Some Lower Bounds for the Numerical Radius of Hilbert Space Operators

We show that if T is a bounded linear operator on a complex Hilbert space, then 1 2 ‖T‖ ≤ √ w2(T ) 2 + w(T ) 2 √ w2(T )− c2(T ) ≤ w(T ), where w(·) and c(·) are the numerical radius and the Crawford number, respectively. We then apply it to prove that for each t ∈ [0, 12 ) and natural number k, (1 + 2t) 1 2k 2 1 k m(T ) ≤ w(T ), where m(T ) denotes the minimum modulus of T . Some other related results are also presented.