Semigroup Forum (2012) 84:1–7 DOI 10.1007/s00233-011-9363-1 TRIBUTE

A tribute to Jerome (Jerry) Arthur Goldstein Rainer Nagel

Accepted: 13 November 2011 / Published online: 9 December 2011 © Springer Science+Business Media, LLC 2011

In 1983, Jerry Goldstein was elected to represent functional analysis and operator theory on the editorial board of Semigroup Forum. Since then, he has rendered not Communicated by László Márki. R. Nagel () Arbeitsbereich Funktionalanalysis, Mathematisches Institut, Auf der Morgenstelle 10, 72076 Tübingen, Germany e-mail: [email protected]

2

R. Nagel

only an enormous service to the journal but has contributed essentially to establishing strongly continuous semigroups of linear and nonlinear operators on Banach spaces as a standard tool for partial differential equations, stochastic processes, quantum mechanics, numerical analysis, and financial mathematics. His 70th birthday on August 5, 2011, has allowed me to review his important and still ongoing contributions to mathematics and, in particular, to semigroup theory and its applications.

His career Jerry earned his Ph.D. in 1967 at Carnegie Mellon University at Pittsburgh with a thesis on Stochastic Differential Equations and Nonlinear Semigroups supervised by Malempati M. Rao. After a year at the Institute of Advanced Studies he joined the faculty of Tulane University in New Orleans. In 1991, he decided to move up the Mississippi and took a professorship at Louisiana State University in Baton Rouge. Since 1996 he has been Professor of Mathematical Sciences at the University of Memphis, where he received, among many other distinctions, the Eminent Faculty Award in 2006, the highest honor bestowed by the University of Memphis.

His mathematics Though educated as a probabilist, Jerry soon realized the enormous potential of the Hille-Yosida theory of operator semigroups for partial differential equations, stochastics, mathematical physics, and other areas. His program is well-expressed by the title of the first paper [1] (see also [2]) he published in Semigroup Forum: A semigroup-theoretic proof of the central limit theorem and other theorems in analysis. Besides extending the theory of operator semigroups and discovering new applications, Jerry soon felt the need to update the 1957 classic Functional Analysis and Semigroups by E. Hille and R. Phillips [3] and to broaden the perspective to include applications. In the introduction to his 1970 Tulane Lecture Notes Semigroups of Operators and Abstract Cauchy Problems he wrote: “A conscious effort was made to solve some nontrivial parabolic and hyperbolic Cauchy problems without doing any of the hard work associated with elliptic theory. . . . We wish to get across some of the main ideas involved in the study of initial value problems for partial differential equations as an easy consequence of semigroup theory.” At this point I shall deviate from my main topic (Jerry) and explain—to the non expert—what these semigroups are and how they can solve parabolic partial differential equations “without doing any of the hard work”. We start with an autonomous linear ordinary differential equation x  (t) = Ax(t)

(1)

with A a complex n × n matrix and x(t) a Cn -valued function. Then, if the initial value x(0) = x0 is prescribed, the unique solution is x(t) = etA x0 ,

t ∈ R,

A tribute to Jerome (Jerry) Arthur Goldstein

3

where etA =

∞ n n  t A

n!

n=0

.

This holds since the matrix valued exponential function R  t −→ etA ∈ Mn (C) is differentiable with derivative d tA e = AetA . dt All this is known since the 19th century. But as soon as Hilbert and Banach spaces appeared on the scene, it became possible to even write partial differential equations in the form (1). Let me explain this idea for the heat equation. Take  a domain in Rn with (smooth) boundary ∂, and describe by x(s, t) the “heat” at the point s ∈  at time t ≥ 0. If x0 (·) is the heat distribution on  at initial time t = 0, then one is led to the equation  n ∂ 2 ∂ j =1 ∂s 2 x(s, t), s ∈ , t ≥ 0 ∂t x(s, t) = (HE) i x(s, 0) = x0 (s), s ∈ , with, e.g., x(s, t) = 0

for s at the boundary ∂.

If we now, instead at the values x(s, t), look for fixed t at the function   s −→ x(s, t), identify it with an element of a Banach space X of functions on , and define a linear operator A on this space by Af :=

n  ∂2 f ∂si2 j =1

for f in some appropriate domain D ⊂ X, then (HE) can be written (formally) as  d dt x(t) = Ax(t), t ≥ 0, (ACP) x(0) = x0 , for an X-valued function x(·). This looks like the initial value problem (1) discussed above, and, since x(t) and A take values in a Banach space, it is called Abstract Cauchy Problem (ACP). Inspired by the finite dimensional case, already Lagrange, Fourier, Boole et al. had the idea that the solution to problems of this form should be something like x(t) = etA x0 But how to define the symbol

“etA ”?

(see [4]).

Already for the simplest differential operator Af = f 

the formal power series for etA yields etA f = f + tf  +

t 2  f + ··· 2!

4

R. Nagel

which exists and converges only for few functions f . So, let us choose an axiomatic approach to the exponential function. Motivated by the fact that the scalar, and also the matrix valued exponential functions are the only continuous functions ξ : R+ → L(Cn ) satisfying the functional equation ξ(t + s) = ξ(t) · ξ(s),

ξ(0) = I

(2)

we introduce the concept of a semigroup of operators as follows. Definition For each t ≥ 0, let T (t) be a bounded linear operator on a Banach space X. We call (T (t))t≥0 a strongly continuous semigroup (short: C0 -semigroup) if (FE) T (t + s) = T (t) · T (s) for all t, s ≥ 0; T (0) = I ; (C) For each x ∈ X, the map t −→ T (t)x is continuous from R+ into X. It is because of this definition and its consequences that Jerry and his followers were connected with Semigroup Forum! The functional equation (FE) combined with the continuity property (C) means that t −→ T (t) is a continuous semigroup homomorphism from (R+ , +) into (Ls (X), ·), the algebra of all bounded linear operators on X endowed with the strong operator topology. Algebraically and topologically this seems to be a quite simple object. However, taking into account the motivation for the above definition, it is no surprise that even differentiability follows in a certain sense. Lemma For all x in a dense subspace D of X the map t −→ x(t) := T (t)x is even continuously differentiable, and if we define Ax := lim

t→0

d T (t)x − x = x(t)|t=0 t dt

then the function x(·) satisfies 

d dt x(t) = Ax(t), x(0) = x0 .

t ≥ 0,

for x ∈ D,

(ACP)

This means that to each C0 -semigroup (T (t))t≥0 one can associate a unique linear operator A, called the generator of (T (t))t≥0 , such that x(t) := T (t)x0 solves the (ACP) corresponding to A. Note that this operator is, in most cases, unbounded and only densely defined. At this point the fundamental question becomes: Which operators are generators of C0 -semigroups? If so, the corresponding (ACP) is well-posed and solved by the semigroup.

A tribute to Jerome (Jerry) Arthur Goldstein

5

Only in 1948, E. Hille and K. Yosida independently and using different approaches found the necessary and sufficient conditions. The idea of Hille was to replace the power series for the exponential function by Euler’s formula  −n t ta e = lim 1 − a . n→∞ n Indeed, if the operator satisfies the so-called Hille-Yosida conditions, it is the generator of a C0 -semigroup (T (t))t≥0 and  −n t T (t)x = lim 1 − A x for all x ∈ X. n→∞ n With this interpretation one can write formally T (t) = etA even for unbounded operators, and, by verifying the Hille-Yosida conditions, one obtains the solutions of concrete or abstract Cauchy problems “as an easy consequence of semigroup theory”. This semigroup approach turned out to be extremely useful not only because it reveals the structure (the exponential function) behind autonomous initial value problems for partial differential equations, but also because it allows us to solve many other and quite diverse problems. By choosing appropriate Banach spaces and operators, initial value problems, e.g., for integrodifferential equations, delay differential equations, difference equations, higher order and nonautonomous partial differential equations can be rewritten as (ACP) and then solved via semigroups. This ubiquity led E. Hille already in 1948 to his famous dictum: “I hail a semigroup whenever I see one and I seem to see them everywhere.” Jerry also seems to have seen these semigroups everywhere when he wrote his Tulane Lecture Notes in 1970. For me it was a most fortunate coincidence that Zentralblatt für Mathematik asked me to review these lecture notes [5]. This was my first contact with semigroup theory and I discovered its beauty and power. So, up to this day, I am proud to consider myself a student of Jerry. By 1985, the original lecture notes had grown into the monograph Semigroups of Linear Operators and Applications [6] in which Jerry not only presented the theory in a most elegant form but also showed its potential for applications. Today, Jerry’s list of publications (papers, monographs, proceedings) comprises more than 250 entries (see [7]). Among the more recent items, I especially want to draw attention to his ongoing cooperation, begun in 1998, with Angelo Favini from Bologna, his wife Gisele R. Goldstein, and Silvia Romanelli from Bari ([8]). Under the acronym FGGR they studied so-called Wentzell (or, dynamic) boundary conditions for second and higher order differential operators. By a detailed analysis they found the correct spaces and domains where these operators generate a C0 - or even an analytic semigroup and therefore give rise to a well-posed Cauchy problem. Numerous applications of this theory with wide ranging scientific interest have been made by Jerry and his co-authors. In a quite different direction I mention his work (jointly with Pierre Baras [9]) on the heat equation with singular potential. Here the Feynman-Kac integral formula is

6

R. Nagel

used to show the instantaneous blowup of the solutions to a Schrödinger parabolic problem. This is his most quoted paper, and the ideas have been applied by many authors to a wide variety of problems. Presently he is working (with G.R. Goldstein, I. Kombe, and A. Rhandi) on similar phenomena involving Ornstein-Uhlenbeck operators.

His teaching Jerry, by his teaching and also by his countless talks at conferences, summer schools, and seminars, has inspired young (and not so young) mathematicians all over the world. According to [10], he has 27 Ph.D. students, but many more were inspired by his talks, his comments on their work, and by his legendary letters of support. As a particular highlight, I will always remember July 2001, when I had the pleasure to teach, together with Jerry as my partner, a group of 20 highly motivated European students in the summer school on Semigroups of Operators at the Palazzone in Cortona (Italy). The perfect combination of a gorgeous “ambiente” together with the contagiously high motivation of the students pushed us all into some sort of “flow” experience, with Jerry as our navigator. But while I had to pay tribute to my age and retreat to rest regularly, Jerry was on duty around the clock, never showing any signs of fatigue. Many of the participants are now well established in the mathematical community, and some of them became Jerry’s co-authors. The sheer fact that so far he has had more than 100 co-authors is further evidence of his influence and charisma.

Jerry Jerry is a person with striking magnetism and seemingly inexhaustible energy. Every time he was expected for a visit at my department in Tübingen, my students became electrified, awaiting inspiration, and they were never disappointed. Unfortunately for us, his favorite place to go in Europe was to become Italy. Among the lore he left in Tübingen is the following short conversation: Jerry: Can you imagine a place in the world more romantic than Paris? Answer: (suspecting a blue joke, responding only after some time, to ease the situation) Paris, Texas? Jerry: ANY town in Italy! He still pays us a visit now and then and is most welcome any time! With his wife Gisele he has found personal happiness together with professional challenge and support: a most desirable combination that hopefully will keep him young for many more years to come! Don’t slow down, Jerry!

A tribute to Jerome (Jerry) Arthur Goldstein

7

References 1. Goldstein, J.A.: A semigroup theoretic proof of the central limit theorem and other theorems of analysis. Semigroup Forum 12, 189–206 (1976) 2. Goldstein, J.A.: A semigroup theoretic proof of the law of large numbers. Semigroup Forum 15, 89–90 (1977) 3. Hille, E., Phillips, R.: Functional Analysis and Semigroups. Am. Math. Soc., Providence (1957) 4. Hahn, T., Perazzoli, C.: A brief history of the exponential function. In: Engel, K.-J., Nagel, R. (eds.) One-Parameter Semigroups for Linear Evolution Equations. Springer, Berlin (2000) 5. Zentralblatt, http://www.zentralblatt-math.org/zbmath/search/?q=an 6. Goldstein, J.A.: Semigroups of Operators and Applications. Oxford University Press, London (1985) 7. Goldstein, J.A.: List of publications, Mathscinet 8. Favini, A., Goldstein, J., Goldstein, G., Romanelli, S.: C0 -semigroups generated by second order differential operators with general Wentzell boundary conditions. Proc. Am. Math. Soc. 128, 1981– 1989 (2000) 9. Goldstein, J., Baras, P.: The heat equation with a singular potential. Trans. Am. Math. Soc. 284, 121– 139 (1984) 10. Ph.D. students of J.A. Goldstein: Mathematics Genealogy Project. http://www.genealogy.ams.org/ id.php?id=15007