Extremality of bounds for numerical radii of Foguel operators

For any operator T on & POUND;2, its associated Foguel operator FT is T 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)IITII. 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)IITII if and only if w(S + T*S*T) = 1 + IITII2. When T is a diagonal operator with nonnegative diagonals, we obtain, among other results, that w(FT) = 1 + (1/2)IITII if and only if w(ST) = IITII. 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).& COPY; 2023 Elsevier Inc. All rights reserved.