Factorization of a class of meromorphic matrix valued functions
Given an \(m\times m\)-matrix function A(z) which is meromorphic on the whole of \({\mathbb{C}}\), the order p of A is defined by \(p=\limsup (\log T(r,A)(\log r)\), where \[ T(r,A)=\frac{1}{2\pi}\int^{2\pi}_{0}\ln^+\| A(re^{i\theta})\| d\theta +\int^{r}_{n}\frac{n(A,t)}{t}dt \] when A has no poles on \(\{z| | z| =r\}\) and \(n(A,t)=\#\) poles of A in the set \(\{\) \(z| | z| \leq t\}\). It is observed that \(p=\max \{p_{ij}\}\); where \(p_{ij}\) is the order of the entry \(a_{ij}(z)\) of A(z). For entire matrix functions A(z) this notion of the order coincides with the usual definition \(p=\inf \{\phi | \| A(z)\| \leq c \exp (b| z|^{\phi})\) for some \(b,c>0\}\). If A is regular (i.e., if \(A(z)^{-1}\) exists) then \(A(z)^{-1}\) has order at most p if A has order p. In analogy to the scalar case it is shown that each regular meromorphic matrix function A(z) on \({\mathbb{C}}\) admits a decomposition \(A(z)=T(z)C(z)\), where T(z) is an entire function such that \(T(z)^{-1}\) exists for all \(z\in {\mathbb{C}}\) and the order of T does not exceed p, and C(z) carries the complete pole- and zero information (i.e., the chain structure) of A(z); in fact \(C(z)=C_ 0(z)C_ p(z)\), where, given the zeros \(| z_ 1| \leq | z_ 2| \leq...\leq | z_ k| \leq..\). of A(z) (that is, the poles of \(A(z)^{-1})\) and the poles \(| w_ 1| \leq | w_ 2| \leq...\leq | w_{\ell}| \leq..\). of A(z), the order of \(C_ p\) is inf\(\{\) \(a| \sum | z_ j|^{-\alpha}<\infty \}\) and the order of \(C_ p\) is inf\(\{\) \(a| \sum | w_ j|^{-\alpha}<\infty \}\) and \(C_ 0,C_ p\) have the structure \[ C_ 0(z)=\prod_{i}(\prod_{j}(I-\frac{z}{z_ j})Q_{ij})\exp (\sum^{p}_{k=1}\frac{z^ k}{kz_ j^ k})Q_{ij}), \] \[ C_ p(z)=\prod_{i}(\prod_{j}I-\frac{z}{w_ j}P_{ij})\exp (\sum^{q}_{k=1}\frac{z^ k}{kw^ k_ j}P_{ij}), \] for appropriate projections \(P_{ij}\), \(Q_{ij}\) associated with the pole, resp. zero chain structure at \(w_ j\), resp. \(z_ j\). The proof of this result is quite direct; it is related to the proofs of similar results, obtained by \textit{D. C. Lay} and the reviewer [Indagationes Math. 44, 303- 309 (1982; Zbl 0533.47016)] for finite-meromorphic operator valued functions on arbitrary regions.
factorisation of meromorphic matrix functions
scientific article; zbMATH DE number 4069631
scientific article