STRUCTURE OF DEFECTIVE VECTORS FOR ASSOCIATED ENTIRE CURVES WITH LINEARLY DEPENDENT COMPONENTS

Abstract

We study the structure of the set of exceptional vectors in the sense of Nevanlinna for associated entire curves of finite order with linearly dependent components. It is shown that even if an entire curve has linearly independent components, its associated entire curve may nevertheless have linearly dependent components. It is known that the set of all Nevanlinna exceptional vectors of a finite-order entire curve with linearly dependent components, together with a given subspace }$A_0=\left\{\overrightarrow{a}\in {\boldsymbol{C}}^p:\overrightarrow{G}\left(z\right)\cdot \overrightarrow{a}?0\right\}$\textit{, can be expressed as at most a countable union of subspaces  }$A_j$\textit{, satisfying }$\omega\le \mathrm{dim} A_j\le p-1$\textit{  and  }$A_j\supset A_0$\textit{. Furthermore, it is established that the corresponding associated entire curve }${\overrightarrow{G}}_l:\boldsymbol{C}\ \to {\boldsymbol{C}}^q$, where $q=C^l_p$, also has $\omega^*\ge C^l_p{-C}^l_{p-\omega}$\textit{ linearly dependent components. For this associated entire curve, the set of its Nevanlinna exceptional vectors, together with the subspace }${\mathrm{A}}^*_0=\left\{\overrightarrow{b}\in {\boldsymbol{C}}^q:\overrightarrow{G}\left(z\right)\cdot \overrightarrow{b}\equiv 0\right\}$\textit{, constitutes at most a countable union of subspaces }${\mathrm{A}}^*_j$\textit{  of dimension not exceeding }$\mathrm{q}\mathrm{-1}$\textit{, where }${{\mathrm{A}}^*_j\supset \mathrm{A}}^*_0$\textit{. In addition, for each }${\mathrm{A}}^*_j$\textit{, there exists a so-called special vector }${\overrightarrow{b}}_j\in {\boldsymbol{C}}^q$\textit{, orthogonal to }${\mathrm{A}}^*_j$.