Abstract
Associated with a boundedg-holed (g≥0) planar domainD are two types of reproducing kernel Hilbert spaces of meromorphic functions onD. We give explicit formulas for the reproducing kernel functions of these spaces. The formulas are in terms of theta functions defined on the Jacobian variety of the Schottky double of the regionD. As applications we settle a conjecture of Abrahamse concerning Nevalinna-Pick interpolation on an annulus and obtain explicit formulas for the curvature (in the sense of Cowen and Douglas) of rank 1 bundle shift operators.
Similar content being viewed by others
References
M.B. Abrahamse, The Pick interpolation theorem for finitely connected domains,Michigan Math. J. 26 (1979), 194–203.
M.B. Abrahamse and R. G. Douglas, A class of subnormal operators related to multiply-connected domains,Advances in Math. 19 (1976), 106–147.
D. Alpay and V. Vinnikov, Analogues d'espaces de de Branges sur des surfaces de Riemann,C.R. Acad. Sci. Paris, Serie I 318 (1994), 1077–1082.
J.A. Ball, Operators of classC 00 over multiply-connected domains,Michigan Math. J. 125 (1978), 183–196.
S. Bergman,The Kernel Function and Conformal Mapping, Math. Surveys No. 5, Amer. Math. Society, New York (1950).
K.F. Clancey, The geometry of representing measures and their critical values, inThe Gohberg Anniversary Collection Vol. II, OT 41 Birkhauser (1989), Basel, pp. 61–75.
K.F. Clancey, Representing measures on multiply connected planar domains,Illinois J. Math. 35 (1991), 286–311.
M.J. Cowen and R.G. Douglas, Complex geometry and operator theory,Acta. Math. 141 (1978), 187–261.
H.M. Farkas and I. Kra,Riemann Surfaces, Springer-Verlag, New York (1980).
J.D. Fay,Theta Functions on Riemann Surfaces, Lecture Notes in Mathematics No. 352, Springer-Verlag, New York (1973).
S. McCullough and L.-C. Shen, On the Szego kernel of an annulus,Proc. Amer. Math. Soc. 121 (1994), 1111–1121.
D. Mumford,Tata Lectures on Theta I, II, Birkhauser, Basel (1983)
D. Sarason,The H p spaces of an annulus, Mem. Amer. Math. Soc. No. 56, Amer. Math. Soc., Providence (1965)
H. Widom, Extremal polynomials associated with a system of curves in the complex plane,Advances in Math. 13 (1969), 127–232.