Paranormal Elements in Normed Algebra

For a normed algebra A and natural numbers k we introduce and investigate the ∥ · ∥ closed classes P k (A). We show that P1(A) is a subset of P k (A) for all k. If T in P1(A), then Tn lies in P1(A) for all natural n. If A is unital, U, V ∈ A are such that ∥U∥ = ∥V∥ = 1, VU = I and T lies in P k (A), then UTV lies in P k (A) for all natural k. Let A be unital, then 1) if an element T in P1(A) is right invertible, then any right inverse element T−1 lies in P1(A); 2) for ssIss = 1 the class P1(A) consists of normaloid elements; 3) if the spectrum of an element T, T ∈ P1(A) lies on the unit circle, then ∥TX∥ = ∥X∥ for all X ∈ A. If A = B(H), then the class P1(A) coincides with the set of all paranormal operators on a Hilbert space H.