By using the concept of the weak subordination, we examine the stability a class of analytic functions in the unit disk is said to be stable if it is closed under weak subordination for a class of admissible functions in complex Banach spaces. The stability of analytic functions in the following classes is discussed: Bloch class, little Bloch class, hyperbolic little Bloch class, extend Bloch c...

We compare the rate of decay singular numbers a given composition operator acting on various Hilbert spaces analytic functions unit disk D. show that for Hardy and Bergman spaces, our results are sharp. also give lower upper estimates with symbol “cusp map” lens maps, weighted Dirichlet spaces.

We show that every complete metric space is homeomorphic to the locus of zeros of an entire analytic map from a complex Hilbert space to a complex Banach space. As a corollary, every separable complete metric space is homeomorphic to the locus of zeros of an entire analytic map between two complex Hilbert spaces. §1. Douady had observed [8] that every compact metric space is homeomorphic to the...

Brennan’s conjecture in univalent function theory states that if τ is any analytic univalent transform of the open unit disk D onto a simply connected domain G and −1/3 < p < 1, then 1/(τ ′)p belongs to the Hilbert Bergman space of all analytic square integrable functions with respect to the area measure. We introduce a class of analytic function spaces La(μp) on G and prove that Brennan’s conj...

Weighted composition operators appear in the study of dynamical systems and also in characterizing isometries of some classes of Banach spaces. One of the most important generalizations of weighted composition operators, are generalized weighted composition operators which in special cases of their inducing functions give different types of well-known operators like: weighted composition operat...

We give some sufficient conditions under which the tuple of the adjoint of weighted composition operators $(C_{omega_1,varphi_1}^*‎ , ‎C_{omega_2,varphi_2}^*)$ on the Hilbert space $mathcal{H}$ of analytic functions is supercyclic‎.