Pei Yuan Wu (吳培元)
(I) The Topic
We begin with our initial research on the topic of numerical range. It all started in 1996 when I was supervising the Ph.D. work of Hwa-Long, at the time a budding graduate student at the applied mathematics department of National Chiao Tung University (國立交通大學). After a few tries, we finally settled on the problem of generalizing a result of James P. Williams (1938 - 1983) from 1971. Williams was one of my mentors back in the early 1970s at Indiana University when I was a graduate student there. His result gives the first construction of the inscribing ellipses of a triangle via the use of numerical range. Thus it contains the genesis of the treatment of the 200-year old Poncelet porism, discovered by the French engineer and mathematician Jean-Victor Poncelet (1788 - 1867) in 1813 (when he was a prisoner of war in the Russian village of Saratov), using the 100-year old notion of numerical range due to Otto Toeplitz (1881 - 1940) and Felix Hausdorff (1868 - 1942).
The Poncelet porism says that if A and B are ellipses in the plane with A contained inside B, and if there is one polygon P with n vertices (that is, P is an n-gon) having all n sides tangent to A (that is, P circumscribes about A) and all n vertices on B (that is, P inscribes in B), then for any point z on B there is always a unique such circumscribing-inscribing n-gon with z as a vertex. This is a porism meaning that some property (the existence of a circumscribing-inscribing n-gon) either fails or, if it holds for one instance, succeeds infinitely many times; in other words, it yields a 0-∞ dichotomy. In fact, more is true, namely, we can even assume that A and B are any two conic sections on the plane, not necessarily ellipses nor satisfying any containment restriction. On the other hand, the notion of the numerical range was established by Toeplitz and Hausdorff in 1918 and 1919 when they proved the convexity of the numerical range of any finite matrix.
We soon found out that via this approach Williams’s result can be generalized from the n = 3 case to more general values of n with the (outer) ellipses normalized as the unit circle in the plane and the (inner) ellipses replaced by the boundaries of numerical ranges of the so-called S_{n-1}-matrices. This latter class of matrices is the finite-dimensional version of a more general class of operators, the compressions of the shift, studied by Donald Sarason (1933 - 2017) back in 1967. So the convergence of these three topics from projective geometry, matrix theory, and analytic operator theory, respectively, makes the subject more exciting and interesting. We have to mention that Boris Mirman also obtained, independently, many deep results along this line, his first published paper appearing even ahead of ours by several months in 1998. These together with other related later developments are all featured in Chapter 7 of our book.
(II) The Book
The first draft of the book, comprising Chapters 1 to 6, was prepared starting from 1999. In the beginning, the intention was just to write a standard graduate-level textbook on numerical ranges for the purpose of preparing my own students for their researches with easy references. Hence no plan was made at the time to actively seek a publisher to have it formally published. By that time, there were already in the market some widely known book chapters or even full-length books devoted to this subject, as for instance Chapter 22 of the second edition of P. R. Halmos’s “A Hilbert Space Problem Book” (1982), Chapter 1 of R. A. Horn and C. R. Johnson’s “Topics in Matrix Analysis” (1991), the two-volume monograph “Numerical Ranges of Operators on Normed Spaces and of Elements of Normed Algebras” (1971) and “Numerical Ranges II” (1973) by F. F. Bonsall and J. Duncan, and the later “Numerical Range, the Field of Values of Linear Operators and Matrices” (1997) by K. E. Gustafson and D. K. M. Rao. I thought a more updated and comprehensive book may help to grab the attention of a wider class of mathematicians to this interesting research area and thus to have the gospel spread.
The draft of the book has been lying there idly for all these years. In the beginning, I did keep abreast of the developments of the subject so as to update the book text from time to time. As time went by, this became more and more difficult for obvious reasons. It remained at this stage until the summer of 2018. While traveling in northeast China as a tourist, I received out of the blue an e-mail from Linglei Meng (孟令磊), a Senior Commissioning Editor of Cambridge University Press (CUP) based in Beijing, China, asking for a meeting with me for the purpose of preparing a book to be published by CUP. As he would be in Taiwan the following month, we agreed to have such a get-together in early July at a Starbucks store close to the campus of National Taiwan University in Taipei. Turned out Linglei himself also majored in mathematics and was responsible of soliciting book manuscripts from the eastern Asian area for publication by CUP. The meeting went so well that I agreed to send in a book to CUP through him. One thing then led to the other. In the ensuing months, I first asked my longtime collaborator Hwa-Long to join me as a co-author of the book. Then I submitted a book proposal, together with a tentative Table of Contents and a sample chapter, to Linglei for a peer review by four expert reviewers. After their (essentially) positive reviews and our response to their comments, the Syndicate of CUP finally agreed to offer a contract for publishing the book. The signing of the book contract was not done until April, 2019. The contract dictates that we deliver the typescript by the end of August, 2020 at a maximum of 480 printed pages.
From this moment on, we started seriously the updating and expanding of the contents of my old draft and also the writing of the additional Chapters 0, 7 and 8. For the past 20 years, Hwa-Long and I have been working together resulting in more than 40 joint papers and thus a division of duties on this project resulted naturally: I was responsible to prepare the manuscript in handwriting and he to type it into a Latex file. Over the years, he has developed into a creative researcher and a competent typist at the professional level. For the present one, he organized the known results for Chapters 7 and 8 from the literature and also played the role of a rescuer by providing the needed technical improvements when I occasionally got stuck at some point. So everything went on as planned. As time got closer to the deadline, we did feel some pressure. This is because the topics of various generalized numerical ranges in Chapter 8 were not in our familiar ground, and a judicious summarization of the major results in this area for the book has to be done carefully. In the end, we worked this out successfully and had a comprehensive research-level book at hand. The manuscript carries viii + 472 printed pages, including Table of Contents, Preface, Chapters 0 to 8 (plus a total of 475 Problems scattered after different chapters), Appendix (on convex set), Bibliography (of 600 items), List of Symbols, and Index. It turns out to have the exact 480 pages dictated by the contract, no more and no less.
As of July, 2020, the world was still swept up by the Covid-19 pandemic. In time, an e-mail from Cambridge arrived, which inquired whether we have been hampered in any way by it. On July 20th, we sent in the pdf and zip files of the book, around 40 days before the deadline. Afterwards, there were still a Marketing Questionnaire and an Author Information Request Form from CUP to be filled up. In the former we provided information on the authors, the book itself and the various marketing activities, and in the latter on our bank details and the interest conflicts declaration. These will be used, respectively, to make the blurbs for the book cover and other promotional gimmicks, and the payment of the royalties of the book. In November, 2020, another mail reached us. This time, again for the promotion purpose, we had to fill the Metadata template providing for each chapter of the book (a total of 8 plus Introduction, which replaces our original Chapter 0, and Appendix) its keywords and abstract. After all these, we hope we can finally wrap up the work which gripped us exclusively for the past one and half years with a sigh of relief.
(III) Epilogue
This article was first posted in my FB blog on August 21, 2020, tentatively closed to the public as it's not yet the final edition and needs some modifications from time to time. The plan was to have it released openly when the book is officially published. However, starting from November 1, 2020, FB has a policy change: it abolishes its Blog section so that I won't be able to access it, not to mention to have it locally revised. Fortunately, an extra copy was found somewhere and I can now proceed with an early release.
One final lingering question: Why does the publisher need one year to have the book ready for publication? It's not like in the old days when the whole book had to be typeset from a manually-typed manuscript in the paper form. In the present case, we submitted the whole typescript of the book in a Latex file. Seems that the job can be done in a much shorter time.
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