부산대학교 도서관

O. Nevanlinna has written extensively about the idea of treating a polynomial $p$ with simple zeroes as a new variable $w=p(z)$; in the resulting ``multicentric calculus'', complex scalar-valued functions $\varphi$ are represented by ${\Bbb C}^d$-valued functions $f$, where $d$ is the degree of the polynomial $p$. The key idea in applications to an operator $A$ is that one can arrange that $p(A)$ is either small, or structurally simple. It is Hilbert's ``lemniscate theorem'' that polynomially convex compact sets can be approximated from outside by ``polynomial lemniscates'' $|p(z)|=\rho$, and hence reduced to a disc. \par In the paper under review, the authors remove the polynomially convex restriction by extending the idea from polynomials $p$ to rational functions $r=p/q$. There is then a Banach algebra for which the original function $\varphi$ appears as the Gelfand transform of its representative $f$. An application of this enhanced calculus is that the solution of the Sylvester equation $AX-XB=C$ can be given by a power series whenever the spectra $\sigma(A)$ and $\sigma(B)$, rather than their polynomially convex hulls, are disjoint. \par Note the related paper [S.-G. Lee and Vũ~Quốc~Phóng, Studia Math. {\bf 222} (2014), no.~1, 87--96; MR3215241].