Hankel operators on planar domains

Let Ω be a bounded domain in the plane whose boundary consists of a finite number of disjoint analytic simple closed curves LetA denote the space of analytic functions on Ω which are square integrable over Ω with respect to area measure and letP denote the orthogonal projection ofL ²(Ω,dA) ontoA. A functionb inA induces a Hankel operator (densely defined) onA by the ruleH _b (g)=(I?P)bg. This paper continues earlier investigations of the authors and others by determining conditions under whichH _b is bounded, compact, or lies in the Schatten-von Neumann idealS _p , 1<p<∞     

On a transplant operator and explicit construction of Cauchy-type integral representations for p-analytic functions

We present a new technique for explicit construction of Cauchy kernels and Cauchy integral representations for a class of generalized analytic functions and p-analytic functions.     

A holomorphic characterization of compact sets in Banach spaces

An extension of Lempert’s result about non approximability by entire functions of analytic functions on some open subsets of ?_∞ is obtained for Banach spaces having a bounding non relatively compact set.We also prove that subsets A that are bounding for analytic functions defined in any of its neighborhoodswhose boundary lies at positive distance from A are relatively compact.     

Starlikeness and subordination properties of a linear operator

In this paper, we introduce a new class of p-valent analytic functions defined by using a linear operator L_k^α. For functions in this class H_k^α(p,λh) we estimate the coefficients. Furthermore, some subordination properties related to the operator L_k^α are also derived.     

Upper and lower bounds on norms of functions of matrices

Given an n by n matrix A, we look for a set S in the complex plane and positive scalars m and M such that for all functions p bounded and analytic on S and throughout a neighborhood of each eigenvalue of A, the inequalities

m·inf{‖f‖_L^∞(S):f(A)=p(A)}?‖p(A)‖?M·inf{‖f‖_L^∞(S):f(A)=p(A)}     

Estimates of the norms of powers of functions in certain Banach spaces

Asymptotic estimates of the norms of powers of analytic functions in certain Banach spaces are obtained. For a function ? analytic in the closed unit disk and satisfyingsup |?(z)|=1, it is shown that there exist constants C, c, and α depending on ? and a Banach space X such that, for every n, $$cn^\alpha \leqslant \left\| {\varphi ^n } \right\|x \leqslant Cn^\alpha .$$ Cases in which X is the space l^p _A or the Besov space are considered. Bibliography: 4 titles.     

A first-order differential double subordination with applications

Let q₁ and q₂ belong to a certain class of normalized analytic univalent functions in the open unit disk of the complex plane. Sufficient conditions are obtained for normalized analytic functions p to satisfy the double subordination chain q₁(z)?p(z)?q₂(z). The differential sandwich-type result obtained is applied to normalized univalent functions and to Φ-like functions.     

A class of B-(p,r)-invex functions and mathematical programming

Invexity of a function is generalized. The new class of nonconvex functions, called B-(p,r)-invex functions with respect to η and b, being introduced, includes many well-known classes of generalized invex functions as its subclasses. Some properties of the introduced class of B-(p,r)-invex functions with respect to η and b are studied. Further, mathematical programming problems involving B-(p,r)-invex functions with respect to η and b are considered. The equivalence between saddle points and optima, and different type duality theorems are established for this type of optimization problems.     

Spaces of analytic functions in a region with an angle

In this paper we consider the space A_p of analytic functions which are p-power integrable in a region with an angle. We find a set of numbers p and q (1/p+1/q=1) (which depend on the magnitude of the angle) for which the spaces A_p and A_q are mutually conjugate. In each of these spaces we introduce the orthonormal system $$e_n = \sqrt {{{\left( {n + 1} \right)} \mathord{\left/ {\vphantom {{\left( {n + 1} \right)} \pi }} \right. \kern-\nulldelimiterspace} \pi }} \varphi \prime \varphi ^n ,n = 0,1, \ldots ,$$ where? is the conformal mapping of the region onto the unit disc. We prove it is dense and determine when it will be a basis.     

Integral Representations and the Behavior of Krasner Analytic Functions Around Singular Points

Given a prime number p and K a compact subset of ? _p , the topological completion of the algebraic closure of the field of p-adic numbers, we are interested in integral representations for a class of Krasner analytic functions defined on ${\mathbb{P}}^1({\mathbb{C}}_p)\setminus K$ with values in ${\mathbb{P}}^1({\mathbb{C}}_p)$ . We apply these results to study the behavior of Krasner analytic functions around singular points.     

Algebraic extension of *-A operator

In this paper, we study various properties of algebraic extension of *-A operator. Specifically, we show that every algebraic extension of *-A operator has SVEP and is isoloid. And if T is an algebraic extension of *-A operator, then Weyl's theorem holds for f(T), where f is an analytic functions on some neighborhood of σ(T) and not constant on each of the components of its domain.     

On some classes of p-valent functions involving Carlson-Shaffer operator

Let , be a class of functions analytic in the open unit disc E. We use Carlson-Shaffer operator for p-valent functions to define and study certain classes of analytic functions. Inclusion results, a radius problem and some other interesting properties are discussed.     

О теореме Ч. Горовица о нулях функций иэ пространств Бергмана

Thedistribution of the absolute values of the zeros z _k (f) is studied, where f belongs to the Bergman space A _α ^p with the power weight t ^α α > ?1, or to near spaces of analytic n functions. The main object of the study is the estimate of the product $ \prod\limits_{k = 1}^n {\left| {z_k (f)} \right|^{ - 1} } $ . Estimates of such products for functions f ε A _α ^p , were given by Ch. Gorovic in 1974. However, a number of essential questions in connection with these estimates have not been answered so far. In the present paper, answers are be given for some of these questions.     

Certain subclasses of p-valent functions involving the Dziok-Srivastava operator

Using the methods of differential subordination, we investigate some inclusion relationships of certain subclasses of analytic and p-valent functions which are defined by means of Dziok-Srivastava operator.     

Subharmonicity for uniform algebras

Let A be a uniform algebra with maximal ideal space M_A. A notion of subharmonicity is defined for functions on M_A. Under certain hypotheses of continuity, it is proved that the notion of subharmonicity is local. A consequence is that the notion of Jensen boundary point is local. The solutions to an abstract Dirichlet problem are studied in the context of uniform algebras. The methods are applied to algebras of analytic functions, and in particular a version of the extended maximum principle is obtained for analytic functions of several complex variables.     

Functions of operators under perturbations of class Sp

This is a continuation of our paper [2]. We prove that for functions f in the Hölder class Λα(R) and 1<p<∞, the operator f(A)−f(B) belongs to Sp/α, whenever A and B are self-adjoint operators with A−B∈Sp. We also obtain sharp estimates for the Schatten-von Neumann norms ‖f(A)−f(B)Sp/α_‖ in terms of ‖A−BSp_‖ and establish similar results for other operator ideals. We also estimate Schatten-von Neumann norms of higher order differences . We prove that analogous results hold for functions on the unit circle and unitary operators and for analytic functions in the unit disk and contractions. Then we find necessary conditions on f for f(A)−f(B) to belong to Sq under the assumption that A−B∈Sp. We also obtain Schatten-von Neumann estimates for quasicommutators f(A)R−Rf(B), and introduce a spectral shift function and find a trace formula for operators of the form f(A−K)−2f(A)+f(A+K).     

On the behavior of some two-variable p-adic L-functions

We give some p-adic integral representations for the two-variable p-adic L-functions introduced recently by G. Fox. For powers of the Teichmüller character, we use the integral representation to extend the L-function to a larger domain, in which it is a meromorphic function in the first variable and an analytic element in the second. These integral representations imply systems of congruences for the generalized Bernoulli polynomials, improving previous results of Fox, Gunaratne, and the author; they also lead to generalizations of some formulas of Diamond and of Ferrero and Greenberg for p-adic L-functions in terms of the p-adic gamma and log gamma functions.     

Iterated vanishing cycles

If A ^? is a bounded, constructible complex of sheaves on a complex analytic space X, and ${f : X \rightarrow \mathbb{C}}$ and ${g : X \rightarrow \mathbb{C}}$ are complex analytic functions, then the iterated vanishing cycles φ _g [?1](φ _f [?1]A ^?) are important for a number of reasons. We give a formula for the stalk cohomology H*(φ _g [?1]φ _f [?1]A ^?) _x in terms of relative polar curves, algebra, and Morse modules of A ^?.     

Density of characters of bounded <Emphasis Type="Italic">p</Emphasis>-adic analytic functions in the topological dual

Let K be an ultrametric complete algebraically closed field, let D be a disk {x ∈ K ‖x| < R} (with R in the set of absolute values of K) and let A be the Banach algebra of bounded analytic functions in D. The vector space generated by the set of characters of A is dense in the topological dual of A if and only if K is not spherically complete. Let H(D) be the Banach algebra of analytic elements in D. The vector space generated by the set of characters of H(D) is never dense in the topological dual of H(D).