Holomorphic T-Monsters and Strongly Omnipresent Operators

Assume that G is a nonempty open subset of the complex plane and that T is an operator on the linear space of holomorphic functions in G, endowed with the compact-open topology. In this paper we introduce the notions of strongly omnipresent operator and of T-monster, which are related to the wild behaviour of certain holomorphic functions near the boundary of G. T-monsters extend a concept introduced by W. Luh and K.-G. Grosse-Erdmann. After showing that T is strongly omnipresent if and only if the set of T-monsters is residual, it is proved in this paper that certain kinds of infinite order differential and antidifferential operators are strongly omnipresent, which improves some earlier nice results due to the mentioned authors.     

Supercyclic sequences of differential operators

Summary Several necessary and sufficient conditions for a sequence of infinite order differential linear operators on spaces of holomorphic functions on a domain of the complex plane to be supercyclic or c-hypercyclic are given in this paper, so completing earlier work of the authors on hypercyclicity, which in turn extended Birkhoff--MacLane--Godefroy--Shapiro's theorems. A new, general eigenvalue criterium for supercyclicity is also provided.     

Holomorphic Functions Having Large Images under the Action of Differential Operators

We prove in this note that, given a simply connected domainGin the complex plane and a sequence of infinite order linear differential operators generated by entire functions of subexponential type satisfying suitable conditions, then there are holomorphic functionsfonGsuch that the image of any open subset under the action of those operators onfis arbitrarily large. This generalizes an earlier result about images of derivatives. A known statement about close orbits is also strengthened.     

Monsters in Hardy and Bergman Spaces

A monster in the sense of Luh is a holomorphic function on a simply connected domain in the complex plane such that it and all its derivatives and antiderivatives exhibit an extremely wild behaviour near the boundary. In this paper, the Hardy spaces H p and the Bergman spaces B p (1 h p < X ) on the unit disk are considered, and it is shown that there are no Luh-monsters in them. Nevertheless, it is proved that T -monsters (as introduced by the authors in an earlier work) can be found in each of these spaces for any finite order linear differential operator T .     

Hypercyclic Sequences of Differential and Antidifferential Operators

In this paper, we provide some extensions of earlier results about hypercyclicity of some operators on the Fréchet space of entire functions of several complex variables. Specifically, we generalize in several directions a theorem about hyper- cyclicity of certain infinite order linear differential operators with constant coefficients and study the corresponding property for certain kinds of “antidifferential” operators which are introduced in the paper. In addition, the existence of hypercyclic functions for certain sequences of differential operators with additional properties, for instance, boundedness or with some nonvanishing derivatives, is established.     

On regular linear relations

For a closed linear relation in a Banach space the concept of regularity is introduced and studied. It is shown that many of the results of Mbekhta and other authors for operators remain valid in the context of multivalued linear operators. We also extend the punctured neighbourhood theorem for operators to linear relations and as an application we obtain a characterization of semiFredholm linear relations which are regular.     

Fundamental Solutions of Some Linear Operator Equations and Applications

A concept of a fundamental solution is introduced for linear operator equations given in some functional spaces. In the case where this fundamental solution does not exist, the representation of the solution is found by the concept of a generalized fundamental solution, which is introduced for operators with nontrivial and generally infinite-dimensional kernels. The fundamental and generalized fundamental solutions are also investigated for a class of Fredholm-type operator equations. Some applications are given for one-dimensional generally nonlocal hyperbolic problems with trivial, finite- and infinite-dimensional kernels. The fundamental and generalized fundamental solutions of such problems are constructed as particular solutions of a system of integral equations or an integral equation. These fundamental solutions become meaningful in a general case when the coefficients are generally nonsmooth functions satisfying only some conditions such as p-integrablity and boundedness.     

Optimal Secant-Type Methods for Operator Equations

We prove an existence and uniqueness theorem for operator equations in Banach spaces with (generally non-differentiable) operators whose divided differences are Lipschitz continuous on operator's domain. The theorem makes possible to apply the concept of entropy optimality of iterative methods introduced in our earlier work to the class of secant-type methods. Using this concept, we show that it is feasible to get a method that needs the same information (one value of the operator) per iteration but exhibits a faster convergence than the secant and secant-update methods.     

Multiplicative Decompositions of Holomorphic Fredholm Functions and ψ*-Algebras

In this article we construct multiplicative decompositions of holomorphic Fredholm operator valued functions on Stein manifolds with values in various algebras of differential and pseudo differential operators which are submultiplicative ψ* - algebras, a concept introduced by the first author. For Fredholm functions T(z) satisfying an obvious topological condition we. Prove (0.1) T(z) = A(z)(I + S(z)), where A(z) is holomorphic and invertible and S(z) is holomorphic with values in an “arbitrarily small” operator ideal. This is a stronger condition on S(z) than in the authors' additive decomposition theorem for meromorphic inverses of holomorphic Fredholm functions [12], where the smallness of S(z) depends on the number of complex variables. The Multiplicative Decomposition theorem (0.1) sharpens the authors' Regularization theorem [11]; in case of the Band algebra L(X) of all bounded linear operators on a Band space, (0.1) has been proved by J. Letterer [20] for one complex variable and by M. 0. Zaidenberg, S. G. Krein, P. A. Kuchment and A. A. Pankov [26] for the Banach ideal of compact operators.     

Strongly omnipresent integral operators

An operatorT on the spaceH(G) of holomorphic functions on a domainG is strongly omnipresent whenever there is a residual set of functionsfH(G) such thatT f exhibits an extremely wild behaviour near the boundary. The concept of strong omnipresence was recently introduced by the first two authors. In this paper it is proved that a large class of integral operators including Volterra operators with or without a perturbation by differential operators has this property, completing earlier work about differential and antidifferential operators.The work of the first two authors has been partially supported by DGES grant PB96-1348 and the Junta de Andalucía.     

ON THE GENERALIZED RESOLVENT OF LINEAR PENCILS IN BANACH SPACES

Utilizing the stability characterizations of generalized inverses of linear operator, we investigate the existence of generalized resolvent of linear pencils in Banach spaces. Some practical criterions for the existence of generalized resolvents of the linear pencil λ→ T λ S are provided and an explicit expression of the generalized resolvent is also given. As applications, the characterization for the Moore-Penrose inverse of the linear pencil to be its generalized resolvent and the existence of the generalized resolvents of linear pencils of finite rank operators, Fredholm operators and semi-Fredholm operators are also considered. The results obtained in this paper extend and improve many results in this area.     

Cyclicity of coefficient multipliers: Linear structure

We characterize various kinds of cyclicity of sequences of coefficient multipliers, which are operators defined on spaces of holomorphic functions. In the case of a single coefficient multiplier we characterize its cyclicity, which contrasts with the fact that such operators are never supercyclic. Moreover, it is proved that for each cyclic function there is a dense part of the linear span of its orbit all of whose vectors are cyclic. The authors have been partially supported by MCYT Grants BFM2003-03893-C02-01, MTM2004-21420-E and Plan Andaluz de Investigación Junta de Andalucía FQM-127.     

Estimates for Elements of Inverse Matrices for a Class of Operators with Matrices of Special Structure

In this paper, we consider questions related to the structure of inverse matrices of linear bounded operators acting in infinite-dimensional complex Banach spaces. We obtain specific estimates of elements of inverse matrices for bounded operators whose matrices have a special structure. Matrices are introduced as special operator-valued functions on an index set. The matrix structure is described by the behavior of the given function on elements of a special partition of the index set. The method used for deriving the estimates is based on an analysis of Fourier series of strongly continuous periodic functions.     

Banach空间中闭线性算子广义预解式存在定理

在Banach空间中研究闭线性算子广义逆扰动问题和广义预解式存在性问题.给出了闭线性算子广义逆在T-有界扰动下的一些稳定特征,这些特征推广了在有界线性算子情形、闭线性算子有界扰动情形以及闭线性算子保值域或保核空间情形的一些已知结果.以此为基础,得到了闭线性算子广义预解式存在的一些充要条件及其广义预解式的显式表达式.作为应用,给出了闭Fredholm算子和闭半-Fredholm算子的广义预解式存在性特征.     

Differential operators associated with holomorphic mappings

Two differential operators which act on holomorphic mappings to complex projective space are studied. One operator is of second order and characterizes projective linear mappings. The other operator is of third order and may be viewed as a curvature. The two operators together play a role analogous to the Schwarzian derivative.A canonical approximation to a holomorphic mapping is defined, and a relationship between the approximation and the operators is derived. In the one variable case, this reduces to a classical result relating the Schwarzian derivative and the best Möbius approximation to a holomorphic function.     

关于无界连续函数逼近

§ 1 .　Introduction　　Itisaveryimportantstudytoapproximatenonboundedcontinuousfunctionsdefinedonalargerange.Aneffectivemethodcalledthemethodofmultiplierenlargementhasbeenputfor wardandgreatlydevelopedbyHSULCandWANGReng Hong [1— 3].Thesecondauthorofthispaperinparticularhasprovedin [2 ]almostuniformlythatanynonboundedcontinuousfunctionsdefinedonawholerealaxiscanbeapproximatedbypolynomialoperators.We’llgen eralizeinthispaperthebasicprincipleofthismethod,andasanexample,Bernsteinpolynomial…     

Universal holomorphic and harmonic functions with additional properties

Based on the concept of so-called (total) omnipresence of operators, several results on the generity of (translation-dilation) universal functions are proved. Mainly to have a unified approach to holomorphic and harmonic functions, in the first part operators on spaces of P-holomorphic functions are considered. The second part is devoted to holomorphic functions having lacunary power series structure and to holomorphic functions which are univalent in certain prescribed sets.     

Outer preserving linear operators

A natural question about linear operators on the Hilbert-Hardy space is answered, motivated by work in geophysical imaging. Namely, which bounded linear operators on the Hardy space preserve the set of all shifted outer functions? A complete characterization is determined, which allows an explicit construction of all such operators. Every operator that preserves the set of shifted outer functions is necessarily a product-composition operator, consisting of composition with a shifted outer function followed by multiplication with a (possibly different) shifted outer function. Such operators represent important physical processes, including the propagation of seismic wave energy through the earth. Applications to seismic imaging are briefly discussed.     

Symmetric anti-eigenvalue and symmetric anti-eigenvector

The idea of symmetric anti-eigenvalue and symmetric anti-eigenvector of a bounded linear operator T on a Hilbert space H is introduced. The structure of symmetric anti-eigenvectors of a self-adjoint and certain classes of normal operators is found in terms of eigenvectors. The Kantorovich inequality for self-adjoint operators and bounds for symmetric anti-eigenvalues for certain classes of normal operators are also discussed.     

Products of integral-type operators and composition operators between Bloch-type spaces

A complete picture on the boundedness and compactness of the products of integral-type operators and composition operators between Bloch-type spaces of holomorphic functions on the unit disk is given in this paper.