Jordan higher derivations on some operator algebras

Motivated by the systemic work of Lu [21, 23] we mainly consider the question of whether any Jordan higher derivation on some operator algebras is a higher derivation. Let A be a torsion free algebra over a commutative ring R, D be the set of all Jordan higher derivations D = {d_n} ^∞_n=0 on A, and Δ be the set of all sequences {δ_n}^∞_n=0 of Jordan derivations on A with δ₀ = 0. Then there is a one to one correspondence between D and Δ. It is shown via this correspondence that every Jordan higher derivation on some operator algebras is a higher derivation. The involved operator algebras include CSL algebras, reexive algebras, nest algebras. At last, we describe local actions of Jordan higher derivations on nest algebras.