G. de Barra

The boundary of the numerical range

In [1] it was shown that for a compact normal operator on a Hilbert space the numerical range was the convex hull of the point spectrum. Here it is shown that the same holds for a semi-normal operator whose point spectrum satisfies a density condition (Theorem 1). In Theorem 2 a similar condition is shown to imply that the numerical range of a semi-normal operator is closed. Some examples are given to indicate that the condition in Theorem 1 cannot be relaxed too much.

Preface to Second Edition

Half a decade after the first publication of From Dust to Stars there has been rapid and enormous progress in the field of stellar formation and evolution warranting a second edition. After a first review it became clear that the vast majority of the materials covered in the first edition remain very much valid; however, the need for many updates and extensions became obvious. In that respect the basic structure and content of the book has remained intact and existing chapters have received updates at various levels necessary. Three chapters have been added covering multiplicity in star formation (Chapter 7), massive star formation (Chapter 9), and a summary on the detection of exoplanets and basic planet formation (Chapter 13). Two chapters have received major additions, others just updates and minor revisions. The overall changes in the framework of the publication also suggested a switch in title to The Formation and Early Evolution of Stars with the subtitle From Dust to Stars and Planets to also acknowledge the extension towards planet formation. Chapter 3 on interstellar matter now treats some basic physics about heating and cooling processes, cosmic rays, and radiative properties of dust. The issue of the mass functions also has a first discussion in that chapter, but it will re-surface at various occasions throughout the book as it now has become a central topic in the field. One may also wonder why there is a distinction of the initial mass function in general and a cluster mass function, which many studies claim to be the same, but after a review of all the material, the jury is still out on that issue. The chapter also received a much-needed extension to a discussion of star formation rates in the Milky Way, in the Local Group, and on cosmological scales. Chapters 4, 5 and 6 have received only minor updates which might come to many as a surprise, given the enormous amount of material that has been provided through the five years of cold Spitzer operations, through the continuation of HST and Chandra, and now with the availability of Herschel. However, for these three chapters, the fundamentals have not really changed and here some updates and selected references to these new studies seemed to suffice. Much of this material has made its way into the book in later chapters. The first new chapter (Chapter 7) has fixed a major shortcoming of the first edition which did not include much on multiplicity in star formation.