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A bounded linear operator T on a Hilbert space H is concave if, for each x ∈ H , ∥T 2 x∥ 2 −2∥T x∥ 2 +∥x∥ 2 ≤ 0. In this paper, it is shown that if T is a concave operator then so is every power of T. Moreover, we investigate the concavity of shift operators. Furthermore, we obtain necessary and sufficient conditions for N-supercyclicity of co-concave operators. Finally, we establish necessary and sufficient conditions for the left and right multiplications to be concave on the Hilbert-Schmidt class.

Using Banach algebra structure of the Hardy space, we describe all finite codimensional invariant subspaces of a cyclic convolution operator on the Hardy space H p of the unit disc for 1 ≤ p ≤ ∞. We also observe that every operator in the commutant of such operators is not weakly supercyclic.