Nonexistence of certain universal polynomials between Banach spaces

A well-known result due to W.B. Johnson (1971) asserts that the formal identity operator from ℓ1ℓ1 into ℓ∞ℓ∞ is universal for the class of non-compact operators between Banach spaces. We show that there is neither a universal non-compact polynomial nor a universal non-unconditionally converging polynomial between Banach spaces.