TY - JOUR

T1 - Inner functions of numerical contractions

AU - Gau, Hwa Long

AU - Wu, Pei Yuan

N1 - Funding Information:
∗ Corresponding author. E-mail addresses: hlgau@math.ncu.edu.tw (H.-L. Gau), pywu@math.nctu.edu.tw (P.Y. Wu). 1 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-008-006. 2 Research supported by the National Science Council of the Republic of China under NSC 96-2115-M-009-013-MY3 and by the MOE-ATU.

PY - 2009/4/15

Y1 - 2009/4/15

N2 - We prove that, for a function f in H∞ of the unit disc with {norm of matrix} f {norm of matrix}∞ ≤ 1, the existence of an operator T on a complex Hilbert space H with its numerical radius at most one and with {norm of matrix} f (T) x {norm of matrix} = 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{symbol}), where φ{symbol} (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) = zn (n ≥ 2).

AB - We prove that, for a function f in H∞ of the unit disc with {norm of matrix} f {norm of matrix}∞ ≤ 1, the existence of an operator T on a complex Hilbert space H with its numerical radius at most one and with {norm of matrix} f (T) x {norm of matrix} = 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{symbol}), where φ{symbol} (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) = zn (n ≥ 2).

KW - Compression of the shift

KW - Numerical contraction

KW - Numerical radius

KW - Numerical range

UR - http://www.scopus.com/inward/record.url?scp=61449220867&partnerID=8YFLogxK

U2 - 10.1016/j.laa.2008.11.020

DO - 10.1016/j.laa.2008.11.020

M3 - 期刊論文

AN - SCOPUS:61449220867

SN - 0024-3795

VL - 430

SP - 2182

EP - 2191

JO - Linear Algebra and Its Applications

JF - Linear Algebra and Its Applications

IS - 8-9

ER -