New proofs of the Haagerup--de la Harpe inequality (Q1889663)

Let \(B(H)\) denote the algebra of all bounded linear operators on a complex Hilbert space \(H\). The numerical range \(W(T)\) of an operator \(T\in B(H)\) is defined by \(W(T)=\{\langle Tx,x\rangle:x\in (H)_1\}\), where \((H)_1\) is the unit sphere in \(H\). The numerical radius \(w(T)\) of an operator \(T\in B(H)\) is given by \(w(T)=\sup\{| \lambda| :\lambda\in W(T)\}\). \textit{U.~Haagerup} and \textit{P.~de la Harpe} [Proc.\ Am.\ Math.\ Soc.\ 115, 371--379 (1992; Zbl 0781.47014)] proved the following sharp estimate for the numerical radius of a nilpotent operator \(T\in B(H)\): \(w(T)\leq\|T\|\cos\frac{\pi}{n+1}\), where \(n\) is the power of nilpotency of \(T\). In this note, the author gives new proofs of the Haagerup--de la Harpe inequality. Indeed, for a nilpotent contraction operator \(T\in B(H)\) with nilpotency degree \(n\leq2\), as is known from [\textit{B.~Sz.--Nagy} and \textit{C.~Foias}, ``Harmonic analysis of operators on Hilbert spaces'' (Akadémiai Kiadó, Budapest; North--Holland, Amsterdam--London) (1970; Zbl 0201.45003)], the operator \(T\) is unitarily equivalent to its model operator \(M_\theta\). By a simple computation, the author deduces that \[ w(M_\theta)\leq\max\left\{\sum_{m=0}^{m-2}a_{m}a_{m+1}:\sum_{m=0}^{m-1}a_{m}^2=1,\;a_{m}\geq0, \;m=0,\dots,n-1\right\}=\cos\frac{\pi}{n+1}. \] This last equality can be found in [\textit{G.~Polya} and \textit{G.~Szegő}, ``Problems and theorems in Analysis, Vol.~II'' (Springer Study Editon, Springer--Verlag, New York--Heidelberg--Berlin) (1976; Zbl 0359.00003)]. The result follows immediately from the fact that \(w(T)=w(M_\theta)\).