Book reviews: “Friedrich Haslinger; Complex analysis. A functional analytic approach”, De Gruyter Graduate, De Gruyter, Berlin 2018, ix + 338 p., ISBN: 978-3-11-041723-4/pbk; 978-3-11-041724-1/ebook.

Abstract

The book is an introduction to complex analysis in one variable and some topics in several complex variables, oriented to applications to Cauchy-Riemann equations studied via the method of Hilbert space operators. The key tool in this study is the ∂¯-Neumann operator viewed as an operator acting on various Hilbert spaces of analytic functions. This part is largely based on author’s papers, being also treated with more details and supplementary material in the related book by F. Haslinger, The ∂¯-Neumann problem and Schr¨odinger operators, De Gruyter Expositions in Mathematics 59, De Gruyter, Berlin, 2014. It is worth to mention that the needed results on Hilbert spaces (e.g. Riesz’ representation theorem) and spectral theory of linear operators (bounded or unbounded) defined on such spaces are included with full proofs in the book, making the book fairly selfcontained.