Generalized operator for Alexander integral operator

Let Tn be the class of functions f which are defined by a power series f\left( z \right) = z + {a_{n + 1}}{z^{n + 1}} + {a_n}2{z^{n + 2}} + \ldots for every z in the closed unit disc \bar {\mathbb{U}}. With m different boundary points zs, (s = 1,2,...,m), we consider αm ∈ eiβ𝒜−j−λf(𝕌), here 𝒜−j−λ is the generalized Alexander integral operator and 𝕌 is the open unit disc. Applying 𝒜−j−λ, a subclass Bn(αm,β,ρ; j, λ) of Tn is defined with fractional integral for functions f. The object of present paper is to consider some interesting properties of f to be in Bn(αm,β,ρ; j, λ).

Examples of adic spaces

This chapter discusses various examples of adic spaces. These examples include the adic closed unit disc; the adic affine line; the closure of the adic closed unit disc in the adic affine line; the open unit disc; the punctured open unit disc; and the constant adic space associated to a profinite set. The chapter focuses on one example: the adic open unit disc over Zp. The adic spectrum Spa Zp consists of two points, a special point and a generic point. The chapter then studies the structure of analytic points. It also clarifies the relations between analytic rings and Tate rings.

THE ESSENTIAL NORMS OF COMPOSITION OPERATORS ON WEIGHTED DIRICHLET SPACES

Let $\unicode[STIX]{x1D711}$ be an analytic self-map of the unit disc. If $\unicode[STIX]{x1D711}$ is analytic in a neighbourhood of the closed unit disc, we give a precise formula for the essential norm of the composition operator $C_{\unicode[STIX]{x1D711}}$ on the weighted Dirichlet spaces ${\mathcal{D}}_{\unicode[STIX]{x1D6FC}}$ for $\unicode[STIX]{x1D6FC}>0$. We also show that, for a univalent analytic self-map $\unicode[STIX]{x1D711}$ of $\mathbb{D}$, if $\unicode[STIX]{x1D711}$ has an angular derivative at some point of $\unicode[STIX]{x2202}\mathbb{D}$, then the essential norm of $C_{\unicode[STIX]{x1D711}}$ on the Dirichlet space is equal to one.

A Geometric Extension of Schwarz’s Lemma and Applications

Let f be a holomorphic function of the unit disc , preserving the origin. According to Schwarz’s Lemma, |f'(0)| ≤ 1, provided that . We prove that this bound still holds, assuming only that f() does not contain any closed rectilinear segment [0, eiϕ], ϕ ∊ [0, zπ], i.e., does not contain any entire radius of the closed unit disc. Furthermore, we apply this result to the hyperbolic density and give a covering theorem.

Composition operators

A study of centered composition operators on l2 is made in this paper. Also the spectrum of surjective composition operators is computed. A necessary and sufficient condition is obtained for the closed unit disc to be the spectrum of a surjective composition operator.