A Hilbert space operator T is said to be a 2-isometric operator if T * 2 T 2 − 2T * T + I = 0. Let d AB ∈ B(B(H)) denote either the generalized derivation δ AB = L A − R B or the elementary operator ∆ AB = L A R B − I, we show that if A and B * are 2-isometric operators, then, for all complex λ, (d AB − λ) −1 (0) ⊆ (d * AB − λ) −1 (0), the ascent of (d AB − λ) ≤ 1, and d AB is polaroid. Let H(σ(d AB)) denote the space of functions which are analytic on σ(d AB), and let H c (σ(d AB)) denote the space of f ∈ H(σ(d AB)) which are non-constant on every connected component of σ(d AB), it is proved that if A and B * are 2-isometric operators, then f (d AB) satisfies the generalized Weyl's theorem and f (d * AB) satisfies the generalized a-Weyl's theorem.