Natural representations of the multiplicity of an analytic operator-valued function at an isolated point of the spectrum
Let \(A :{\mathcal D}\rightarrow L(E)\) be an analytic operator function where \({\mathcal D}\subset \mathbb{C}\) is an open set and \(L(E)\) denotes the Banach algebra of bounded linear operators on a complex Banach space \(E\). The spectrum of \(A\) is the set \(\Sigma\) of points \(z\in{\mathcal D}\) at which \(A(z)\) is not invertible (i.\,e., there is no inverse operator from \(L(E)\)). Let \(\Omega\subset\mathcal D\) be a bounded open set whose closure lies in \(\mathcal D\) and whose boundary is disjoint with \(\Sigma\). In this case, the pair \((A,\Omega)\) is called admissible. The multiplicity \(m(A,\Omega)\) of the admissible pair is then the isomorphism class of the Banach space \({\mathcal O}(\Omega,E)/A{\mathcal O}(\Omega,E)\) where \({\mathcal O}(\Omega,E)\) denotes the space of all analytic \(E\)-valued mappings with domain \(\Omega\). If the quotient is finite-dimensional and of dimension \(d\), then the multiplicity ``is'' the finite number \(d\). This general definition of multiplicity was given by \textit{J.~Arason} and \textit{R.~Magnus} [Math.\ Scand.\ 82, No.~2, 265--286 (1998; Zbl 0934.47008)]. In the paper under review, the multiplicity of an analytic operator-valued function at the isolated point \(z_0\) of the spectrum, i.\,e., \(m(A,\Omega)\) where \(\Sigma = \{z_0\}\) and \(\Omega\) contains only the point \(z_0\) of the spectrum of \(A\), is characterized in terms of kernels and ranges of infinite block Hankel or block Toeplitz matrices whose entries are derived from the Taylor coefficients of \(A\) and the Laurent coefficients of \(A^{-1}\) about \(z_0\). In two special cases, the results are expressed in terms of finite block matrices: when \(A\) is a polynomial and when \(A^{-1}\) has a pole at \(z_0\). The latter case leads to the theory of Jordan chains [see \textit{A.~S.\ Markus} and \textit{E.~I.\ Sigal}, Мат.\ Исслед.\ 5, No.~3(17), 129--147 (1970; Zbl 0234.47013); \textit{I.~Gohberg} and \textit{L.~Rodman}, Acta Sci.\ Math.\ 45, 189--199 (1983; Zbl 0545.47011); \textit{J.~A.\ Ball, I.~Gohberg} and \textit{L.~Rodman}, ``Interpolation of rational matrix functions'' (Operator Theory: Advances and Applications 45, Birkhäuser Verlag, Basel etc.) (1990; Zbl 0708.15011)].
scientific article; zbMATH DE number 5003825