We give a complete description of the possible ranges of real Smirnov functions (quotients of two bounded analytic functions on the open unit disk where the denominator is outer and such that the radial boundary values are real almost everywhere on the unit circle). Our techniques use the theory of unbounded symmetric Toeplitz operators, some general theory of unbounded symmetric operators, classical Hardy spaces, and an application of the uniformization theorem. In addition, we completely characterize the possible valences for these real Smirnov functions when the valence is finite. To do so we construct Riemann surfaces we call disk trees by welding together copies of the unit disk and its complement in the Riemann sphere. We also make use of certain trees we call valence trees that mirror the structure of disk trees.
Copyright © 2019 Springer International Publishing. Article first published online: February 2018.
The definitive version is available at: https://link.springer.com/article/10.1007%2Fs13324-018-0212-1
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Ferguson, Timothy and William T. Ross. "The Range and Valence of a Real Smirnov Function." Analysis and Mathematical Physics 9, no. 1 (March 2019): 497-521. https://link.springer.com/article/10.1007%2Fs13324-018-0212-1
Ferguson, Timothy and Ross, William T., "The Range and Valence of a Real Smirnov Function" (2019). Department of Math & Statistics Faculty Publications. 224.