Tosio Kato (1917–1999)

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Heinz Cordes, Arne Jensen, S. T. Kuroda, Gustavo Ponce, Barry Simon, and Michael Taylor

Tosio Kato was born August 25, 1917, in Kanuma City, Tochigi-ken, Japan. His early training was in physics. He obtained a B.S. in 1941 and the degree of Doctor of Science in 1951, both at the University of Tokyo. Between these events he published papers on a variety of subjects, including pair creation by gamma rays, motion of an object in a fluid, and results on spectral theory of operators arising in quantum mechanics. His dissertation was entitled “On the convergence of the perturbation method”. Kato was appointed assistant professor of physics at the University of Tokyo in 1951 and was promoted to professor of physics in 1958. During this time he visited the University of California at Berkeley in 1954–55, New York University in 1955, the National Bureau of Standards in 1955–56, and Berkeley and the California Institute of Technology in 1957–58. He was appointed professor of mathematics at Berkeley in 1962 and taught there until his retirement in 1988. He supervised twenty-one Ph.D. students at Berkeley and three at the University of Tokyo. Kato published over 160 papers and 6 monographs, including his famous book Perturbation Theory for Linear Operators [K66b]. Recognition for his important work included the Norbert Wiener Prize in Applied Mathematics, awarded in 1980 by the AMS and the Society for Industrial and Applied Mathematics. He was particularly well known for his work on Schrödinger equations of nonrelativistic quantum mechanics and his work on the Navier-Stokes and Euler equations of classical fluid mechanics. His activity in the latter area remained at a high level well past retirement and continued until his death on October 2, 1999. —Michael Taylor

Atomic Hamiltonians Barry Simon Kato’s most celebrated result is undoubtedly his Tosio Kato founded the modern theory of proof, published in 1951 [K51a], of the essential Schrödinger operators and dominated the field self-adjointness of atomic Hamiltonians: for its first twenty-five years. A Schrödinger oper- Xn Xn X ator is one of the form 1 1 1 (2) (2i) i Z|xi| + |xi xj | . (1) + V i=1 i=1 i

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Rellich knew the Kato-Rellich theorem by the The second involved mid-1930s (Kato, unaware of Rellich’s work, redis- some work I did. The covered it as part of the proof of his theorem), and general wisdom by Sobolev inequalities were also discovered by then 1970 was that for gen- (although Kato did have to understand using them eral Schrödinger opera- on only a subset of the variables and integrating out tors of the form (1) on the remaining variables). I would have expected Rn, the right local con- Rellich or K. O. Friedrichs to have found the result dition on V to assure by the late 1930s. self-adjointness is that One factor could have been von Neumann’s it be in Lp with p n/2 attitude. V. Bargmann told me of a conversation he (at least if n is 5 or had with von Neumann in 1948 in which von more). It was known Neumann asserted that the multiparticle result that one could not im- p was an impossibly hard problem and even the case prove this as far as L of hydrogen was a difficult open problem (even properties alone were though the hydrogen case can be solved by sepa- concerned. As an off- ration of variables and the use of H. Weyl’s 1912 shoot of work I had theory, which von Neumann certainly knew!). done in quantum field Perhaps this is a case like the existence of the Haar theory, I realized in integral, in which von Neumann’s opinion stopped 1972 that this was only work by the establishment, leaving the important the right property for Tosio Kato, February 15, 1997. discovery to the isolated researcher unaware of the negative part of V, von Neumann’s opinion. but that as far as local 2 Another factor surely was the Second World behavior was concerned, L was the proper con- War. My generation and later generations are dition for the positive part. sufficiently removed from the dislocations of the I conjectured that for positive V’s, a sufficient war and its aftermath that we often forget its condition (it is clearly necessary) for essential self- adjointness was that V be locally L2; because of the effect. In [K80] Kato remarks dryly: “During World nature of my proof, I could get the result for locally War II, I was working in the countryside of Japan.” L2 positive potentials only under the unnatural In fact, from a conversation I had with Kato one additional condition that the L2 norm over a ball evening at a conference, it was clear that his of radius R does not grow any faster than exp(cR2). experiences while evacuated to the countryside Within weeks of my mailing out my preprint, a and in the chaos immediately after the war were letter arrived from Kato and shortly afterwards a horrific. He barely escaped death several times, brilliant paper [K72] that is my personal favorite and he caught tuberculosis. (Charles Dolph once among all his works. He not only settled the gen- told me that he regarded his most important con- eral case but did it by introducing a totally new tribution to American mathematics was that when idea—a distributional inequality now called Kato’s he learned Kato was having trouble getting a visa inequality—that for all functions u such that u because of an earlier bout with tuberculosis, Dolph and its distributional Laplacian are both locally in- contacted physicist Otto Laporte, then American tegrable, scientific attaché in Tokyo, to get Kato a waiver.) Formally trained as a physicist, Kato submitted (3) |u|(sgn u)u. his paper to Physical Review, which could not He had results (later improved by H. Leinfelder figure out how and who should referee it, and that and C. Simader) for magnetic fields; indeed, Kato’s journal eventually transferred it to the Transactions version of (3) with magnetic fields led others to of the American Mathematical Society. Along the what are now called universal diamagnetism and way the paper was lost several times, but it diamagnetic inequalities, as well as to abstract finally reached von Neumann, who recommended its results on comparison of semigroups. Kato also acceptance. The refereeing process took over three took advantage of this paper to redo the situation years. for negative potentials, work that led to a class of functions now known as the Kato class, which Later Self-Adjointness Results turns out to be the natural class from a path- Kato returned to the issue of self-adjointness sev- integral point of view. eral times after his initial work, most notably once in the early 1960s and once in the early 1970s. The first of these cases was a paper with his student Eigenvalue Perturbation Theory T. Ikebe [KI], which among other things estab- Kato also returned many times to the issue of eigen- lished the proper (x2) borderline for situations value perturbation theory. Independently, but later where the potential was allowed to go to minus in- than Rellich, he developed the theory of regular finity at spatial infinity. operator perturbations (what he later called type A).

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Ph.D. Students of Tosio Kato The first three students were at the As early as 1955 he ap- Finally, in the early 1970s, Kato together with Kuroda University of Tokyo, and the others preciated the significance [KK] made important contributions to the develop- were at the University of California, of quadratic forms [K55] ment of the limited absorption principle in scatter- Berkeley. and developed what he ing theory, a subject raised to high art by S. Agmon later called type B pertur- and L. Hörmander. Hiroshi Fujita bations. All these situa- Teruo Ikebe tions are regular ones for Other Work S. T. Kuroda which the eigenvalues are It is a tribute to Kato’s depth and breadth that some Erik Balslev (1963) analytic in the perturba- of his other, “less important”, papers would be con- Charles Samuel Fischer (1964) tion parameters. But there sidered major parts of the oeuvre of many math- Ponnaluri Suryanarayana (1964) are many standard quan- ematicians. Not only did Kato write the first sig- Ronald Cameron Riddell (1965) tum examples to which the nificant mathematical analysis of the quantum James S. Howland (1966) theory does not apply, and adiabatic theorem [K50], but the ideas presented Francis McGrath (1966) Kato developed methods in it remain to this day central to most further work Charles Sue-chin Lin (1967) for proving that eigenvalue on the subject, including most mathematical pre- Joel L. Mermin (1968) perturbation series are sentations of M. Berry’s phase. Howard Swann (1968) often asymptotic. He un- At the same time as his fundamental paper on Frank J. Massey III (1971) derstood the critical no- self-adjointness, he proved that the helium atom Charles Francis Amelin (1972) tion of stability used by Hamiltonian in the limit of infinite nuclear mass Preben Kjeld Alsholm (1972) many later authors. had an infinity of bound states [K51b]. It was later Hugh Bruce Stewart (1972) While Kato’s perturba- realized by others that as long as one is careful Gilles François Darmois (1974) tion-theoretic work ini- about using the right coordinate system, Kato’s Gary Childs (1975) tially and often focused on C. Y. Lai (1975) method extends to finite nuclear masses and eigenvalue perturbations, to arbitrary atoms (recovering a theorem of G. Pinchuk (1976) the notion of controlling G. Zhislin proven later than Kato’s work). Rafael José Iório (1977) things perturbatively was Kato understood the nature of the singularities N. X. Dung (1981) a major theme in much of of atomic wave functions (“Kato cusp conditions”) Baoswan Wong-Dzung (1981) his other work, including and wrote about these in 1957. In a 1950 paper he Masomi Nakata (1983) semigroup perturbations, found the right way of formulating G. Temple’s lower the adiabatic theorem, bounds on eigenvalues, and in 1959 he obtained scattering theory, and his some of the earliest results on the absence of later work on nonlinear equations. And, of course, embedded positive energy eigenvalues under any discussion of perturbation theory has to men- suitable assumptions on the decay of the potential. tion Kato’s masterpiece Perturbation Theory for Lin- ear Operators [K66b], used as a bible by a generation Tosio Kato leaves a rich mathematical legacy to of mathematical physicists and operator theorists. everyone working on mathematical problems associated with nonrelativistic quantum mechanics.

Scattering Theory From 1956 until 1980 a major theme in Kato’s work was scattering theory and the understanding of the Gustavo Ponce absolutely continuous spectra of Schrödinger Kato was one of the most influential figures in operators. Among his many papers in this area, I the study of nonlinear evolution equations. He shall focus on three topics. The first, in 1957, formulated a general abstract approach to the involves finite-rank and trace-class perturbations well-posedness of the initial value problem that [K57]; M. Rosenblum shares the credit for develop- isolated and illuminated the basic features of a broad ing the initial theory. Later critical developments class of problems. As Kato stated in his acceptance were made by S. T. Kuroda, M. Birman, and Kato of the Wiener Prize [K80], “I have been fascinated again when he wrote an important paper on the by the Hille-Yosida-Phillips theory of operators semi- invariance principle [K65]. One can also see the groups, and continuously have tried to apply it to interplay between time-dependent and time- solving various evolution equations, linear and non- independent methods, a theme that recurred. linear.” However, his work in this area went far be- The slickest marriage of time-dependent and yond this general local existence theory. Kato de- time-independent methods occurred in what has veloped the basic tools and the framework for the come to be called the theory of Kato smoothness qualitative study of many fundamental [K66a]. The link is essentially the Plancherel theorem! problems in mathematical physics: nonlinear sym- I regard this paper, published in 1965 when Kato metric hyperbolic systems, the equations of was forty-eight, as his most beautiful and in some ways his deepest. There have been applications to Gustavo Ponce is professor of mathematics at the Uni- positive commutator theory (the Putnam-Kato the- versity of California, Santa Barbara. His e-mail address orem), which was a precursor to Mourre theory. is [email protected].

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motions for incompressible and compressible anticipated by more than ten years a celebrated fluids, and nonlinear dispersive models with a inequality due to R. Strichartz. wide variety of boundary conditions. Later, in [KY], in collaboration with K. Yajima, I first met Kato in the fall of 1982 at Berkeley. Kato quantified this effect by showing that for As a postdoc I attended his lectures in nonlinear any fixed q ∈ Ln(Rn), functional analysis. After one of his classes he in- Z ∞ vited me to come to his office. On the board he had it 2 2 (5) kqe k 2 dt ckk 2 . written a mathematical statement, and he asked ∞ L L me whether or not I believed it was true. After a Moreover, Kato showed that the estimate in (5) few minutes of thought I admitted that I had no still holds with the operator idea. With a worried look on his face he showed me a paper where this result was used and (1 + |x|2)1/2(1 )1/4 credited to one of his papers. He said that it was not really there, and in fact he did not know how instead of multiplication by q. to prove it. Some days later he told me that the re- This smoothing effect or gain of derivatives is sult as stated was false, but happily he was able fundamentally related to the dispersive character to get around this point in the paper. Perhaps of the equation. In particular, it does not hold for without realizing it, the author had stumbled on hyperbolic equations. a novel method of mathematical proof by appeal- The results in [K83] together with questions raised ing to Kato’s honesty. there led to the great activity in the last decade on From this period I enjoyed a mathematical and the problem of the optimality of the initial function personal relationship with Kato. I was privileged space X to guarantee the local well-posedness for var- to have the opportunity to collaborate with him and ious nonlinear evolution equations. to know his pure and straightforward intellectual In [KF] Kato and his student H. Fujita consider integrity. Our communication continued until his the Navier-Stokes system, namely, (4) with A = , j death. f =(f1,...,fn), and fj (u)=P((u ∇)u ). Here x is in Of Kato’s many contributions in this area I will Rn, u =(u1,...,un), is > 0, and P denotes the only comment on a select few that seem especially projection onto divergence-free vector fields. They important to me. Following his general framework, introduced an argument (weighted-in-time norms) consider the problem that in the 3-dimensional case allowed them to ob- du tain the local well-posedness in the critical Sobolev (4) = Au + f (u),u(0) = ∈ X, space H˙ 1/2(R3), the dot indicating that the space dt is invariant under the map carrying u(x, t) to where A is a linear operator on a function space X u(x, t). In particular, the criticality of the space and f is a nonlinear function on X. Kato’s notion of guarantees the existence of global solutions for well-posedness in X includes: existence, unique- small data. The existence of global classical solu- ness, the continuity of the map from the data to the tions with large data has remained as an open solution, and persistence (i.e., the question whether problem since J. Leray’s work in 1934. In 1984 the solution describes a continuous curve in X). Kato extended his results to the space L3(R3). The In [K83] Kato studied the initial value problem further extensions and applications of the associated to the Korteweg de Vries equation, for techniques introduced by Kato in these works have 3 which A = ∂x and f (u)=u∂xu in (4). Kato estab- generated a long list of interesting results. lished a smoothing effect, which almost contradicts One of my favorite of Kato’s papers is [K86]. In the the time reversibility of the equation. Roughly, he 2-dimensional case the global well-posedness for showed that if u(0) = is in L2(R), then for any the Navier-Stokes and Euler equations are due to 2 r>0,∂xu(t) is in L (r,r) for almost every time Leray and W. Wolibner respectively. In [K86] Kato t ∈ R . From the time reversibility one has that if gave a unified and extremely simple proof of these 2 ∂xu(0) = ∂x is not in L (R), then ∂xu(t) is not in global results. His proof is based on a logarithmic L2(R) for all t ∈ R. His proof of this result, which type of inequality for the boundedness in L∞(R2) was new even for the associated linear case f 0, was of singular integral operators. As often in Kato’s extremely simple. This was quite a surprise, since the works, [K86] was a fountain of ideas. The extension Korteweg de Vries equation is one of the most famous of this logarithmic inequality to the 3-dimensional nonlinear evolution equations. However, this was case led to a joint result of Kato with J. T. Beale and not the first regularizing effect established by Kato. A. Majda. Questions raised there initiated my The solution u(t)=eit of the linear collaboration with Kato and yielded the results in a Schrödinger equation, namely, (4) with A = i and joint paper of mine with Kato in 1988; this concerned f 0 and data ∈ L2(Rn), describes a continuous sharp energy estimates involving fractional deriva- curve in L2(Rn) . In [K66a] Kato showed that for tives and the extension of his abstract approach to almost every time t ∈ R , eit takes values in a the Ls,p(Rn)-well-posedness of the Navier-Stokes small set (of first category). This smoothing effect and Euler systems.

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paper perhaps led to my later fondness for abstract operator theory. Also, I learned much from his lecture notes from Berkeley [K55]. The simple directness of these notes is unforgettable. Immediately after his return from his first visit to the United States in 1954–55 (University of California at Berkeley and New York University), he focused his attention on perturbations of the continuous spectrum and the associated scattering theory. As he completed his papers on scattering theory for perturbations of finite rank and then of trace class, he passed the manuscripts to us to read and check. I still keep a carbon copy of these papers. He then suggested that I investigate the case of relative trace class, with a view to applications to Schrödinger operators. With some of my effort Tosio Kato (from family photo album). and ideas, this resulted in my doctoral thesis. At that time the group led by Kato was a small I last saw Kato in the spring of 1999. He was one in the physics department. Kôsaku Yosida had emptying his house at Berkeley and asked me to just moved to Tokyo (to the mathematics depart- help him get rid of some publications and books. ment) from Osaka and was building up a func- It was a hard and sad time for him. However, Kato tional analysis group. It was apparent even to the was willing to talk mathematics. His enthusiasm eyes of a beginning student that two giants of for it was intact. He was full of ideas and questions. functional analysis in Japan had very close acade- mic ties and respect for each other. So, these were S. T. Kuroda days when Kato initiated a tradition of mathematical physics in Japan. In this article I concentrate on personal remi- As far as I recall, the last three visits of Kato to niscences about the old days when Tosio Kato was Japan were in 1989 (on the occasion of a belated still in Japan. celebration of his seventieth birthday), 1991, and My recollection of Kato goes back to my younger 1992. During these visits he actively participated days when I attended his course on mathematical in conferences and symposia and gave lectures on physics at the Department of Physics, University varied subjects in operator theory, evolution equa- of Tokyo. It was 1953–54. The course covered, tions, and nonlinear partial differential equations. thoroughly but efficiently, most of the standard Recognizing persistent influences of his ideas in material from the theory of functions through par- mathematical physics and partial differential equa- tial differential equations. The style of his lecture tions, we in Japan were looking forward to never gave an impression that he went quickly, but another occasion. We mourn that the chance has at the end of each class I was surprised by how gone forever. much he had covered within one session. The course could vividly live today, and a plan of edit- ing materials from old notes has been slowly going on. I deeply regret that it could not be completed Arne Jensen while he was alive. Tosio Kato passed away on October 2, 1999. His Two years later I was a graduate student under the death came as a shock to me and was wholly supervision of Kato. It was several years after the unexpected. publication of his first major work on self-adjoint- I spent two and a half years in Berkeley as a ness of Hamiltonians of Schrödinger type and his visitor, from October 1976 to March 1979. I came comprehensive analysis of perturbations of the dis- to Berkeley on a Danish grant, after studying at the crete spectrum. His work on self-adjointness is now University of Aarhus. During my last year there more famous as a turning point in mathematical S. T. Kuroda was a visitor and gave a course on physics or “functional analysis of quantum me- scattering theory. So it was natural for me to go to chanics” (this is the title of an essay he wrote in Berkeley to learn more from Tosio Kato. I learned a Japanese). Nevertheless, I recall that I put more great deal from him. As preprints came in, I was energy into reading his big paper on perturbation asked to read through those he thought would be of theory [K51c]. He gave me a thick reprint, which was interest to me and to give my opinions. This often precious in those days before photocopiers. This resulted in lengthy discussions.

S. T. Kuroda is professor of mathematics at Gakushuin Uni- Arne Jensen is professor of mathematics at Aalborg versity, Tokyo, Japan. His e-mail address is kuroda@ University, Denmark. His e-mail address is matarne@ math.gakushuin.ac.jp. math.auc.dk.

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Some of these discussions resulted in two joint Q satisfies kQxkkBxk and kQxkkAxk for papers. I still recall vividly his attention to detail. all x, then Every proof had to be discussed and distilled to |(Qx, y)|kB xkkA1 yk its essence. Often we would discuss a mathematical problem, and he would think for a while and for all x, y, and 0 1. then give a reply which could not be improved on. There is an analogue for unbounded closed oper- In several cases I found the next morning in my ators, but the inequality is already interesting for mailbox a note complementing the discussion. matrices. The discussions took place both in his office in Kato and I first met at CalTech in 1957, and it Berkeley and during walks in the surrounding area. was in 1962 that he came permanently to Berkeley. We often took walks in Tilden Park or Strawberry Our early contacts at Berkeley were mainly in the Canyon. It was during these walks that I learned PDE seminar, organized jointly by C. Morrey, M. Prot- of his interest in botany. He could identify a large ter, H. Lewy, Kato, and me. We worked indepen- number of plants and trees and knew the Latin dently, and never wrote a joint paper, although our names as well. He had great respect for the backgrounds were quite close. Kato had been raised classification system introduced in the eighteenth as a physicist in the glorious age after quantum me- century by the Swedish botanist Carl von Linné. chanics was created, when J. von Neumann and Thus many years later, in 1993, when he visited the F. Riesz introduced the spectral theory of unbounded Mittag-Leffler Institute, a visit to Uppsala and in operators as a generalization of Hilbert’s spectral particular to Linné’s garden was a must. At home theory, to fit the needs of a mathematically rigor- in Berkeley he tried to grow many kinds of plants, ous quantum theory. In the 1940s Rellich had laid with varying success. In particular, the drought in out perturbation theory for the spectral resolution 1977 was hard on his plants. of unbounded self-adjoint operators, and I was a Of his work I particularly like the paper [K66a]. member of Rellich’s group. It has been highly influential, including on work Rellich had an “analytic” approach to pertur- in the 1980s on the many-body problem. After bation theory, mainly working with power 1980 Tosio Kato concentrated his work on non- series. It may have been B. Sz. Nagy who intro- linear partial differential equations. duced us to resolvent methods, which proved so Finally, I would like to mention an intriguing powerful. Kato entered this area, and perturba- problem that he left for posterity to solve. It is tion theory became “his field” after Rellich passed sometimes referred to as Kato’s square root away and Heinz turned to nonlinear problems. problem. Briefly, the problem can be stated as Kato’s book Perturbation Theory of Linear Operators follows: Let L = div(A grad) , where A(x) is a appeared in 1966. Not only did it extend many matrix with complex-valued bounded entries. The techniques from Hilbert spaces to Banach spaces, operator L is the maximally accretive operator on but it also discussed semigroups and their per- L2(Rn) defined via the quadratic form on the Sobolev turbations. It was carefully and patiently laid out space H1(Rn). Then one can define the square root and easy to digest and proved to be the standard of L, and the problem is whether its domain equals reference in the field. H1(Rn). The answer is not known in general today. In Berkeley in the early 1960s there were several In a 1998 article P. Auscher and P. Tchamitchian strong currents in analysis. Morrey was building on gave a survey and wrote about some recent results the work of J. Nash on estimates for divergence- concerning the problem. form second-order elliptic operators. I made some progress on the nondivergence case (a problem finally settled in the late 1970s by N. Krylov and Heinz Cordes M. Safonov). There was also strong interest in Bourbaki-style functional analysis, with J. Kelley. I miss my friend Tosio Kato. After twenty-five In the wake of the movement crowned by the years of work side by side at UC Berkeley and an- Atiyah-Singer index theorem, I turned my inter- other eleven years together after his retirement, we ests to C-algebras of pseudodifferential operators. had grown close. Kato never veered from classical analysis into this I first heard of Kato in 1952 in Göttingen. My terrain. “Doktorvater” Franz Rellich showed me the work Nevertheless, Kato’s work and mine did wind up of a young Japanese scholar who had improved influencing each other. One example is his paper some very nontrivial estimates in the thesis of [K58] on nullity and deficiency of operators Rellich’s student E. Heinz. An example, in terms of between Banach spaces. As another example, in 1972 matrices: for self-adjoint positive A, B, if a matrix P. Chernoff gave his spectacularly short proof of my result on essential self-adjointness of powers ∞ Heinz Cordes is professor emeritus of mathematics at the of the Laplace operator on C0 (M), when M is a University of California, Berkeley. His e-mail address is complete Riemannian manifold. Kato gave a further [email protected]. improvement, extending Chernoff’s argument to

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treat some non- R. Anderson, a mathematical companion with a semibounded oper- recent Ph.D. from Princeton, told me that the ators. problem of studying fractional powers of elliptic Our most inten- differential operators was hot. Looking into it quickly sive contact may led me to seek out faculty advice, and Kato was have been in mat- willing to provide it. He told me I should learn 2 ters of L bound- interpolation theory. He put me onto the recent edness of pseudo- works of J.-L. Lions and E. Magenes. Their books, in differential French, on boundary problems were not out yet, operators. His en- and of their papers some were in French and some couragement and in Italian. But a $2 paperback on Italian for kind words helped beginners from Moe’s Bookstore helped make them me in my work on accessible. compactness of I took Kato’s PDE (partial differential equations) commutators and course in the fall of 1968. It followed S. Agmon’s boundedness of book on elliptic boundary problems, which empha- pseudodifferential sizes the realization of elliptic operators as closed, Kato and wife Mizue (about 1984). operators, pub- unbounded (sometimes self-adjoint) operators via lished in 1975. In turn, he extended my version to a proof of the the study of quadratic forms on a Hilbert space. This Calderon-Vaillancourt theorem for symbols of type approach evolved from fundamental work of K. O. (,), for <1 (in the Hörmander category), while Friedrichs, and it was also close to Kato’s heart, I had looked at =0. Subsequently, in an im- playing a role in his work on Schrödinger operators. provement of a result of Schulenberger and Wilcox, At the end of the course Kato invited the students I made use of his ideas from a 1976 paper. We also to lunch at the Golden Bear. There were only about had a joint Ph.D. student (G. Childs) who studied four students in the course: Frank Massey and I and L2 boundedness under a weaker (Hölder-type) con- maybe two others. In those days at Berkeley a course dition on the symbol. on topological vector spaces might get twenty In the late 1960s our personal acquaintance students. Students would fill a large room to attend grew, and our families grew closer together. We S. S. Chern’s course on index theory, but the study spent time together in Hamburg in 1968. In 1972 of PDE itself was not popular. the Katos visited us in the Sierra Nevada. We made Fortunately for the handful of us who needed a habit of arranging joint picnics on Sundays, at it, the department ran a three-quarter sequence of places such as Mount Diablo or a Sonoma winery, courses on PDE. The second and third quarters were frequently accompanied by another mathemati- taught that year by I. Kupka and H. O. Cordes. cian visiting the department. They both treated the theory of pseudodifferen- In 1996 Kato’s wife Mizue became ill. After a tial operators. This theory, evolving from classical while they moved into a retirement home in studies of layer potentials, had gained panache Oakland, and things stabilized to the point that he from its role in the Atiyah-Singer index theorem. could start to work again. L. Hörmander was in the midst of producing spec- His plans then aimed at a 3-dimensional tacular results in the area, and I was seduced by extension of his work on the 2-dimensional Euler the microlocal side of analysis. equations. He almost got me to the point of To be sure, Kato’s work had influences on working with him then. microlocal analysis, both direct and indirect. I first His death came suddenly. On a Saturday at mid- mention an indirect influence. Around 1970 there night I was called by a nurse—he had passed away arose definitive solutions to some classes of mixed of heart failure. initial-boundary problems for hyperbolic systems. Important progress was made by H. Kreiss, who constructed “symmetrizers” for such boundary Michael Taylor problems satisfying an analogue of what in the elliptic case were termed Lopatinsky conditions; I first met Tosio Kato in the summer of 1968, near these symmetrizers were pseudodifferential the end of my first year as a graduate student at operators. At this time, Kato’s student Massey Berkeley. Berkeley was hosting a summer sympo- conducted a study of mixed problems for a class sium on global analysis, and the place was abuzz of symmetric hyperbolic systems, while J. Rauch, with activity. Over lunch on Telegraph Avenue, working with P. Lax, produced important refine- ments of Kreiss’s results. Then Massey and Rauch Michael Taylor is William R. Kenan Jr. Professor of Math- got together and produced sharp regularity ematics at the University of North Carolina, Chapel Hill. results for solutions to certain symmetric- His e-mail address is [email protected]. hyperbolic mixed problems.

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More direct was Kato’s role in a lovely proof of the provable error bounds. That there is more to Stirling’s Calderón-Vaillancourt theorem. This result is an formula than an application of the Laplace asymp- estimate on pseudodifferential operators with totic method is a piece of lore that, thanks to Kato, symbols of Hörmander type (,), in the borderline I will not forget. cases = ∈ [0, 1). Proven by A. P. Calderón and R. Vaillancourt in 1971, it quickly had a spectacular References application in work of R. Beals and C. Fefferman, in [K50] KATO, T., On the adiabatic theorem of quantum me- the case = =1/2. Such operators are also use- chanics, J. Phys. Soc. Japan 5 (1950), 435–439. ful in the study of hypoelliptic operators with [K51a] ——— , Fundamental properties of Hamiltonian operators of Schrödinger type, Trans. Amer. Math. Soc. double characteristics, such as arise in treatments 70 (1951), 195–211. of the ∂ -Neumann problem. On the other hand, [K51b] ——— , On the existence of solutions of the helium estimates on operators of type (0, 0) have been wave equation, Trans. Amer. Math. Soc. 70 (1951), important in work on semiclassical asymptotics for 212–218. Schrödinger operators. In 1975 Cordes produced a [K51c] ——— , On the convergence of the perturbation new proof of the theorem, in the case = =0. One method, J. Fac. Sci. Univ. Tokyo 6 (1951), 145–226. particularly interesting feature of the proof was that [K55] ——— , Quadratic forms in Hilbert spaces and as- it required estimates on relatively few derivatives ymptotic perturbation series, Technical Report No. 7, of the symbol. This paper was followed by Kato’s University of California, 1955. [K57] ——— , On perturbations of self-adjoint operators, paper [K76] extending Cordes’s method to treat the J. Math. Soc. Japan 9 (1957), 239–249; Perturbation ∈ general case = [0, 1). Kato’s paper contains a of continuous spectra by trace class operators, Proc. general lemma that makes clear what the basic Japan Acad. 33 (1957), 260–264. mechanism is behind this approach to the proof. In [K58] ——— , Perturbation theory for nullity, deficiency representation-theoretic language one would say it and other quantities of linear operators, J. d’Analyse is the fact that the representation of the Heisenberg Math. 6 (1958), 261–322. group defined by U(p, q)f (x)=eiqxf (x + p) is [K65] ——— , Wave operators and unitary equivalence, square integrable, a feature also emphasized by Pacific J. Math. 15 (1965), 171–180. [K66a] , Wave operators and similarity for some non- R. Howe in work published a few years later. ——— selfadjoint operators, Math. Ann. 162 (1966), Kato made other contributions to microlocal 258–279. analysis in the course of various investigations on [K66b] ——— , Perturbation Theory for Linear Operators, nonlinear evolution equations. His 1984 paper with Springer-Verlag, New York, 1966. T. Beale and A. Majda on the Euler equations had to [K72] ——— , Schrödinger operators with singular poten- deal with a situation where things would be easy if tials, Israel J. Math. 13 (1972), 135–148. pseudodifferential operators, which act nicely on [K80] ——— , Remarks on receiving the 1980 Wiener Prize, Lp for 1

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