Free Holomorphic Functions and Interpolation
نویسنده
چکیده
In this paper we obtain a noncommutative multivariable analogue of the classical NevanlinnaPick interpolation problem for analytic functions with positive real parts on the open unit disc. Given a function f : Λ → C, where Λ is an arbitrary subset of the open unit ball Bn := {z ∈ Cn : ‖z‖ < 1}, we find necessary and sufficient conditions for the existence of a free holomorphic function g with complex coefficients on the noncommutative open unit ball [B(H)n]1 such that Re g ≥ 0 and g(z) = f(z), z ∈ Λ, where B(H) is the algebra of all bounded linear operators on a Hilbert space H. The proof employs several results from noncommutative multivariable operator theory and a noncommutative Cayley transform (introduced and studied in the present paper) acting from the set of all free holomorphic functions with positive real parts to the set of all bounded free holomorphic functions. All the results of this paper are obtained in the more general setting of free holomorphic functions with operator-valued coefficients. As consequences, we deduce some results concerning operator-valued analytic interpolation on the unit ball Bn.
منابع مشابه
A 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 کاملA remark on boundedness of composition operators between weighted spaces of holomorphic functions on the upper half-plane
In this paper, we obtain a sucient condition for boundedness of composition operators betweenweighted spaces of holomorphic functions on the upper half-plane whenever our weights are standardanalytic weights, but they don't necessarily satisfy any growth condition.
متن کاملUnivalent holomorphic functions with fixed finitely many coefficients involving Salagean operator
By using generalized Salagean differential operator a newclass of univalent holomorphic functions with fixed finitely manycoefficients is defined. Coefficient estimates, extreme points,arithmetic mean, and weighted mean properties are investigated.
متن کاملIntegral Representation with Weights Ii, Division and Interpolation
Let f be a r × m-matrix of holomorphic functions that is generically surjective. We provide explicit integral representation of holomorphic ψ such that φ = fψ, provided that φ is holomorphic and annihilates a certain residue current with support on the set where f is not surjective. We also consider formulas for interpolation. As applications we obtain generalizations of various results previou...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2009