Extremality of bounds for numerical radii of Foguel operators

Abstract

For any operator T on ℓ2, its associated Foguel operator FT is [S⁎0TS] on ℓ2⊕ℓ2, where S is the (simple) unilateral shift. It is easily seen that the numerical radius w(FT) of FT satisfies 1≤w(FT)≤1+(1/2)‖T‖. In this paper, we study when such upper and lower bounds of w(FT) are attained. For the upper bound, we show that w(FT)=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(FT)=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(FT)=1 is compact. Examples of various T's are given to illustrate such attainments of w(FT).