Independence from the subspace parameter, related to finite meromorphic operator-functions (Q2266425)

Let X, Y, Z be complex Banach spaces, \(M(Z,\lambda_ 0)\) and \(H(Z,\lambda_ 0)\) be sets of functions \(\phi: {\mathbb{C}}\to Z\) which are meromorphic or holomorphic, resp., at \(\lambda_ 0\). Denote by \(\phi_ n\) the n-th Laurent coefficient of a function \(\phi\), put \(\nu (\phi,\lambda_ 0)=\inf \{n: \phi_ n\neq 0\}\). For an operator-function \(A: {\mathbb{C}}\to L(X,Y)\) that is finite-meromorphic at \(\lambda_ 0\) (i.e. \(A\in M(L(X,Y),\lambda_ 0)\) and \(A_ n\) are finite-dimensional if \(n<0)\), introduce the following sets: \({\mathfrak N}_ m(A,\lambda_ 0)=\{\phi_{m-1}: \phi \in H(X,\lambda_ 0)\), \(\nu (A\phi,\lambda_ 0)\geq m\},\) \({\mathfrak M}_ m(A,\lambda_ 0)=\{y\in Y: \nu (A\phi -y,\lambda_ 0)\geq m\) for a \(\phi \in H(X,\lambda_ 0)\},\) \({\mathfrak N}_ c(A,\lambda_ 0)=\{\phi_ n: \phi \in H(X,\lambda_ 0)\), \(A\phi(\lambda)\equiv 0\) in a neighbourhood of \(\lambda_ 0\},\) \({\mathfrak M}_ c(A,\lambda_ 0)=\{y\in Y:\) there is \(\phi \in H(X,\lambda_ 0)\) so that \(A\phi(\lambda)\equiv y\) in a neighbourhood of \(\lambda_ 0\},\) \({\mathfrak M}_ r(A,\lambda_ 0)=\{y\in Y:\) there is \(\phi \in M(X,\lambda_ 0)\) so that \(A\phi(\lambda)\equiv y\) in a neighbourhood of \(\lambda_ 0\},\) \({\mathfrak N}(A,\lambda_ 0)=\cup^{\infty}_{m=1}{\mathfrak N}_ m(A,\lambda_ 0)\), \({\mathfrak M}(A,\lambda_ 0)=\cap^{\infty}_{m=1}{\mathfrak M}_ m(A,\lambda_ 0).\) Theorem 1. Let A be finite-meromorphic in a domain \(G\subset {\mathbb{C}}\), let the space Im \(A_ 0\) be closed and the stability number k(A,\(\lambda)\) be finite for every \(\lambda\in G\). Then both \(\overline{{\mathfrak N}_ c(A,\lambda)}\) and \({\mathfrak M}r(A,\lambda)\) are independent of \(\lambda\in G\) and moreover, \({\mathfrak N}(A,\lambda)={\mathfrak N}_ c(A,\lambda)\) and \({\mathfrak M}(A,\lambda)={\mathfrak M}_ c(A,\lambda)={\mathfrak M}_ r(A,\lambda)\) if \(\lambda\in G\setminus \Gamma\) where \(\Gamma =\{\lambda \in G: \lambda\) is a pole of A or \(k(A,\lambda)\neq 0\};\) if \(\lambda\in \Gamma\) then both \(\dim {\mathfrak N}(A,\lambda)/{\mathfrak N}_ c(A,\lambda)\) and \(\dim {\mathfrak M}_ r(A,\lambda)/{\mathfrak M}_ c(A,\lambda)\) are finite. In the case of holomorphic (or linear) functions, the precise estimates of dimensions are given. (Without proofs)