Unbounded symmetric analytic functions on $\ell_1$


  • Iryna Chernega
  • Andriy Zagorodnyuk




We show that each $G$-analytic symmetric function on an open set of $\ell _1$ is analytic and construct an example of a symmetric analytic function on $\ell _1$ which is not of bounded type.


Ansemil, J. M., Aron, R. M., and Ponte, S., Behavior of entire functions on balls in a Banach space, Indag. Math. (N.S.) 20 (2009), no. 4, 483–489. https://doi.org/10.1016/S0019-3577(09)80021-9

Chernega, I., Homomorphisms of the algebra of symmetric analytic functions on $ell _1$, Carpathian Math. Publ. 6 (2014), no. 2, 394–398. https://doi.org/10.15330/cmp.6.2.394-398

Chernega, I., Symmetric polynomials and holomorphic functions on infinite dimensional spaces, Journal of Vasyl Stefanyk Precarpathian National University 2 (2015), no. 4, 23–49. https://doi.org/10.15330/jpnu.2.4.23-49

Chernega, I., Galindo, P., and Zagorodnyuk, A., The convolution operation on the spectra of algebras of symmetric analytic functions, J. Math. Anal. Appl. 395 (2012), no. 2, 569–577. https://doi.org/10.1016/j.jmaa.2012.04.087

Chernega, I., Galindo, P., and Zagorodnyuk, A., Some algebras of symmetric analytic functions and their spectra, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 1, 125–142. https://doi.org/10.1017/S0013091509001655

Chernega, I., Galindo, P., and Zagorodnyuk, A., A multiplicative convolution on the spectra of algebras of symmetric analytic functions, Rev. Mat. Complut. 27 (2014), no. 2, 575–585. https://doi.org/10.1007/s13163-013-0128-0

González, M., Gonzalo, R., and Jaramillo, J. A., Symmetric polynomials on rearrangement-invariant function spaces, J. London Math. Soc. (2) 59 (1999), no. 2, 681–697. https://doi.org/10.1112/S0024610799007164

Gould, H. W., The Girard-Waring power sum formulas for symmetric functions and Fibonacci sequences, Fibonacci Quart. 37 (1999), no. 2, 135–140.

Mujica, J., Complex analysis in Banach spaces, North-Holland Mathematics Studies, vol. 120, North-Holland Publishing Co., Amsterdam, 1986.

Nemirovskiĭ, A. S. and Semenov, S. M., The polynomial approximation of functions on Hilbert space, Mat. Sb. (N.S.) 92(134) (1973), 257–281, English translation Mat. USSR Sbornik 21 (1973), no. 2, 255–277. https://doi.org/10.1070/SM1973v021n02ABEH002016




How to Cite

Chernega, I., & Zagorodnyuk, A. (2018). Unbounded symmetric analytic functions on $\ell_1$. MATHEMATICA SCANDINAVICA, 122(1), 84–90. https://doi.org/10.7146/math.scand.a-102082