp} \hat {f}\). Then, in this paper, we obtain the following local spectral formula for the integral operator I on \(L_{p}({\mathbb T})\), the space of 2π-periodic functions belonging to L p (−π,π): \( \lim _{n\rightarrow \infty } \|I^{n} f\|_{p,{\mathbb T}}^{1/n}= \sigma ^{-1}, \) where \(\sigma =\min \{ |k| : k \in \text {supp} \hat {f} \}, If(x)={{\int }_{0}^{x}} f(t) dt -c_{f}, x\in \mathbb {R}\) and the constant c f is chosen such that \({\int }_{0}^{2\pi } If (x) dx=0\). The local spectral formula for polynomial integral operators on \(L_{p}({\mathbb T})\) is also given.