On the Nevanlinna's Theory for Vector-Valued Mappings

The purpose of this paper is to establish the first and second fundamental theorems for anE-valued meromorphic mapping from a generic domainD⊂ℂto an infinite dimensional complex Banach spaceEwith a Schauder basis. It is a continuation of the work of C. Hu and Q. Hu. Forf(z)defined in the disk, we will prove Chuang's inequality, which is to compare the relationship betweenT(r,f)andT(r,f′). Consequently, we obtain that the order and the lower order off(z)and its derivativef′(z)are the same.

The spectrum and eigenspaces of a meromorphic operator-valued function

SynopsisIt is shown how to associate eigenvectors with a meromorphic mapping defined on a Riemann surface with values in the algebra of bounded operators on a Banach space. This generalises the case of classical spectral theory of a single operator. The consequences of the definition of the eigenvectors are examined in detail. A theorem is obtained which asserts the completeness of the eigenvectors whenever the Riemann surface is compact. Two technical tools are discussed in detail: Cauchy-kernels and Runge's Approximation Theorem for vector-valued functions.

On the deficiencies of meromorphic mappings of Cn into PNC

Let f(z) be a non-degenerate meromorphic mapping of the n-dimensional complex Euclidean space Cn into the N-dimensional complex projective space PNC. A generalization of results of Edrei-Fuchs [2] for meromorphic mappings of C into PNC was given by Toda [5], and an estimate of K(λ) for meromorphic mappings of Cn into PNC was done by Noguchi [4]. In this note we generalize several results of Edrei-Fuchs [2] in the case of meromorphic mappings of Cn into PNC.

A relation between order and defects of meromorphic mappings of Cn into Pn(C)

Let f be a meromorphic mapping of the n-dimensional complex plane Cn into the N-dimensional complex projective space PN(C). We denote by T(r,f) the characteristic function of f and by N(r,f*H) the counting function for a hyperplane H ⊂ PN(C). The purpose of this paper is to establish the following results.

On Meromorphic Mappings into Taut Complex Analytic Spaces

In this paper, we study a certain difference between meromorphic mappings and holomorphic mappings into taut complex analytic spaces. We prove in §2 that, for any complex analytic space X, there exists a unique proper modification of X with center Sg (X) which is minimal with respect to the property that M(X) is normal and, for any T-meromorphic mapping f: X → Y (see Definition 1.3) into a complex analytic space Y, there exists a unique holomorphic mapping such that except some nowhere dense complex analytic set, where Sg(X) denotes the set of all singular points of X.