Extremality of bounds for numerical radii of Foguel operators

For any operator T on ℓ², its associated Foguel operator F_T is [Formula presented] on ℓ²⊕ℓ², where S is the (simple) unilateral shift. It is easily seen that the numerical radius w(F_T) of F_T satisfies 1≤w(F_T)≤1+(1/2)‖T‖. In this paper, we study when such upper and lower bounds of w(F_T) are attained. For the upper bound, we show that w(F_T)=1+(1/2)‖T‖ if and only if w(S+T^⁎S^⁎T)=1+‖T‖². When T is a diagonal operator with nonnegative diagonals, we obtain, among other results, that w(F_T)=1+(1/2)‖T‖ if and only if w(ST)=‖T‖. As for the lower bound, it is shown that any diagonal T with w(F_T)=1 is compact. Examples of various T's are given to illustrate such attainments of w(F_T).