a tribute to henry helson - american mathematical … tribute to henry helson donald sarason...

A Tributeto Henry Helson

Donald SarasonHenry Helson, a leading figure in harmonic analysiswhose ideas have had an enormous impact, diedon January 10, 2010. Born on June 2, 1927, Henrygrew up in Bryn Mawr, where his father Harry wasa professor, carrying out the research that wouldmake him one of the eminent psychologists of hisera. His mother Lida, despite her upbringing asa fundamentalist Lutheran, found her way to theQuaker faith when Henry and his sister Marthawere children. All three of them became andremained committed Quakers.

Henry’s exceptional mathematical abilitiesshone through early. Although his father Harryhoped Henry would go into physics, Henry washooked on mathematics. He entered Harvardas an undergraduate, graduating in 1947. Hiseducation had been interrupted for thirteenmonths by military service, which for him, as aQuaker, was a loathsome experience. On the eveof his graduation Henry was awarded a Harvardtraveling fellowship. The fellowship enabled himto fulfill an intense desire he had nurtured tovisit Europe. In the academic year 1947–1948 hevisited London, Paris, Prague, and Vienna, but hespent most of the year in Poland, first in Warsaw,then in Wroclaw.

Why Poland? The circumstances are explainedin Henry’s essay [5]. He writes: “. . . [I] was crazy tosee the destruction caused by the war in Europe. . . .The most desperate place in Europe seemed to bePoland, and there were mathematicians in Poland.”Another passage from the same essay reads:

Donald Sarason is professor of mathematics at University

of California, Berkeley. His email address is sarason@

math.berkeley.edu.

Young Henry, location and date uncertain.

In college I had spent much timebrowsing in the mathematics li-brary, and even in the stacks,and I found Fundamenta Mathe-

maticae and Studia Mathematica

of special interest. There weremagic names in them: Sierpinski,Steinhaus, Kuratowski, Ulam, Os-trowski, Szegö, Borsuk, Tarski, andof course Banach, and a namethat appeared in two forms, some-times as Szpilrajn and sometimesas Szpilrajn–Marczewski, which Icould not explain. Many papersseemed related, many were jointwork, all were quite short. I could

274 Notices of the AMS Volume 58, Number 2

not understand most of them, butI tried.

In spring 1948 Henry was in Wroclaw. He writes:

Szpilrajn had become Marczewskiby this time. In Polish law, you hadto keep both names for a while ifyou wanted to change. I supposethat Szpilrajn was a Jewish name.Marczewski took me in hand andgave me good problems, in anal-ysis, to work on. I solved two ofthem and they made my first pub-lications, and he was pleased. (Sowas I.)

The publications referred to are [2] and [3], whichappeared in Studia Math. and Colloquium Math.,respectively.

In fall 1948 Henry enrolled in the Harvardmathematics Ph.D. program.His thesis advisor wasLynn Loomis, but by happenstance Arne Beurlingturned out to be the source of his mathematicalinspiration. In an essay that did not receive widecirculation [6], Henry writes:

When I came back to Harvard in1948 Arne Beurling came to visitthe department. His course lookedhard and unrewarding but my men-tor, Lynn Loomis, told me I had totake it. Of course the departmenthad to get together some studentsto listen to Beurling! So I went,and that set my subsequent math-ematical trail. . . . Beurling was fullof wonderful ideas, and I soakedthem up. In this time I was thetwin of my friend John Wermer,whom I had known for years asan undergraduate. At the conclu-sion of this year, John went backto Uppsala with Beurling; I stayedanother year to write my thesis.

Henry received his Ph.D. in spring 1950 butwas late getting on the job market. An offer fromUCLA eventually came, but this was during theCalifornia loyalty oath controversy; all Universityof California faculty were required to sign theoath, which Henry would not do. He wrote to Wer-mer in Uppsala, requesting that John ask Beurlingif he could support another student. A positiveanswer came back, so Henry spent the academicyear 1950–1951 in Uppsala, his stay there inter-spersed with trips elsewhere in Europe. One tripwas to Nancy, France, which then housed LaurentSchwartz, Jean Dieudonné, Roger Godement, andAlexander Grothendieck (then a student), amongothers. The Nancy trip resulted in the offer of a fel-lowship for the following year, but Henry’s parentsprevailed on him to stay in the United States. He

Henry carving turkey, Berkeley, 1950s.

Henry, Ravenna, and children Ravenna, David,and Harold (on Ravenna’s lap), Berkeley, 1960.

accepted an instructorship at Yale and was even-tually promoted to assistant professor. However,Yale at that time, despite housing an impressivegroup of young nontenured mathematics faculty,regarded their positions as strictly temporary. In1955 Henry accepted an assistant professorshipoffer from Berkeley. (The loyalty oath had by thenbeen declared unconstitutional.) Henry remainedat Berkeley for the remainder of his career, be-coming associate professor in 1958, professor in1961, and emeritus professor in 1993.

Henry and his wife Ravenna were married in1954. Ravenna had received a Ph.D. in psychologyfrom Berkeley and was eager to return there,which enhanced Berkeley’s intrinsic attractionsfor the couple. They raised three children, bornbetween the years 1956 and 1960. Ravenna hada distinguished career as research psychologistat Berkeley’s Institute of Personality and SocialResearch, including serving as director of the MillsLongitudinal Study, which she founded.

February 2011 Notices of the AMS 275

Henry’s early research, which quickly gainedrecognition, is discussed below by Izzy Katznel-son and John Wermer. I’ll dwell here a bit onHenry’s two most influential papers, [8] and [9],coauthored with David Lowdenslager. Bill Arvesonwill describe how those papers were instrumentalin the development of the theory of nonself-adjoint operator algebras, a direction likely notenvisioned by the paper’s authors. The pieces byKathy Merrill and Jun-ichi Tanaka concern anotherdirection initiated in [8] and [9], involving invariantsubspaces.

As the titles of [8] and [9] inform us, thepapers arise from certain questions in probability.Attached to a stationary stochastic process withdiscrete time is its so-called spectral measure,a positive Borel measure µ on the unit circle.Certain questions about the process can be recastas questions about the geometry of the (complex)Hilbert space L2(µ). Thus questions in probability

Henry, early 1960s.

are transformed into questionsin function theory. Prior tothe work of Helson and Low-denslager, a number of thosequestions had been answeredthrough the work of, for ex-ample, Gabor Szegö, NorbertWiener, Andreı N. Kolmogorov,and Mark G. Kreın. Helsonand Lowdenslager propelled thistheory in two new directions.

First, Helson and Low-denslager replaced the classicalone-parameter stationary sto-chastic processes referred to in

the preceding paragraph by multiparameter sta-tionary stochastic processes of a certain kind. Inthose processes, the unit circle from the classicaltheory is replaced by a compact abelian group Bdual to a dense discrete subgroup of R—a torusof dimension larger than 1, possibly of infinitedimension. The Bohr compactification of R, thedual of the entire discrete real line, is a primeexample. Fourier analysis on B leads to the Fourierseries in several variables of the papers’ titles.

Helson and Lowdenslager established versionsin their setting of some basic theorems fromthe classical setting, for instance, Szegö’s theoremfrom prediction theory and the famous theorem ofF. and M. Riesz on analytic measures. This requirednew ideas, because Helson and Lowdenslager didnot have available the classical techniques.

While Helson and Lowdenslager emphasized thegroup setting in which they worked, it was quicklyrealized that their ideas extended far beyond thatcontext. In particular, the ideas were employedby people working in the then emerging theoryof function algebras, providing decisive impetus.The ideas also suggested new ways to approachthe study of Hardy spaces in the unit disk, ideas

adopted by Kenneth Hoffman in his well-knownbook [11].

Helson and Lowdenslager’s F. & M. Riesztheorem attracted particular attention. Earlier,Solomon Bochner [1] had established a generaliza-tion of the original F. & M. Riesz theorem in whichthe circle is replaced by the two-torus. A maximaltheorem underlies Bochner’s method, which canbe applied to give a proof of the original F. & M.Riesz theorem, a proof different from the severalthat had been previously found. Bochner’s F. & M.Riesz theorem is a very special case of Helson andLowdenslager’s theorem, which they proved withwhat were at the time completely new methods,an elegant blend of function theory and operatortheory. Those methods can be applied to give yetanother and a remarkably simple proof of theoriginal F. & M. Riesz theorem.

In a second direction, Helson and Lowdenslagerstudied multivariate stochastic processes, replac-ing the scalar-valued processes from the classicaltheory by vector-valued processes. This led toquestions about factoring matrix-valued functionson the unit circle and corresponding new results.This part of their work hooks up with what even-tually became the theory of operator models ofBelá Sz.-Nagy and Ciprian Foias, a central part ofoperator theory.

The brief remarks above are but a very roughsketch of the basic papers of Helson and Low-denslager. The papers are too rich to reviewadequately in a few lines. They are well worthstudying even today; in them one will find theseeds of much of Henry’s subsequent research.

David Lowdenslager died tragically in Septem-ber 1963 at the age of thirty-three. This must havebeen a terrible blow to Henry. But Henry neverspoke of it to me, and of course I never pressedhim to do so.

As anyone who has read some of his papersknows, Henry was a masterful expositor. His styleis exceptionally clear, economical, vigorous, andfluid, altogether worthy of emulation.

In 1992 Henry started his own publishing firm,Berkeley Books, motivated by his distaste for ourpresent-day crop of calculus textbooks, by the highcost of mathematics books at all levels, and alsoby what he terms “a passion for entrepreneurialactivity”. Berkeley Books was a one-man operationexcept for the actual printing of the preparedtext, which Henry contracted to a local printer. Hepublished mostly his own textbooks, but also twoby Berkeley colleagues (George Bergman and me).Clearly, Henry’s overhead was far below that ofmainstreampublishers, but also clearly, Henry wasat a severedisadvantage when it came to marketinghis products. Henry describes his experiences as apublisher in [7]. As he states there, Berkeley Booksfinished in the black every year (at least until [7]was written).


Mathematics was not Henry’s only passion.Another was music. He was an accomplishedviolinist and violist, and after his retirement, freedfrom many prior responsibilities, he practiceddaily for two hours to enhance his skills. He joineda local music group, Amphion, and performedpublicly, especially with a talented and congenialquartet he assembled. Henry’s last performance,at Amphion, and the last time he played the violin,was in September 2009, just a few months beforehis death. He bravely fought off the terrible effectsof his disease and chemotherapy and performedBrahms’s First Violin Sonata.

Travel was another of Henry’s passions. Hewas keenly interested in other places and othercultures. While on the Berkeley faculty he spentsabbaticals in Sweden, Orsay, Ghana, Montpellier,Florence, Bonn, Marseille, and India. India held aspecial fascination for Henry; between 1980 and2000 he made four extensive visits there.

My own indebtedness to Henry is profound,both for the mathematics I learned from him andfor his support and kindness during our forty-six years as colleagues. I have illustrated the lastremark elsewhere [13], but the story will bearrepeating here, in abbreviated form.

The story involves a fundamental theorem ofZeev Nehari about Hankel operators [12]. Forpresent purposes, a Hankel operator is an operatoron ℓ2(Z+) induced by a Hankel matrix, a complexmatrix with constant cross diagonals (the (j, k)th

entry depends only on j + k). Nehari’s theoremanswers the question of when such an operator isbounded. The answer, in qualitative form, is thatboundedness happens if and only if the matrixentries in the first column are the forward Fouriercoefficients of a function in L∞ of the circle. Thequantitative version adds that the L∞ function canbe so chosen that its norm equals the norm of theoperator.

As a by-product of his joint paper with Szegö[10], Henry found an elegant and insightful proofof Nehari’s theorem using a duality argumentfacilitated by a factorization result of FrigyesRiesz for functions in the Hardy space H1. Henryshowed me his proof in fall 1965, during one of ourfrequent mathematical chats. The following day itsuddenly dawned on me that Henry’s techniquewas exactly what I needed to complete the proofof a theorem I had been working on for two years.The paper that grew out of this breakthrough [14],I would guess, was a big help in my getting tenureat Berkeley, as was no doubt Henry’s supportbehind the scenes.

Henry’s final paper [4] appeared recently inStudia Mathematica, sixty-two years after his firstpaper appeared there. In it, Henry addresses thequestion of whether Nehari’s theorem can be gen-eralized from its original one-dimensional settingto an infinite-dimensional setting. He surmises

that the answer is negative, and he maps out apossible way to produce the needed counterex-ample. At the end he states: “But we cannot gofurther.”

The paper was submitted on October 20, 2009,less than three months before Henry died. He musthave feared death was imminent and known thathe had better tell other harmonic analysts what heknew before it came to pass.

Henry is survived by his wife Ravenna, sister,Martha Wilson, daughter Ravenna Helson Lipchik,sons David and Harold, and three grandchildren.

(I was guided in composing the preceding re-marks, in addition to the sources cited, by a sketchof Henry’s life distributed at his memorial serviceon February 20, 2010, at the Berkeley FriendsMeeting Hall, and by advice from Ravenna, SteveKrantz, and several of my coauthors.)

References[1] S. Bochner, Boundary values of analytic func-

tions in several variables and of almost periodicfunctions, Ann. of Math. (2) 45 (1944), 708–722.

[2] H. Helson, Remarks on measures in almost-independent fields, Studia Math. 10 (1948), 182–183.

[3] , On the symmetric difference of sets asa group operation, Colloquium Math. 1 (1948),203–205.

[4] , Hankel forms, Studia Math. 198 (2010), 79–84.

[5] , Mathematics in Poland after the war, Notices

Amer. Math. Soc. 44 (1997), 209–212.[6] , Colloquium at St. Mary’s, unpublished.[7] , Publishing Report, Mathematical Intelli-

gencer 24 (2002), 5–7.[8] H. Helson and D. Lowdenslager, Prediction the-

ory and Fourier series in several variables, Acta

Math. 99 (1958), 165–202.[9] , Prediction theory and Fourier series in

several variables II, Acta Math. 106 (1961), 175–213.[10] H. Helson and G. Szegö, A problem in prediction

theory, Ann. Mat. Pura Appl. (4) 51 (1960), 107–138.[11] K. Hoffman, Banach Spaces of Analytic Functions,

Prentice Hall, Inc., Englewood Cliffs, NJ, 1962.[12] Z. Nehari, On bounded bilinear forms, Ann. Math.

65 (1957), 153–162.[13] D. Sarason, Commutant lifting, Operator Theory:

Advances and Applications 207 (2010), 351–357.[14] , Generalized interpolation in H∞, Trans.

Amer. Math. Soc. 127 (1967), 179–203.

Y. Katznelson

Some of Henry’s Early-Fifties WorkMuch of Henry’s work in the early 1950s dealt

with two problems that, retrospectively, turnedout to be very closely related.

Y. Katznelson is emeritus professor of mathematics at

Stanford University. His email address is katznel@math.

stanford.edu.


1) The description of homomorphisms of groupalgebras (in terms of homomorphisms of theunderlying groups).

Beurling and Helson [5] prove that automor-phisms of L1(R) are trivial, i.e., given by a linearchange of variables on R and, using some earlierresults of Beurling and Henry’s paper [4], show:

If G and H are locally compact abelian groups

at least one of which has a connected dual, and iftheir group algebras are isomorphic, then G andHare isomorphic.

2) The description of idempotent measures(under convolution) on locally compact abeliangroups.

A theorem of Szegö [2], states that if a functionf =

∑anzn is holomorphic in the unit disc and

admits some analytic continuation across somearc, and if its Taylor coefficients {an} assumeonly a finite number of distinct values, then f isrational: f (z) = P(z)/Q(z), with all the zeros ofthe denominator on the unit circle. The conclusionis equivalent to the statement that the sequence{an} is ultimately periodic to the right.

Henry [1] extended the theorem, showing thatthe condition “holomorphic f with some analyticcontinuation” is too strict and can be replaced bya “harmonic function u(r , t) =

∑∞

−∞ anr|n|eint for

which there exists an arc (α, β) on the unit circle

such that∫ βα |u(r , t)|dt is bounded for {r : r < 1}”.

The statement now is that if the coefficientsassume only a finite number of distinct values,then the sequence {an} is ultimately periodic tothe right and ultimately periodic to the left.

In the special case, which Henry proved in1953 [6], the arc (α,β) is the entire circle, theboundedness of

∫T|u(r , t)|dt implies that u is the

Poisson integral of a measure µ on the unit circlewhose Fourier coefficients µ(n) = an assume onlya finite set of distinct values,

The stage is now taken by measures whoseFourier coefficients assume only a finite number ofdistinct values, and in particular idempotent mea-sures (whose Fourier coefficients are necessarily 0or 1).

The ultimate periodicity of µ is shown to implythat µ is the sum of a discrete measure carried bya finite subgroup of the circle and an absolutelycontinuous part whose density is a trigonometricpolynomial.

This description of idempotent measures (onT) was the basis for Rudin’s work on isomor-phisms of group algebras and paved the way forCohen’s eventual complete description of idem-potent measures (on general groups) and solutionof the homomorphism problem.

Helson Sets

Another theorem that Henry proved in 1953 [3]inspired much interest and research activity. It

Top, left to right: Hans Lewy, Ravenna, Henry;bottom, left to right: Harold Helson and

Michael Lewy. Drakes Bay near Berkeley, early1960s.

has to do with perfect sets E ⊂ T (the circle group)that have the property that every continuouscomplex-valued function on E is the restrictionto E of an absolutely convergent Fourier series.(Such sets are now commonly referred to asHelson sets.) Henry proved that these are setsof uniqueness for measures, that is, a nontrivialmeasure µ carried by such sets cannot have itsFourier-Stieltjes coefficients µ(n) tend to zero asn→∞.

References[1] Henry Helson, On a theorem of Szegö, Proc. Amer.

Math. Soc. 6 (1955), 235–242.[2] G. Szegö, Über Potenzreihen mit endlich vielen

verschiedenen Koeffizienten, Sitzungsgerichte der

preussischen Akademie der Wissenschaften, 1922,88–91.

[3] Henry Helson, Fourier transforms on perfect sets,Studia Math. 14 (1953), 209–213.

[4] , Isomorphisms of abelian group algebras, Ark.

Mat. 2 (1952), 475–487.[5] Arne Beurling and Henry Helson, Fourier-Stieltjes

transforms with bounded powers, Math. Scand. 1

(1953), 120–126.[6] Henry Helson, Note on harmonic functions, Proc.

Amer. Math. Soc. 4 (1953), 686–691.

John WermerHenry Helson’s work has played a central rolein modern harmonic analysis and its applicationssince the 1950s.

In [1] Henry proved a conjecture of Steinhaus.The conjecture states that if the partial sums of atrigonometric series are all nonnegative, then the

John Wermer is emeritus professor of mathematics at

Brown University. His email address is wermer@math.

brown.edu.


coefficients of the series tend to 0. Henry in factproved something stronger: If the trigonometricseries

∑∞

−∞ aneinx satisfies

supN>0

∫ 2π

0

∣∣∣∣∣∣

N∑

n=−N

aneinx

∣∣∣∣∣∣dx <∞,

then an → 0 as |n| → ∞.Another early theorem concerns the Banach

algebra A of Fourier–Stieltjes transforms f (t) =∫∞−∞ exp(itx)dσ(x) with ‖f‖ =

∫∞−∞ |dσ(x)|, where

σ is a finite measure on −∞,∞. The problem isto find all f ǫA such that ‖f n‖ is bounded for−∞ < n <∞. The obvious such f are the functionsexp(a + bt)i, where a, b are constants. Are theseall? Henry worked on this question jointly withArne Beurling during his stay in Uppsala in 1950–51. Making use of a very ingenious argument Henryfound back in the States, they jointly showed thatthe answer is Yes [2].

In a related paper [3], Henry classified iso-morphisms of group algebras of certain abeliantopological groups.

During the next decade, Henry studied the fol-lowing problem: Let µ be a finite positive measureon [0,2π]. Form the weighted L2 space on this,with weight µ. Under what conditions on µ isthe conjugation operator on this space a boundedoperator? Jointly with Gabor Szegö, Henry founda necessary and sufficient condition on µ [4]. Thisresult has been important in further developmentsin real analysis.

Moving to a different terrain, Henry and FrankQuigley made a contribution to the study offunction algebras. They gave some results on max-imal function algebras and found conditions on asmooth closed curve γ in Cn in order that everycontinuous function onγ can be uniformly approx-imated by polynomials in the complex coordinates[5].

AmongHelson’smost influentialpaperswere [6]and [7], written jointly with David Lowdenslager.They deal with a generalization of classical har-monic analysis from the circle to a class of compactabelian topological groups. Observe that the inte-gers Z form a discrete, ordered abelian group withdual group the circle Γ . The characters of Γ are thefunctions exp(inθ) on Γ , labeled by n in Z . Usingonly half of the characters, we can form functions∑∞

0 cn exp(inθ), whose Fourier coefficients withnegative index are 0. Such functions are bound-ary functions of holomorphic functions in the unitdisk. The closure of these functions in the Lp normon the circle, using the measure dθ, are the Hardyspaces Hp. A beautiful, powerful theory of theseHardy spaces was developed in the 1920s and1930s, based on the classical theorems of Hardy,F. and M. Riesz, Szegö, Nevanlinna, Beurling, andothers.

In the 1950s people started to extend Fourierseries theory by replacing the circle group by acompact abelian group G with suitably ordereddual group and using the characters χλ on G,labeled by λ in the dual group: the counterpartsof the series

∑∞

0 cn exp(inθ) on Γ are the series∑λ≥0 cλχλ, where λ ≥ 0 is taken in the sense of the

ordering on the dual group of G.Using Haar measure on G to replace the mea-

sure 12πdθ on Γ , one then definesHp spaces onG to

replace the Hardy spaces on Γ . An example is pro-vided by takingG to be the 2-torus, with dual groupthe group Z+Z , and (p, q) > 0 in Z+Z if (p, q) liesabove the line n+αm = 0, where α is a fixed pos-itive irrational number. The characters χλ now arethe functions exp(inθ)exp(imφ) on the torus G.

Henry with violin bow,Berkeley, 1980s.

Helson and Low-denslager succeededin generalizing classi-cal Hp theory to thisnew context in a re-markably elegant andcomplete way. Theirwork had importantconsequences both forstochastic processesand for the study of aclass of commutativeBanach algebras, the“sup norm” algebras.Kenneth Hoffman in[8] was able to use their ideas to make significantprogress in this field.

I myself came to interact with their work inmy effort to solve a problem in the theory ofsup norm algebras posed by the work of AndrewGleason in [9]. Gleason had introduced the notionof “parts” in the maximal ideal space of a sup normalgebra and conjectured the existence of complexanalytic structure in parts that are not a singlepoint. I was able, using methods from the Helson–Lowdenslager theory, to prove this conjecture forthe case of “Dirichlet” algebras. These are supnorm algebras on a compact space X such thatthe real parts of the functions in the algebra areuniformly dense in the real continuous functionson X [10].

My own personal connection with Henry goesback a long way. We were classmates in theHarvard class of 1947, and for some years in theearly 1950s, we were fellow instructors at Yale. Inthat period we shared an apartment in New Haven.Henry was a quite competent cook, and I was aloyal assistant, and we ran some quite successfuldinner parties. In the spring of 1952 I got married,and Christine and I needed an apartment. Henry,very generously, was willing to move out and allowme to trade apartment mates. He also said to meat that time, “You are lucky, John. She thinks youare funny.” I took it as a compliment.


References[1] H. Helson, Solution of a problem of Steinhaus, Proc.

Nat. Acad. Sci. 40 (1954).[2] A. Beurling and H. Helson, Fourier–Stieltjes

transforms with bounded powers, Math. Scand. 1

(1953).[3] H. Helson, Isomorphisms of abelian group alge-

bras, Ark. Math. 2 (1952).[4] H. Helson and G. Szegö, A problem in prediction

theory, Ann. Mat. Pura Appl. 51 (1960).[5] H. Helson and F. Quigley, Existence of maximal

ideals in algebras of continuous functions, Proc.

AMS 8 (1957).[6] H. Helson and D. Lowdenslager, Prediction the-

ory and Fourier series in several variables, Acta

Math. 99 (1958).[7] H. Helson and D. Lowdenslager, Prediction the-

ory and Fourier series in several variables, II, Acta

Math. 106 (1961).[8] K. Hoffman, Analytic functions and log-modular

Banach algebras, Acta Math. 108 (1962).[9] A. Gleason, Function Algebras, Seminars on

Analytic Functions, IAS, Princeton, 1957.[10] J. Wermer, Dirichlet algebras, Duke Math. J. 27

(1960).

William A. VeechHenry Helson was a wonderful human being. An

invitation to contribute to this memorial articleimmediately brought to mind a very importantepisode from the early part of one career.

After spending a first academic year, in this case1966–67, in the legendary permanent temporarybuilding, T4, a newcomer to Berkeley would berelocated to Campbell Hall, as often as not in theoffice of a permanent faculty member on leavefor the year. In 1967–68 Jack Feldman’s officewas made available thanks to his being on leavefor a year in Russia. However, Jack cut short hisleave and returned to Berkeley at midyear withthe prospect of being cramped in a small officewith a visitor. While Jack accepted this impositionwith grace, Henry Helson saved the day with anunexpected invitation to join him in his muchlarger office. This was a little disconcerting toa young mathematician who barely knew Henryand who, having read Helson, especially Helson-Lowdenslager, in his graduate school days, hadviewed Henry as something of a hero.

During the several months of joint officing,my “hero” became a great friend. Henry offeredconstant encouragement to an officemate whowas absorbed, even fixated, with a problem aboutZ2-skew products. The relaxed atmosphere cre-ated by daily conversations with Henry aboutall matters, mathematical and nonmathematical,probably contributed as much to the solution asdid my own efforts. Thank you and Godspeed,Henry!

William A. Veech is professor of mathematics at Rice

University. His email address is [email protected].

William Arveson

Helson and Subdiagonal OperatorAlgebrasCommutative Origins

Henry Helson is known for his work in harmonicanalysis, function theory, invariant subspaces, andrelated areas of commutative functional analy-sis. I don’t know the extent to which Henryrealized, however, that some of his early workinspired significant developments in noncommu-tative directions—in the area of nonself-adjointoperator algebras. Some of the most definitive re-sults were obtained quite recently. I think he wouldhave been pleased by that—while vigorously dis-claiming any credit. But surely credit is due; and inthis note I will discuss how his ideas contributed tothe noncommutative world of operator algebras.

It was my good fortune to be a graduate studentat UCLA in the early 1960s, when the place wasbuzzing with exciting new ideas that had grownout of the merger of classical function theoryand the more abstract theory of commutativeBanach algebras as developed byGelfand,Naimark,Raikov, Silov, and others. At the same time, theemerging theory of von Neumann algebras andC∗-algebras was undergoing rapid and excitingdevelopment of its own. One of the directionsof that noncommutative development—though itwent unrecognized for many years—was the roleof ergodic theory in the structure of von Neumannalgebras that was pioneered by Henry Dye [Dye59],[Dye63]. That Henry would become my thesisadvisor. I won’t say more about the remarkabledevelopment of noncommutative ergodic theorythat is evolving even today, because it is peripheralto what I want to say here. I do want to describethe development of a class of nonself-adjointoperator algebras that relates to analytic functiontheory, prediction theory, and invariant subspaces:subdiagonal operator algebras.

It is rare to run across a reference to NorbertWiener’s book on prediction theory [Wie57] in themathematical literature. That may be partly be-cause the book is directed toward an engineeringaudience and partly because it was buried as aclassified document during the war years. Like allof Wiener’s books, it is remarkable and fascinating,but not an easy read for students. It was inspira-tional for me and was the source from which I hadlearned the rudiments of prediction theory that Ibrought with me to UCLA as a graduate student.Wiener was my first mathematical hero.

Dirichlet algebras are a broad class of functionalgebras that originated in efforts to understand

William Arveson is professor emeritus at Univer-

sity of California, Berkeley. His email address is

[email protected].


the disk algebra A ⊆ C(T) of continuous complex-valued functions on the unit circle whose negativeFourier coefficients vanish. Several paths throughharmonic analysis or complex function theory orprediction theory lead naturally to this functionalgebra. I remind the reader that a Dirichlet algebrais a unital subalgebraA ⊆ C(X) (X being a compactHausdorff space) with the property that A+A∗ ={f + g : f , g ∈ A} is sup-norm-dense in C(X);equivalently, the real parts of the functions in Aare dense in the space of real valued continuousfunctions. One cannot overestimate the influenceof the two papers of Helson and Lowdenslager([HL58], [HL61]) in abstract function theory andespecially Dirichlet algebras. Their main resultsare beautifully summarized in Chapter 4 of KenHoffman’s book [Hof62].

Along with a given Dirichlet algebra A ⊆ C(X),one is frequently presented with a distinguishedcomplex homomorphism

φ : A→ C,

and because A + A∗ is dense in C(X), one findsthat there is a unique probability measure µ onX (of course I really mean unique regular Borelprobability measure) that represents φ in thesense that

(1.1) φ(f) =

∫

Xf dµ, f ∈ A.

Here we are more concerned with the closelyrelated notion of weak∗-Dirichlet algebra A ⊆

L∞(X, µ), in which uniform density of A + A∗ inC(X) is weakened to the requirement that A+A∗

be dense inL∞(X, µ) relative to the weak∗-topologyof L∞. Of course we continue to require that thelinear functional (1) should be multiplicative on A.

Going Noncommutative

Von Neumann algebras and C∗-algebras of opera-tors on a Hilbert space H are self-adjoint—closedunder the ∗-operation of B(H). But most opera-tor algebras do not have that symmetry; and fornonself-adjoint algebras, there was little theoryand few general principles in the early 1960sbeyond the Kadison-Singer paper [KS60] on trian-gular operator algebras (Ringrose’s work on nestalgebras was not to appear until several yearslater).

While trolling the waters for a thesis topic, I wasstruck by the fact that so much of prediction theoryand analytic function theory had been captured byHelson and Lowdenslager, while at the same timeI could see diverse examples of operator algebrasthat seemed to satisfy noncommutative variationsof the axioms for weak∗-Dirichlet algebras. Therehad to be a way to put it all together in anappropriate noncommutative context that wouldretain the essence of prediction theory and containimportant examples of operator algebras. I worked

on that idea for a year or two and produced aPh.D. thesis in 1964—which evolved into a moredefinitive paper [Arv67]. At the time I wantedto call these algebras triangular; but Kadison andSinger had already taken the term for their algebras[KS60]. Instead, these later algebras became knownas subdiagonal operator algebras.

Here are the axioms for a (concretely acting)subdiagonal algebra of operators in B(H). It is apair (A, φ) consisting of a subalgebraA of B(H)that contains the identity operator and is closed inthe weak∗-topology of B(H), all of which satisfy

SD1: A+A∗ is weak∗-dense in the von Neu-mann algebra M it generates.

SD2: φ is a conditional expectation, mappingMonto the von Neumann subalgebraA∩A∗.

SD3: φ(AB) = φ(A)φ(B), for all A,B ∈A.

What SD2 means is that φ should be an idem-potent linear map from M onto A ∩ A∗ thatcarries positive operators to positive operators, iscontinuous with respect to the weak∗-topology,and is faithful in the sense that for every positiveoperator X ∈M, φ(X) = 0 =⇒ X = 0.

We also point out that these axioms differslightly from the original axioms of [Arv67] butare equivalent when the algebrasare weak∗-closed.

Examples of subdiagonal algebras:

(1) The pair (A,φ),A being the algebra of alllower triangular n × n matrices, A∩A∗

is the algebra of diagonal matrices, andφ :Mn →A∩A∗ is the map that replacesa matrix with its diagonal part.

(2) Let G be a countable discrete group whichcan be totally ordered by a relation ≤ sat-isfying a ≤ b =⇒ xa ≤ xb for all x ∈ G.There are many such groups, includingfinitely generated free groups (commuta-tive or noncommutative). Fix such an order≤ on G and let x ֏ ℓx be the natural (leftregular) unitary representation of G on itsintrinsic Hilbert space ℓ2(G), let M be theweak∗-closed linear span of all operatorsof the form ℓx, x ∈ G, and let A be theweak∗-closed linear span of operators ofthe form ℓx, x ≥ e, e denoting the identityelement of G. Finally, let φ be the state ofM defined by

φ(X) = 〈Xξ, ξ〉, X ∈M, ξ = χe.

If we view φ as a conditional expectationfromM to the algebra ofscalarmultiplesofthe identity operator by way ofX ֏ φ(X)1,then we obtain a subdiagonal algebra ofoperators (A,φ).

(3) There are natural examples of subdiagonalalgebras in II1 factors M that are basedon ergodic measure-preserving transfor-mations that will be familiar to operatoralgebraists (see [Arv67]).


In order to formulate the most importantconnections with function theory and predic-tion theory, one requires an additional propertycalled finiteness in [Arv67]: there should be a dis-tinguished tracial state τ on the von NeumannalgebraM generated byA that preservesφ in thesense that τ ◦φ = τ . Perhaps we should indicatethe choice of τ by writing (A,φ, τ) rather than(A,φ), but we shall economize on notation by notdoing so.

Recall that the simplest form of Jensen’s in-equality makes the following assertion aboutfunctions f ≠ 0 in the disk algebra: log |f | is in-

tegrable around the unit circle, and the geometricmean of |f | satisfies(2.1)

|1

2π

∫

T

f (eiθ) dθ| ≤ exp1

2π

∫

T

log |f (eiθ)|dθ.

In order to formulate this property for subdiag-onal operator algebras, we require the determinantfunction of Fuglede and Kadison [FK52]—definedas follows for invertible operators X in M:

∆(X) = expτ(log |X|),

|X| denoting the positive square root of X∗X.There is a natural way to extend the definition of∆ to arbitrary (noninvertible) operators in M. Forexample, when M is the algebra of n× n complexmatrices and τ is the tracial state, ∆(X) turns outto be the positive nth root of |detX|.

Corresponding to (2.1), we will say that a finitesubdiagonal algebra (A, φ) with tracial state τsatisfies Jensen’s inequality if

(2.2) ∆(φ(A)) ≤ ∆(A), A ∈A,

and we say that (A,φ) satisfies Jensen’s formulaif

(2.3) ∆(φ(A)) = ∆(A), A ∈A∩A−1.

It is not hard to show that (2.2) =⇒ (2.3).Finally, the connection with prediction theory

is made by reformulating a classical theorem ofSzegö, one version of which can be stated asfollows: For every positive function w ∈ L1(T, dθ)one has

inff

∫

T

|1+ f |2w dθ = exp

∫

T

logw dθ,

f ranging over trigonometric polynomials of theform a1eiθ + · · · + aneinθ . In the noncommutativesetting, there is a natural way to extend thedefinition of determinant to weak∗-continuouspositive linear functionals ρ onM, and the properreplacement for Szegö’s theorem turns out to bethe following somewhat peculiar statement: Forevery weak∗-continuous state ρ on M,

(2.4) inf ρ(|D +A|2) = ∆(ρ),the infimum taken over D ∈ A∩A∗ and A ∈ Awith φ(A) = 0 and ∆(D) ≥ 1.

In the 1960s there were several important ex-amples for which I could prove properties (2.2),(2.3), and (2.4), but I was unable to establish themin general. The paper [Arv67] contains the resultsof that effort. Among other things, it was shownthat every subdiagonal algebra is contained in aunique maximal one and that maximal subdiago-nal algebras admit factorization: Every invertiblepositive operator in M has the form X = A∗A forsome A ∈ A∩A−1. Factorization was then usedto show the equivalence of these three propertiesfor arbitrary maximal subdiagonal algebras.

Resurrection and ResurgenceI don’t have to say precisely what maximalitymeans because, in an important developmenttwenty years later, Ruy Exel [Exe88] showed thatthe concept is unnecessary by proving the fol-lowing theorem: Every (necessarily weak∗-closed)subdiagonal algebra is maximal. Thus, factoriza-tion holds in general and the three properties (2.2),(2.3), and (2.4) are always equivalent.

Encouraging as Exel’s result was, the theoryremained unfinished because no proof existedthat Jensen’s inequality, for example, was true ingeneral. Twenty more years were to pass beforethe mystery was lifted. In penetrating work ofLouis Labuschagne and David Blecher [Lab05],[BL08], [BL07a], [BL07b], it was shown that notonly are the three desired properties true ingeneral but virtually all of the classical theoryof weak∗-Dirichlet function algebras generalizesappropriately to subdiagonal operator algebras.

I hope I have persuaded the reader that thereis an evolutionary path from the original ideasof Helson and Lowdenslager, through forty yearsof sporadic progress, to a finished and eleganttheory of noncommutative operator algebras thatembodies a remarkable blend of complex functiontheory, prediction theory, and invariant subspaces.

Henry with Jun-ichi Tanaka, Berkeley campus,May 1984.


References[Arv67] W. Arveson, Analyticity in operator algebras,

Amer. J. Math. 89(3) (1967), 578–642.[BL07a] D. Blecher and L. Labuschagne, Noncom-

mutative function theory and unique exten-sions, Studia Math. 178(2) (2007), 177–195,math.OA/0603437.

[BL07b] D. Blecher and L. Labuschagne, Von NeumannalgebraicHp theory, In Function Spaces, volume435 of Contemp. Math., pages 89–114, Amer.Math. Soc., 2007, math.OA/0611879.

[BL08] D. Blecher and L. Labuschagne, A Beurlingtheorem for noncommutative Lp, J. Oper. Th. 59

(2008), 29–51, math.OA/0510358.[Dye59] H. A. Dye, On groups of measure preserving

transformations. I, Amer. J. Math. 81(1) (1959),119–159.

[Dye63] , On groups of measure preservingtransformations. II, Amer. J. Math. 85 (1963),551–576.

[Exe88] R. Exel, Maximal subdiagonal algebras, Amer. J.

Math. 110 (1988), 775–782.[FK52] B. Fuglede and R. V. Kadison, Determinant

theory in finite factors, Ann. Math. 55 (1952),520–530.

[HL58] H. Helson and D. Lowdenslager, Predictiontheory and Fourier series in several variables,Acta Math. 99 (1958), 165–202.

[HL61] , Prediction theory and Fourier series inseveral variables. II, Acta Math. 106 (1961), 175–213.

[Hof62] K. Hoffman, Banach Spaces of Analytic Func-

tions, Prentice-Hall, Englewood Cliffs, 1962.[KS60] R. V. Kadison and I. M. Singer, Triangular

operator algebras, Amer. J. Math. 82 (1960),227–259.

[Lab05] L. Labuschagne, A noncommutative Szegötheorem, Proc. Amer. Math. Soc. 133 (2005),3643–3646.

[Wie57] N. Wiener, Extrapolation, Interpolation, and

Smoothing of Stationary Time Series, Technol-ogy Press of M.I.T., John Wiley and Sons, NewYork, 1957.

Kathy MerrillI first met Henry Helson on a bus commutingto a regional AMS meeting in Salt Lake City onespring morning in 1983. Later that morning, Henrytold the group attending his talk in our specialsession that I knew more about cocycles of anirrational rotation than he did. This of course wasnot true, but I was to learn that it was typicalof Henry’s kindness and generosity. I was justfinishing my Ph.D. at the University of Coloradoand was on the job market, hungry for praisebut not so deserving of it. Henry, on the otherhand, was already well known for using cocyclesto mine information about invariant subspaces offunctions on compact groups with ordered duals

Kathy Merrill is professor emerita of mathematics at

Colorado College. Her email address is kmerrill@

coloradocollege.edu.

[2]. Many of his results had important applicationsto ergodic theory and Diophantine approximationas well.

Our conversations in Salt Lake City began amathematical friendship that added joy as wellas practical support to the beginning of my ca-reer. Henry helped me get a (non-tenure-track) joboffer at Berkeley, which was quite helpful to myconfidence even though I had to turn it down.He went on to grace me with a regular stream ofcorrespondence over the next ten years, includingconjectures, examples, references, and odd bitsof mathematical beauty. One result of this inter-change was a joint publication [4], leading to myenviable Helson number of 1. Henry also wrote tome about his philosophy of teaching: “It is simplynot true that ordinary students are incapable ofabstraction, and must be jollied with a constantstream of pseudo-applications.” He sent reassur-ance when I expressed self-doubt. When I told

Henry before a shave, Evans Hall, Berkeley,1984 (from George Bergman’s collection, withhis permission).

Henry after a shave, Evans Hall, Berkeley, 1984(from George Bergman’s collection, with hispermission).


him I was afraid I would never have another newidea, he told me he always feared that each paperhe wrote would be his last, but so far it never was.When I apologized for being slow to understandsomething he had sent me, he assured me that hewas even slower. He said that while he could writemathematical papers, he could not read them.

I spent part of my first sabbatical in 1991 visit-ing Henry at Berkeley. He arranged for me to speakin the functional analysis seminar there and after-ward chided me for making cocycles appear toosimple. We spoke of difficulties faced by women inmathematics, and particularly of the controversybrewing at Berkeley at the time. Henry’s perspec-tive, as fit his character, was thoughtful and kind,avoiding angry rhetoric and supporting a valuedcolleague. During that visit, he introduced me tohis student Herbert Medina, with whom I begana fruitful and pleasurable collaboration. My workwith Herbert and fellow coauthor Larry Baggettsoon moved from cocycles to wavelets. HenryHelson’s work was lurking under the central ideathere as well. Our approach to wavelets dependedon applying the multiplicity function of a spectralmeasure to the representation of the integers astranslations in the core subspace of a multireso-lution analysis. In his book, The Spectral Theorem[3], Henry gives a simple and elegant account ofthe multiplicity function, using range functionsinstead of direct integrals. Indeed, a referee forour wavelet paper [1] commented that he had toread Henry Helson to understand our approach.Many other wavelet researchers have also foundHelson’s range functions a useful tool for under-standing the shift-invariant subspaces associatedwith wavelets and multiresolution analyses.

References[1] L. Baggett, H. Medina, and K. Merrill, Generalized

multi-resolution analyses and a construction proce-dure for all wavelet sets in Rn, J. Fourier Anal. Appl.

5 (1999), 563–573.[2] H. Helson, Analyticity on compact abelian groups, in

Algebras in Analysis, ed. J. H. Williamson, AcademicPress, 1975, 1–62.

[3] , The Spectral Theorem, Springer-Verlag,Berlin, 1986.

[4] H. Helson and K. Merrill, Cocycles on the circle II,in Operator Theory: Advances and Applications 28,Birkhauser Verlag Basel, 1988, 121–124.

Jun-ichi Tanaka

Recollections of Henry HelsonIn early spring of 1980, Ravenna and Henrystopped at Tokyo on their way to Calcutta. M. Ha-sumi and I met them at the airport, which was the

Jun-ichi Tanaka is professor of mathematics at the

Waseda University. His email address is jtanaka

@waseda.jp.

first time I saw Henry in person. At that time I wasstudying function algebras and got interested infunction theory associated with flows, developedby F. Forelli and P. Muhly. This research directionwas pioneered primarily by the series of works byHelson and Lowdenslager.

A few years later, 1984–1985, I had a chanceto stay at Berkeley as a postdoc, and Henry waswilling to take me in as an advisee. It was afantastic experience, although my poor Englishcaused a lot of trouble. I joined in the Arvesonseminar and occasionally attended colloquia withHenry. I remember vividly D. Sarason’s lecture onthe Bieberbach conjecture, P. Jones’s one on thecorona problem, and so on.

I felt rather shy around Henry, but he wasalways kind and friendly. He would look at what Iwas reading over my shoulder, then explain to merelated topics. I felt very lucky to learn functiontheory on groups directly from one of its founders.Henry sometimes talked about mathematicianswhom he had met in the past. I was surprised alittle to learn that he knew S. Kakutani and hisfamily well.

Helson and Lowdenslager developed their the-ory on a compact abelian group K dual to adense discrete subgroup Γ of R. (This setting ismore general than it may look.) Let σ be thenormalized Haar measure on K. The Hardy spaceHp(σ),1 ≤ p ≤ ∞, is defined to be the space ofall functions in Lp(σ) whose Fourier coefficientswith negative indices vanish. Since H∞(σ) is aweak*-Dirichlet algebra in L∞(σ), we are led tostudy the structure of invariant subspaces, whichcan be investigated in detail by virtue of the groupstructure. An ergodic flow x→ x+et is introducednaturally onK, called an almost periodic flow. Thenφ lies in Hp(σ) if and only if for almost every x,t → φ(x+ et) lies in Hp(dt/π(1+ t2)), the closureof H∞(R) in Lp(dt/π(1 + t2)). This observationenables us to extend such Hardy spaces to thecase of general ergodic flows. Recall that a closedsubspace M of Lp(σ) is invariant if H∞(σ) ·M iscontained in M. Denote by H

p0 (σ) the subspace of

all functions in Hp(σ) with mean value zero. Thisinvariant subspaceH

p0 (σ) plays an important role.

Dealing with invariant subspaces, we may restrictour attention to the case of p = 2. There is acorrespondence between invariant subspaces andcertain unitary functions on K×R, called cocycles.

Henry raised repeatedly the problem of whetherevery invariant subspace can be represented asthe smallest one containing a definite element,the so-called single generator problem (SGP). Andhe showed me a lot of equivalent and sufficientconditions. In his survey article [5, 1975], he writes“Is every simply invariant subspace ofL2 generatedby one of its elements? HasH2

0 a single generator?These problems are old, and apparently untouchedby all that precedes. Answers will be interesting


and probably difficult.” Henry once told me heraised the problem in 1957, but it seems to appearin [6, p. 183] for the first time, in an equivalentform related to evanescent stochastic processes.The difficulty of SGP seemed to center on the caseof H2

0(σ).This case had been the main subject of my

research for a long time. (Yes, this sentence iswritten in the past perfect tense. I believe thatI finally succeeded recently in settling it.) Theanswer is negative in the context of Helson andLowdenslager: In the case of almost periodic flows,H2

0 has no single generator, and there are manykinds of simply invariant subspaces of L2 withno generator. In view of these facts it may soundstrange to say that it can be shown that there is anergodic flow on which H2

0 has a single generator.Thus, in the setting of weak*-Dirichlet algebras, itis meaningful to characterize the extreme pointsof the unit ball of H1

0 , which is due to Gamelin.Henry seemed to be very satisfied with this report.

In the process of investigating SGP, I learned alot of things and had some results on flows. M.Nadkarni once wrote me that he had also gotteninterested in SGP soon after he finished his Ph.D.in 1965, and had obtained a number of resultsthat he had been able to publish. We seemed toshare the same experience in this regard.

Henry was very encouraging in spite of myrather slow progress and was always pleasedwhen I could obtain new ideas to attack theproblem. As an example of by-products of myendeavor, let me explain how one can settle, byusing my ideas, both questions of Forelli’s statedin his report [2] for the Nice Congress, which sitsback-to-back with Henry’s [3]. At the end of the1980s, I was eager to look for an ergodic flowon which H2

0 is singly generated. The shift on theintegers Z induces a homeomorphism τ on theStone-Cech compactification βZ of Z. Let X be thequotient space of βZ× [0,1], by identifying (ω,1)with (τ(ω),0). Then we have a continuous flowon X being an extension of the translation onR. I am still interested in this flow in connectionwith the corona theorem. Indeed, by a normalfamilies argument, one can show that H∞(R) isisometrically isomorphic to H∞(X), the algebra ofall bounded Borel functions which are analytic oneach orbit. Then X × (0,∞) represents concretelya certain part of the fiber of the maximal idealspace of H∞(R). Notice that the upper half-planeR × (0,∞) is a dense open subset of X × (0,∞).There are a lot of representing measures forH∞(X) that are invariant, which suggests thatthere may exist a relation between the coronaproblem and the individual ergodic theorem, viaWiener’s Tauberian theorem. Somehow, I felt asif I understood vaguely why Kakutani had saidoccasionally, “There should be a simple directproof of the corona theorem” (I learned Kakutani’s

remark from O. Hatori). In any event, I had notonly a desired flow in a quotient of X but also anegative answer to one of Forelli’s questions: Thereis a minimal flow on which the induced uniformalgebra is not a Dirichlet algebra, which holds oneach minimal set inX. The other question was alsosettled in the course of looking into the absolutevalues of single generators. It runs “What makesa positive measure the total variation measure ofan analytic measure?” (This is also a title of oneof Forelli’s papers [1].) The answer turned outto be: the necessary and sufficient condition forthis to be true is that the distant future equals{0}. Of course, both results heavily relied on thepreceding work of Muhly.

What do you imagine from these key words:“analytic almost periodic functions”, “the behavioraround infinity”, and “the distribution of zeros”?One always confronts these phrases when dealingwith the structure of invariant subspaces in thealmost periodic setting. In the autumn of 1984,the rumor that a Japanese mathematician stayingin France at the time had solved the RiemannHypothesis (RH) spread fast in the Berkeley math-ematics community. Henry seemed to get excitedabout it and wanted to know about this mathe-matician and the group of number theorists inJapan, although I did not know anything at allabout them. As it happened, the rumor turned outto be premature, and the story died down soonthereafter. I felt that just like A. Beurling, Henrymight have had RH as one of his problems heshould challenge, as Dirichlet series was central tohis life’s work, and Cassels’s book was always hisfavorite. He once provided another proof of theconvergence of ζ(1)−1, which seems roughly tohave the depth of the prime number theorem [4].I learned from him how to extend ζ(s) and ζ(s)−1

to outer functions in H2(T∞), and the similaritybetween Carleson’s convergence theorem and theconvergence of ζ(s)−1 on the critical strip, whichis equivalent to RH.

Although I had no chance to visit Berkeleyagain after that, we became close friends, keepingin touch until his death. Henry was a good corre-spondent and a masterly writer. After discussingmathematical ideas, he usually wrote about therecent situation of himself and his family. I re-member how he was delighted when his tenderfor a fine old Italian instrument was accepted (hewas a well-known violinist) and how he was happywhen his elder son David got married to an Englishlady in London. He also reported his regrets whenit became too hard to continue making wine byhimself.

Last autumn, in connection with Rudin’s au-tobiography [7], we corresponded about polydiscalgebras. The closed subalgebra of H∞(R) ob-tained by restricting A(T2) to one orbit is thesimplest one for which the corona theorem fails. A


Henry with camera, Berkeley, early 1990s.

cocycle argument provides easily the fact that anentire inner function in A(Tn) is a monomial (thetheorem of Bojanic and Stoll), as it gives rise to acocycle of the form exp(iλ t), a weight at infinity.

At the end of last year, I sent season’s greetingsas usual, together with my new idea to attackthe corona problem on polydiscs, depending ontensor products. However, I had no answer fromhim, and his poor health lay heavily on my mind.

Then, early in the new year, I received thesad news from Hasumi and Nadkarni one afteranother.

I feel so lucky to have met such a first-rate mindat an early stage of my career and to have receivedhis affectionate interest in my research. He will begreatly missed by everyone who has known him.

References[1] F. Forelli, What makes a positive measure the total

variation measure of an analytic measure? J. London

Math. Soc. (2) 2 (1970), 713–718.[2] , Fourier theory and flows, Actes Congrès

Intern. Math. 2 (1970), 475–476.[3] H. Helson, Cocycles in harmonic analysis, Actes

Congrès Intern. Math. 2 (1970), 477–481.[4] , Convergence of Dirichlet series, in L’Analyse

Harmonique dans le Domaine Complexe, LectureNotes in Mathematics, 336, Springer, 1973, 153–160.

[5] , Analyticity on compact abelian groups, in Al-

gebras in Analysis, Academic Press, London, 1975,1–62.

[6] H. Helson and D. Lowdenslager, Prediction theoryand Fourier series of several complex variables. II,Acta Math. 106 (1961), 175–213.

[7] W. Rudin, The Way I Remember It, Amer. Math. Soc.,1997.

John E. McCarthyHenry Helson was a transforming influence onme. In my last year of graduate school, his workon the Smirnov class [5] had led him to ask the

John E. McCarthy is professor of mathematics at Wash-

ington University. His email address is mccarthy

@math.wustl.edu.

following question: What functions in the Hardyspace H2 are in the range of every coanalyticToeplitz operator?

In other words, what functions f inH2 have theproperty that for every nonzero functionm inH∞,there is a function g in H2 such that

(1) Tmg := PL2→H2 mg = f .

D. Sarason observed that such functions musthave a certain regularity. Indeed, let m(z) = 1− zand g(z) =

∑∞

n=0 anzn in (1). Then

f (z) =∞∑

n=0

(an − an+1)zn.

So, in particular, f must have a convergent Fourierseries at 1.

Replacing m by other functions, one can getmore and more regularity conditions that f needsto satisfy. Conversely, one can show that if thecoefficients of f decay rapidly enough, then itwill be in the range of every coanalytic Toeplitzoperator. The trick was to find the right rate ofdecay. Several people worked on this problemaround the time 1988–89.

Why was this problem transformative? Becauseof the way Henry discussed it with me. Eventhough I was just a graduate student, he treatedme as an equal, with courtesy and respect. He wasgenuinely interested in what I had to say, or if Imade ε progress. This gave me the confidence Ineeded to think I might be able to solve it, andin the summer of 1989, just after I graduatedfrom Berkeley, I succeeded [7]. My work reliedheavily on earlier results of N. Yanagahira [8], butnonetheless I was proud of it. I saw that paper, farmore than my dissertation, as my transition fromstudent to mathematician.

After leaving Berkeley, I kept up a frequentcorrespondence with Henry. He gave me a lot ofencouragement and advice both mathematical andpersonal. We wrote one small paper together [6],but mainly wrote letters about what we were think-ing about. (My missives were mainly electronic; hisfrequently handwritten.) He had strong opinionsabout most things, including grammar and style.To this day, I cannot use the word “factorization”,since, as he pointed out, there is no verb “tofactorize”; the correct noun is “factoring”.

One piece of advice he gave me was “Everymathematician should look at the ζ-function. It isa mirror.” I looked, and having spent much of myearly career studying the Hardy space and relatedobjects, I decided to study spaces of Dirichletseries

∑ann−s with square-summable coefficients.

This was not a new idea—a beautiful paper on thisspace had been written by Hedenmalm, Lindqvist,and Seip [1]—but it gave me a mental hook topull myself into this fascinating area. Henry him-self had pioneered the introduction of functionalanalysis into the study of Dirichlet series [2, 3, 4],


and he returned to the subject in the past fewyears. The last letter I received from him was aboutthe generalization of Nehari’s theorem on bound-edness of Hankel operators to infinitely manyvariables. (As Harald Bohr had observed, Dirichletseries provide a convenient setting for studyingfunctions of infinitely many variables—essentially,for each prime p, the function p−s plays the roleof a variable.)

When I think of Helson’s work, the adjectivethat comes to mind is “elegant”. His proofs werealways clever, insightful, and graceful. As a person,he was honest and clear-seeing. Like all those whowere privileged to know him, I shall miss him.

References[1] H. Hedenmalm, P. Lindqvist, and K. Seip, A Hilbert

space of Dirichlet series and systems of dilatedfunctions in L2(0; 1), Duke Math. J. 86 (1997), 1–37.

[2] H. Helson, Convergent Dirichlet series, Ark. Mat. 4

(1963), 501–510.[3] , Foundations of the theory of Dirichlet series,

Acta Math. 118 (1967), 61–77.[4] , Compact groups and Dirichlet series, Ark.

Mat. 8 (1969), 139–143.[5] , Large analytic functions, II. In Analysis and

Partial Differential Equations (Cora Sadosky, ed.),Marcel Dekker, Basel, 1990.

[6] H. Helson and J. E. McCarthy, Continuity of semi-norms. In SLNM, (B. S. Yadav and D. Singh, eds.) 1511,88–90, Springer, Berlin, 1992.

[7] J. E. McCarthy, Common range of co-analyticToeplitz operators. J. Amer. Math. Soc. 3(4) (1990),793–799.

[8] N. Yanagihara, Multipliers and linear functionalsfor the class N+, Trans. Amer. Math. Soc. 180 (1973),449–461.

Herbert Medina

Reminiscences of a Former StudentHenry Helson was a mathematician who wantedhis students to learn to appreciate, to work on, andto love “true analysis”. He was eager for us to tackleconcrete problems that usually required delicateestimates. He did not believe in analysis thatused “sledgehammer” theorems not requiring theestimates and inequalities that are so importantto an analyst. I remember him telling me that hethought the real interesting problems were notones with general hypotheses (e.g., “let G be atopological group”), but ones in specific domains,function spaces, or with specific operators. Heshared these views with his students, and the wayin which I view mathematics has very much beeninfluenced by Helson’s views.

As far as my specific dissertation work, hewas the mathematician who introduced me to

Herbert Medina is professor of mathematics at Loyola

Marymount University. His email address is hmedina@

lmu.edu.

cocycles and operators that arose from irrationalrotations on the circle and showed me how to usethe spectral theorem to study and classify these.Work on problems related to these topics was thefocus of the first few years of my mathematicalcareer.

There are many memories that I have of Pro-fessor Helson, but I’ll only relate one anecdote. Iremember one time going into his office feelingnot terribly excited about something that I wasreading and communicated this to him. He simplysaid, “We are interested in what we work on.” Inother words, I simply should get to work on themathematics in the paper and that would get meexcited about the topic. He was right! I have usedthat phrase with my own students several times.

Farhad ZabihiI was one of Henry Helson’s last students. I beganthe research on my dissertation in the summerof 1991 amid personal and financial difficultiesthat had troubled me for several years. Helsonwas aware of my circumstances but neverthelesswanted to start working together once he returnedfrom Toulouse in late spring. What followed weretwo years full of excitement and mathematicaldiscovery.

Helson proposed that I work on a problemthat interpolates between the Riemann mappingtheorem and a theorem of Szegö. The Riemannmapping theorem produces an analytic functionon the unit disk whose boundary values lie onthe boundary curve of a simply connected region.Szegö’s theorem, when interpreted geometrically,says that an analytic function in H2 can take itsvalues on different circles centered at the origin.Helson thought these theorems could be marriedtogether, and perhaps one could find analyticfunctions that take their values on prescribedcurves of a general type.

In our initial approach, Helson suggested that Ifind analytic functions whose values lie on curvesgiven by formulas, and he thought we should tryfixed point methods. Indeed, using Banach’s fixedpoint theorem, I was able to solve the problemfor families of lines, hyperbolas, and ellipsesunder strict conditions. Helson was particularlyinterested in the ellipse case because it generalizedSzegö’s theorem (in a narrow sense) and indicatedthat a solution may exist for a more general classof curves.

With much greater effort, I was able to findsolutions for a class of curves that bound convexregions. The proof involved a complicated con-struction of analytic functions, the limit of which

Farhad Zabihi is lead software engineer at Automatic

Data Processing, Inc. and professor of mathematics at

Diablo Valley College. His email address is fzabihi@

dvc.edu.


Henry and Ravenna, Berkeley, circa 2007.

yielded a solution. I was convinced the construc-tion worked, but Helson was not satisfied. Hethought that the proof was unwieldy and compu-tational. “The hardest problems,” he told me, “aresolved with the simplest ideas.” So I continued towork on my proof and found a new approach us-ing ideas from the proof of the Riemann mappingtheorem. It cracked the problem wide open andled to the main theorem in my dissertation.

Helson’s insistence on brevity and simplicityalso carried over to writing the dissertation. Hiswriting style is clean, clear, and engaging, and heexpected no less from his students. It had taken mea year to complete my research, and it took almostas long to simplify the proofs and distill them totheir final form. We published these results, alongwith some extensions that I discovered later, ina joint paper [1]. In the course of writing mydissertation and the paper, Helson taught me asmuch about the art of presentation as he hadabout rigorous research.

By the time I graduated in 1993, the difficultiesI faced early on were mostly behind me. I am sureHelson was in part responsible, for he had made aprofound impression on me. I kept in touch withhim through the years after my graduation. Weoften had lunch at his favorite Indian restaurantin Berkeley, and I later introduced him to Persiancuisine, to his delight. On reflection, he once saidthat he could have been more supportive of hisstudents. I disagreed, as my own debt of gratitudeto Henry Helson is immeasurable.

References[1] Henry Helson and Farhad Zabihi, A geomet-

ric problem in function theory, Illinois Journal of

Mathematics 51, No. 3 (2007), 1027–1034.

AcknowledgmentsUnless otherwise noted, all photos are from theHelson family collection, courtesy of RavennaHelson.

We thank Mary Jennings for her work on thephotos for the article.

Ph.D. Students of Henry Helson (all are fromUC Berkeley)

Frank Forelli Jr. 1961

Michael Cambern 1963

Ernest Fickas 1964

Malcolm Sherman 1964

Stephen Gerig 1966

Iri Yale 1966

Shirley Jackson Jr. 1967

Malayattil Rabindranathan 1968

Udai Tewari 1969

Samuel Ebenstein 1970

Chester Jacewitz 1970

Bonnie Saunders 1978

Geon Choe 1987

Carlos Carbonera 1990

Trent Eggleston 1990

Zachary Franko 1990

Herbert Medina 1992

Adrian Zidaritz 1992

Farhad Zabihi 1993

Nicholas James 1996


a tribute to henry helson - american mathematical … tribute to henry helson donald sarason...

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