Zero Sets for Spaces of Analytic Functions
نویسندگان
چکیده
We show that under mild conditions, a Gaussian analytic function F that a.s. does not belong to a given weighted Bergman space or Bargmann–Fock space has the property that a.s. no non-zero function in that space vanishes where F does. This establishes a conjecture of Shapiro (1979) on Bergman spaces and allows us to resolve a question of Zhu (1993) on Bargmann–Fock spaces. We also give a similar result on the union of two (or more) such zero sets, thereby establishing another conjecture of Shapiro (1979) on Bergman spaces and allowing us to strengthen a result of Zhu (1993) on Bargmann–Fock spaces.
منابع مشابه
Zeros of Random Analytic Functions
Zeros of Random Analytic Functions The dominant theme of this thesis is that random matrix valued analytic functions, generalizing both random matrices and random analytic functions, for many purposes can (and perhaps should) be effectively studied in that level of generality. We study zeros of random analytic functions in one complex variable. It is known that there is a one parameter family o...
متن کاملComposition operators acting on weighted Hilbert spaces of analytic functions
In this paper, we considered composition operators on weighted Hilbert spaces of analytic functions and observed that a formula for the essential norm, gives a Hilbert-Schmidt characterization and characterizes the membership in Schatten-class for these operators. Also, closed range composition operators are investigated.
متن کاملA special subspace of weighted spaces of holomorphic functions on the upper half plane
In this paper, we intend to define and study concepts of weight and weighted spaces of holomorphic (analytic) functions on the upper half plane. We study two special classes of these spaces of holomorphic functions on the upper half plane. Firstly, we prove these spaces of holomorphic functions on the upper half plane endowed with weighted norm supremum are Banach spaces. Then, we investigate t...
متن کاملSystems of germs and theorems of zeros in infinite-dimensional spaces
Systems of germs of sets in infinite-dimensional spaces are introduced and studied. Such a system corresponds to a local zero-set of an ideal of the ring of analytic functions of infinite number of variables. Conversely, this system of germs defines the ideal of germs of analytic functions vanishing on it. A theorem of zeros is proved, stating that this ideal is the radical (in the complex case...
متن کاملZero sets in pointfree topology and strongly $z$-ideals
In this paper a particular case of z-ideals, called strongly z-ideal, is defined by introducing zero sets in pointfree topology. We study strongly z-ideals, their relation with z-ideals and the role of spatiality in this relation. For strongly z-ideals, we analyze prime ideals using the concept of zero sets. Moreover, it is proven that the intersection of all zero sets of a prime ideal of C(L),...
متن کامل