Generalized von Neumann inequalities and applications
by
Gilles Cassier
University of Lyon, France

A celebrated theorem of von Neumann assets that, for T a linear contraction on a complex hilbert space H, the inequality ||p(T)|| <= sup{| p(z)| :| z| <= 1} holds. A more recent inequality due to Uffe Haagerup and Pierre de la Harpe asserts that, for any contration on H such that Tⁿ=0 for some n >= 2, the inequality w₂(T)=sup{| < Tx | x > | ;||x|| = 1} <= cos(\frac\pin+1)  holds. Apparently, there is no relationship between them. The following result show that there is a commun way to get both inequalities.

Recall that an operator T belongs to the classical class C_\rho (\rho > 0) of Nagy-Foias if there exists a larger Hilbert space K contains or equal H and a unitary operator U on K such that Tⁿx=P_HUⁿx (n >= 1) for any x in H. The associated radius is defined by setting w_\rho(T)=inf{a > 0:T/a in C_\rho}.

Theorem 1: Let T be a Hilbert space contraction and q a polynomial such that q(T ^* , T)=0 . For any polynomial p in two variables and any \rho in ]0, 2[, we have

w\rho(p(T * , T)) <= w\rho(p(SE, SE * ))

 where S is the usual shift on H², the subspace E is invariant for S ^*  and S_E in B(E) is defined by setting S_E ^* =S ^* | E. When q(e^it, e^-it) =/= 0, we have dim(E) <= 2d ^o (q) (dim(E)=d ^o (q) if q =/= 0 and depends in one variable).

Remarks: Since w₁(T)=|| T|| , w₂(T)=sup{| < Tx | x > | ;|| x|| = 1} and w₂(S ^* | ker(S ^{* n}))=cos(\frac\pin+1), we can recover the inequalities of  von Neumann and  of  Haagerup - Harpe.

We will give now a example of estimate, extending Fejer-Egervary-Szasz and Boas-Kac inequalities, which shows the links with harmonic analysis.

Theorem 2: Let F=P/Q be rational function without principal part wich is positive on the torus. D For any k in N ^* , the coefficient c_k of order k in Laurent developpment of F satisfies the following inequality  

| ck| <= c0w2(Rk),

where R is defined by setting R ^* =S ^* | ker(Q(S) ^* ). The positive real number w₂(R^k) is the better constant.

We will also present other extensions of Haagerup-Harpe inequality, other generalized von Neumann inequalities and a few other results of [BaCa].

[BaCa] C. BADEA AND G. CASSIER, Constrained von Neumann inequalities, preprint.

[Hof62] U. HAAGERUP AND P. DE LA HARPE, The numerical radius of a nilpotent operator on a Hilbert space , Proc. Amer. Soc., 115 (1992), 371-379.

[vNe] J. VON NEUMANN, Eine Spektraltheorie für allgemeine operatoren eines unitären Raumes, Math Nachr., 4, (1951) 258-281.

Date received: June 5, 2000