On the numerical radius of the truncated adjoint shift

Abstract

A celebrated theorem of Fejer (1915) asserts that for a given positive trigonometric polynomial Σⁿ⁻¹ j=n+1 c jeͥʲͭ, we have |c₁|6≤ c0 cos π /n+1 . A more recent inequality due to U. Haagerup and P. de la Harpe [9] asserts that, for any contraction T such that Tn = 0, for some n ≥ 2, the inequality ω2(T) ≤ cos π n+1 holds, and ω2(T) = cos π /n+1 when T is unitarily equivalent to the extremal operator S * n = S * jCn = S * jKer(un(S*)) where un(z) = zn and S * is the adjoint of the shift operator on the Hilbert space of all square summable sequences. Apparently there is no relationship between them. In this mathematical note, we show that there is a connection between Taylor coefficients of positive rational functions on the torus and numerical radius of the extremal operator S * (ϕ) = S * jKer(ϕ(S*)) for a precise inner function ϕ. This result completes a line of investigation begun in 2002 by C. Badea and G. Cassier [1]. An upper and lower bound of the numerical radius of S * (ϕ) are given where ϕ is a finite Blashke product with unique zero