Inner functions of numerical contractions

✍ Scribed by Hwa-Long Gau; Pei Yuan Wu

• Publisher: Elsevier Science
• Year: 2009
• Tongue: English
• Weight: 146 KB
• Volume: 430
• Category: Article
• ISSN: 0024-3795
• DOI: 10.1016/j.laa.2008.11.020

✦ Synopsis

We prove that, for a function f in H ∞ of the unit disc with f ∞ 1, the existence of an operator T on a complex Hilbert space H with its numerical radius at most one and with f (T )x = 2 for some unit vector x in H is equivalent to that f be an inner function with f (0) = 0. This confirms a conjecture of Drury [S.W. Drury, Symbolic calculus of operators with unit numerical radius, Linear Algebra Appl. 428 (2008) 2061-2069]. Moreover, we also show that any operator T satisfying the above conditions has a direct summand similar to the compression of the shift S(φ), where φ(z) = zf (z) for |z| < 1. This generalizes the result of Williams and Crimmins [J.P. Williams, T. Crimmins, On the numerical radius of a linear operator, Amer. Math. Monthly 74 (1967) 832-833] for f (z) = z and of Crabb [M.J. Crabb, The powers of an operator of numerical radius one, Michigan Math. J. 18 (1971) 253-256] for f (z) = z n (n 2).