Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

On the operator ranges of analytic functions


Author: J. S. Hwang
Journal: Proc. Amer. Math. Soc. 87 (1983), 90-94
MSC: Primary 47A60; Secondary 30C45
DOI: https://doi.org/10.1090/S0002-9939-1983-0677239-3
MathSciNet review: 677239
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Following Doob, we say that a function $ f(z)$ analytic in the unit disk $ U$ has the property $ K(\rho )$ if $ f(0) = 0$ and for some arc $ \gamma $ on the unit circle whose measure $ \left\vert \gamma \right\vert \geqslant 2\rho > 0$,

$\displaystyle \mathop {\lim \inf }\limits_{j \to \infty } \left\vert {f({z_j})}... ...lant 1\quad {\text{where}}\;{z_j} \to z \in \gamma \;{\text{and}}\;{z_j} \in U.$

Let $ H$ be a Hilbert space over the complex field, $ A$ an operator whose spectrum is included in $ U$, $ \vert\vert A \vert\vert$ the operator norm of $ A$, and $ f(A)$ the usual Riesz-Dunford operator. We prove that there is no function with the property $ K(\rho )$ satisfying

$\displaystyle (1 - \vert\vert A \vert\vert)\vert\vert {f'(A)} \vert\vert \leqslant 1/n\quad {\text{for}}\;{\text{all}}\;\vert\vert A \vert\vert < 1,$

where $ n > N(\rho ) = \log (1/(1 - \cos \rho ))$. We also show that if $ f$ has the property $ K(\rho )$ then the operator range of $ f(A)$ covers a ball of radius $ k(\rho ) = \sqrt 3 /(4N(\rho ))$. These two results generalize our previous solutions of two long open problems of Doob [1]. Finally, we prove that the operator range of any $ 4$-fold univalent function is not convex. This extends our solution to Ky Fan's Problem [4].

References [Enhancements On Off] (What's this?)

  • [1] J. L. Doob, The ranges of analytic functions, Ann. of Math. (2) 36 (1935), 117-126. MR 1503212
  • [2] -, Conformally invariant cluster value theory, Illinois J. Math. 5 (1961), 521-549. MR 0186821 (32:4276)
  • [3] N. Dunford and J. T. Schwartz, Linear operators, Part I: General theory, Interscience, New York, 1958. MR 1009162 (90g:47001a)
  • [4] Ky Fan, Analytic functions of a proper contraction, Math. Z. 160 (1978), 275-290. MR 0482310 (58:2383)
  • [5] E. Hille, Analytic function theory. II, Ginn, Boston, Mass., 1962. MR 0201608 (34:1490)
  • [6] J. S. Hwang and D. C. Rung, Proof of a conjecture of Doob, Proc. Amer. Math. Soc. 75 (1979), 231-234. MR 532142 (81i:30061)
  • [7] -, An improved estimate for the Bloch norm of functions in Doob's class, Proc. Amer. Math. Soc. 80 (1980), 406-410. MR 580994 (81i:30058)
  • [8] J. S. Hwang, On an extremal property of Doob's class, Trans. Amer. Math. Soc. 252 (1979), 393-398. MR 534128 (80i:30057)
  • [9] -, On the ranges of analytic functions, Trans. Amer. Math. Soc. 260 (1980), 623-629. MR 574804 (81g:30039)
  • [10] -, On estimate for Bloch and Doob norm and covering problem in Doob's class, J. Math. Anal. Appl. (to appear).
  • [11] -, A problem on Riesz-Dunford operator and convex univalent function, Glasgow Math. J. (to appear).

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A60, 30C45

Retrieve articles in all journals with MSC: 47A60, 30C45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1983-0677239-3
Keywords: Analytic function, Riesz-Dunford operator, convex function and operator range
Article copyright: © Copyright 1983 American Mathematical Society

American Mathematical Society