Numerical ranges of the powers of an operator

The numerical range W ( A ) of a bounded linear operator A on a Hilbert space is the collection of complex numbers of the form ( A v , v ) with v ranging over the unit vectors in the Hilbert space. In terms of the location of W ( A ) , inclusion regions are obtained for W ( A k ) for positive integers k, and also for negative integers k if A − 1 exists. Related inequalities on the numerical radius w ( A ) = sup { | μ | : μ ∈ W ( A ) } and the Crawford number c ( A ) = inf { | μ | : μ ∈ W ( A ) } are deduced.