In this paper, we introduce a new norm, christened the t-operator norm, on the space of all bounded linear operators defined on a complex Hilbert space H as (Formula Prasented), where x, y ϵ H and t ϵ [ 0, 1 ]. This norm satisfies 1/2 Tt ≤ w (T) ≤ Tt 12 and we explore its properties. This norm characterizes those invertible operators that are also unitary. We obtain various inequalities involving the t-operator norm and the usual operator norm. We show that w (T) ≤ min t ϵ improves the existing bounds w (T) ≤ 1/2 w(T) (see [F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 2003, 1, 11-17]) and w (T) ≤ 1/2 T ∗ (see [F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168 2005, 1, 73-80]). We show that (T) ≤ min | λ | = 1 . Further, we study the t-operator norm of operator matrices.