Improved bounds for the numerical radius via a new norm on B(H)

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 parallel to T parallel to(t) := sup(parallel to y parallel to=1 parallel to x parallel to=1) (t vertical bar < Tx, y >vertical bar + (1 - t)vertical bar < x, Ty >vertical bar), where x, y is an element of H and t is an element of [ 0, 1]. This norm satisfies 1/2 parallel to T parallel to(t) <= w(T) <= parallel to T parallel to(t) 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 is an element of[0,1]) parallel to T parallel to(t) improves the existing bounds w(T) <= 1/2 (parallel to T parallel to + root T-2 parallel to) (see [F. Kittaneh, A numerical radius inequality and an estimate for the numerical radius of the Frobenius companion matrix, Studia Math. 158 (2003), no. 1, 11-17]) and w(T) <= root 1/2 parallel to T*T + TT*parallel to (see [F. Kittaneh, Numerical radius inequalities for Hilbert space operators, Studia Math. 168 (2005), no. 1, 73-80]). We show that parallel to T parallel to - w(T) <= min(vertical bar lambda vertical bar=1) parallel to T+lambda T*/2 parallel to. Further, we study the t-operator norm of operator matrices.