Isometric Equivalence of Integration Operators

We are interested in the isometric equivalence problem for the Cesàro operator $${C(f) (z) =\frac{1}{z} \int_{0}^{z}f(\xi) \frac{1}{1-\xi}d \xi}$$ and an operator $${T_{g}(f)(z)=\frac{1}{z}\int_{0}^{z}f(\xi) g^{\prime}(\xi) d \xi}$$ , where g is an analytic function on the disc, on the Hardy and Bergman spaces. Then we generalize this to the isometric equivalence problem of two operators $${T_{g_{1}}}$$ and $${T_{g_{2}}}$$ on the Hardy space and Bergman space. We show that the operators $${T_{g_{1}}}$$ and $${T_{g_{2}}}$$ satisfy $${T_{g_{1}}U_{1}=U_{2}T_{g_{2}}}$$ on H p , 1 ≤ p < ∞, p ≠ 2 if and only if $${g_{2}(z) =\lambda g_{1}(e^{i\theta}z) }$$ , where λ is a modulus one constant and U i , i = 1, 2 are surjective isometries of the Hardy Space. This is analogous to the Campbell-Wright result on isometrically equivalence of composition operators on the Hardy space.