In Memory of Jacob Schwartz

In Memory of Jacob Schwartz Sal Anastasio, Coordinating Editor —The outstanding contributions Jack had made to many fields and his ability to enter a field new to him, master it, make significant contributions, and move on. —Jack’s exceptional kindness and generosity. Eugenio Omodeo spoke of his “deep empathy with others”; Edmond Schonberg said that “his generosity had changed the lives of many;” and his sister, Judith, said that, to Jack, “acts of exceptional kindness were perfectly routine” and that his “stunning generosity…did not look for thanks” and “he was puzzled and even annoyed when they came.” —Finally, and in a lighter vein, Jack’s addiction to particular Chinese restaurants whose quality ranged from, at best, “mediocre,” to, at worst, “loathsome.” Jacob Theodore Schwartz was born on January 9, 1930, in the Bronx, New York City, of immigrant parents: his father, Ignatz, from Hungary, and his Photo courtesy of NYU Publicity for Courant. mother, Hedwig, from Germany. At every level of Jack Schwartz. his life, Jack was precocious. As a child he was an omnivorous reader. He attended Stuyvesant, a high school for gifted students. At age nineteen A few months after Jack Schwartz’s death on he graduated from the City College of New York (a producer of many Nobel Prize winners and March 2, 2009, a memorial gathering was held frequently called, at that time, “the poor man’s for him at the Courant Institute of New York Harvard”). While there, he studied with E. L. Post. University, his home base for almost fifty years. His friend and colleague Martin Davis recalls The well-planned program included a few piano that Jack interpreted one of Post’s remarks as pieces and almost a dozen speakers, each of whom expressing a need for a proof that all recursive had collaborated with Jack in one or more of his functions are Turing computable, and Jack sent many activities. such a proof to Post. The letter we have reproduced Those present would have heard the speakers was Post’s response to Jack. It shows that Post talk about: recognized Jack’s exceptional ability and that he was correct in his prediction of Jack’s future. Sal Anastasio is professor emeritus of mathematics, SUNY- Jack began graduate study at Yale and completed New Paltz. His email address is anastass@hawkmail. his PhD at age twenty-two. He then began to collab- newpaltz.edu. orate with his thesis advisor, Nelson Dunford, on Personal and family photos of J. Schwartz provided by and what was to become the monumental three-volume used with permission of the Schwartz family. Linear Operators, known simply to mathemati- DOI: http://dx.doi.org/10.1090/noti1246 cians as “Dunford-Schwartz.” Jack left Yale for the May 2015 Notices of the AMS 473 Courant Institute in 1957 and continued there until his retirement in 2005. In the mid-1960s he became interested in computer science, eventually being instrumental in founding the computer science department at Courant in 1969 and chairing the department. In addition to developing the programming language SETL, he made major contributions to computer architecture, parallel computation, and compiler optimization. In the early 1980s Jack became interested in robotics and founded the robotics laboratory at New York University. His work there consisted not only of theoretical aspects but of such prac- tical issues as robot programming and machine manufacturing. Toward the end of his life he became interested in the field of multimedia. His coworker in the field was Ken Perlin. Ken said that “Jack brought the same high level of engagement, original thinking, and intellectual rigor to this topic that he had brought to all his previous work. Jack saw, before nearly anyone else, the shift in the US Jack at a favorite Chinese food restaurant, Los from a manufacturing economy to an information Angeles, California, 1989. Photo by Diana economy. He realized that this would bring about Schwartz. a profound cultural change in which computers and interactivity would become an integral part of both the creation and the consumption of media. theory” and “concrete applications” is present in As usual, Jack was right.” many of Jack’s contributions to operator theory, This sketch of his achievements is fleshed as well as in the monumental encyclopedic three out in the following articles and recollections. volume treatise Linear Operators, by N. Dunford For a more complete biography, we refer the and J. T. Schwartz, a project seemingly initiated by reader to the memoir written by Martin Davis and Dunford, in which several of his doctoral students Edmond Schonberg for the National Academy of Sci- (including Jack) were involved. Jack, with his great ences: www.nasonline.org/member-directory/ mathematical power and working stamina, became deceased-members/50702.html. the main driving force behind the project and, in For more thorough essays on his work from the particular, directed the choice and presentation time of his entry into the field of computer science, of the rich mathematical content of the treatise. we refer the reader to From Linear Operators to Moreover, the books present a detailed history of Computational Biology, M. Davis and E. Schonberg, the development of operator theory up to that eds., Springer, 2013. time, in which, as the authors indicate, they were assisted by W. G. Bade and R. G. Battle. Ronald G. Douglas and A good illustration of Jack’s approach to linear Ciprian Foias operator theory—namely, it should play the role of a catalyst for understanding the structure and the properties of the linear maps arising in other Contributions to Operator Theory branches of mathematical study—is the extended Jacob (Jack) T. Schwartz was one of the very presentation in Volume II of Linear Operators of few contemporary mathematicians who had solid the boundary value problems for finite systems knowledge of many fields of modern and classical of ordinary differential equations using operator mathematics, to which he made outstanding con- theory (that is, the use of semigroups of linear tributions from “abstract mathematics” all the way operators as the catalyst). This is also illustrated by to “concrete applications” in “applied mathemat- the extension in Volume III of Dunford’s spectral ics.” This organic connection between “abstract theory from bounded operators to unbounded ones, as well as by Jack’s studies of the (“full” Ronald G. Douglas is Distinguished Professor of Math- ematics, Texas A&M University. His email address is or “partial”) preservation of the “good” spectral [email protected]. features of an operator under certain perturbations Ciprian Foias is Distinguished Professor of Mathe- which occur in applications. Many of Jack’s results matics, Texas A&M University. His email address is in this direction were first published in Volume [email protected]. III of Linear Operators and constitute the most 474 Notices of the AMS Volume 62, Number 5 original part of the whole treatise. It is this is called the center of A. Let M be a µ-finite part which, due to the progress made since then measure space, L2(M) the Hilbert space of square- in “functional analysis”, provides inspiring new integrable functions g on M. Then L1(M), the set paradigms and opportunities for “pure and applied of all essentially bounded µ-measurable functions operator theory.” f corresponding to mf : g(x) ! f (x)g(x), is a ∗ On a more per- commutative W -algebra on L2(M), and the center sonal side, Jack of L1(M) is equal to itself. On the other end of the was a very gener- spectrum, a W ∗-algebra A is called a factor if c(A) ous mathematician. is just C, the field of complex numbers, i.e., scalar As an example, multiples of the identity operator I. An operator in the second author A is a projection if E = E∗ = EE. Two projections of this eulogical in A are equivalent, E ∼ F, if there is a partial article is very in- isometry u in A such that uu∗ = E, u∗u = F. This debted to him. defines a partial order among projections of A, and While Jack was this partial order is a linear order if A is a factor. A still working on projection E is finite if there is no projection F ∼ E Volume II of Lin- with 0 < F < E; E is minimal if simply no such F ear Operators, he exists. Murray and von Neumann proved [2] that encouraged I. Colo- there are three and only three different types of joara and the factors: second author to Type I: has minimal projections, In; n = 1; 2;::: , write a monograph the n × n matrix algebra, and I1, B(H) for infinite on their slightly dimensional H; Jack, age five, with sister, more abstract spec- Type II: II1 if the identity operator I is finite, II1 Judith Dunford, 1935. tral decomposition if the identity operator I is infinite; approach. After they sent Jack their manuscript, Type III: all projections are infinite. the “mail connection” concerning that work was In the final paper [5] of the rings of operators se- interrupted by the local authorities (who else?), ries, von Neumann proved that every W ∗-algebra A so, in particular, they never got the galley proofs. can be uniquely expressed, up to isomorphism, as a Nevertheless, Jack rewrote parts of their (rather direct integral of factors over a measure space aris- clumsy) introduction and published the work in his ing from a measure space M associated with c(A). Gordon & Breach Mathematics and Applications This can be regarded as an infinite-dimensional series. extension of the classical Wedderburn Structure Theorem for semisimple finite-dimensional rings.

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