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.     

Singular parts of contractive analytic operator functions

One considers the class _G of holomorphic functions in a domain G, whose values are contractions in a separable Hilbert space. It is proved that if T(·) _G , T(z₀) is a weak contraction, its singular part T^s(z₀) is complete, and the increments T(z)–T(z₀) are not too large (for example, finite-dimensional), then the operator T^s(z₀) is complete for almost all zG. If, however, T(z₀) is, in addition, completely nonunitary and satisfies definite smoothness conditions, then in the nontrivial case the spectrum [z] of the contraction T^s(z) (zG) is a thin set: The proof of the mentioned results is based on the investigation of the formulas obtained in the paper, connecting the characteristic functions of the contractions T(z) for different values of zG.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 157, pp. 30–44, 1987.     

Some stability properties for analytic operator functions

Let be a connected, finite-dimensional, complex analytic manifold; let T() be an analytic function defined on , whose values are J-biexpanding operators on a J-space H. Let (A) denote the range of A. The following assertions are proved: 1. The lineals and do not depend on . 2. For arbitrary we have Translated from Matematicheskie Zametki, Vol. 20, No. 4, pp. 511–520, October, 1976.     

Convolution operator in the space of real analytic functions

Translated from Matematicheskie Zametki, Vol. 49, No. 3, pp. 57–65, March, 1991.     

Some classes of analytic functions involving Noor integral operator

The object of the present paper is to investigate some inclusion properties of certain subclasses of analytic functions defined by using the Noor integral operator. The integral preserving properties in connection with the operator are also considered. Relevant connections of the results presented here with those obtained in earlier works are pointed out.     

Classes of analytic functions associated with the Choi-Saigo-Srivastava operator

We investigate several properties of the linear Choi-Saigo-Srivastava operator and associated classes of analytic functions which were introduced and studied by J.H. Choi, M. Saigo and H.M. Srivastava [J.H. Choi, M. Saigo, H.M. Srivastava, Some inclusion properties of a certain family of integral operators, J. Math. Anal. Appl. 276 (2002) 432-445]. Several theorems are an extension of earlier results of the above paper.     

Counterexample to a uniqueness theorem for analytic operator functions

It is proved that there exists a bounded holomorphic operator-function z→F(z), ¦z¦<1, with compact values (in a separable Hilbert space) and such that its boundary values F(S), ¦S¦=1 are compact on one (given) arc of the circle and not compact on the other. The corresponding example is constructed with the help of vectorial Hankel operators.     

Cyclic elements of the differentiation operator in spaces of analytic functions

Mathematical Notes -     

On maximal ideals in two algebras of operator valued analytic functions

The maximal ideals in two algebras of operator valued analytic functions in the unit disc are described.     

On the best approximations and rate of convergence of decompositions in the root vectors of an operator

We establish upper bounds of the best approximations of elements of a Banach space B by the root vectors of an operator A that acts in B. The corresponding estimates of the best approximations are expressed in terms of a K-functional associated with the operator A. For the operator of differentiation with periodic boundary conditions, these estimates coincide with the classical Jackson inequalities for the best approximations of functions by trigonometric polynomials. In terms of K-functionals, we also prove the abstract Dini-Lipschitz criterion of convergence of partial sums of the decomposition of f from B in the root vectors of the operator A to f