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A Survey of Chandler Davis

Man-Duen Choi and Peter Rosenthal* Department of Mathematics Toronto, Ontario, Canada M5S 1Al

Dedicated to Chandler Davis

Submitted by Hans Schneider

ABSTRACT

Chandler Davis is a very unusual, interesting, and complex person. This short sketch, however, is restricted to his contributions to linear algebra and to operator theory, although we begin with a brief biography.

1. INTRODUCTION

Chandler Davis was born on 12 August 1926 in Ithaca, New York, where his father, Horace Bancroft Davis, was then an economics instructor at Cornell. Chandler’s mother, Marian Rubins Davis, was also an economist. Several of Chandler’s paternal ancestors traveled from England to Amer- ica on the Mayflower. The name Davis goes back to Dolor Davis, who landed in 1634. On his mother’s side, Chandler’s ancestors immigrated to America from and Sweden in the 1800s. Chandler was a true “red-diaper baby”: his parents were actively involved in a number of left wing organizations, including the Communist Party U.S.A. Horace wrote a two-volume study of the relationship between Marxism and nationalism, in addition to other academic and popular tracts. Both parents participated in many efforts to organize unions and to combat fascism and racism.

*We are grateful to Carol Kitai and Pat Broughton for their help in preparing this article.

LINEAR ALGEBRA AND ITS APPLICATIONS 208/209:3-18(1994) 3

0 Elsevier Science Inc., 1994 655 Avenue of the Americas, New York, NY 10010 0024-3795/94/$7.00 4 MAN-DUEN CHOI AND PETER ROSENTHAL

They also raised five children; Chandler is the eldest. Political causes and academic jobs led them to live in various places, including Brazil, where young Chandler learned to speak Portuguese. When Chandler’s mother died in 1960, her family and friends started the Marian Davis Scholarship Fund, which provides stipends to socially commit- ted young people working towards undergraduate degrees. Several years ago, a new source of support led to this fund being renamed the Davis-Putter Scholarship Fund; it is still going strong. Chandler received his Ph.D. from in 1950; his thesis, written under the supervision of , was entitled “Lattices of Modal Operators.” He then accepted a position as Instructor at the Univer- sity of Michigan, and began writing papers on linear algebra and other mathematical subjects (see [2-71). Chandler’s normal career path was rudely interrupted several years later by the House Un-American Activities Committee.’ Chandler refused to answer HUACs questions, and, to compound his sin, he based his refusal on the First Amendment (on the grounds of political freedom) rather than on the Fifth Amendment (i.e., on the grounds that it might incriminate him). He knew that a refusal on either ground was likely to end his academic career, and that relying on the “wrong” amendment was likely to lead to the additional punishment of imprisonment. He nonetheless refused to answer as a matter of conscience, in spite of the fact that he had previously left the Communist Party and thus might have been able to save his career if he’d been willing to testify against his former colleagues in the Party. In May 1954, as soon as his testimony was concluded, Chandler was suspended with pay from his instructorship. He was fired shortly afterward, following what University committees considered to be due process. He was also cited for contempt of Congress. He held various jobs over the next six years, including part-time teaching and working for an ad agency. His most satisfactory position during this period was at Mathematical Reviews. Chandler was totally blacklisted by American universities. For example, a small midwest university had no mathematicians at all on its staff. A professor who knew something about mathematics was greatly impressed with Chan- dler’s credentials and easily convinced the dean to hire him. However, the required approval of the higher administrators, previously always routine, was never forthcoming, so neither was the job. In 1960, after appeals against his conviction for contempt of Congress were exhausted, Chandler served six months in jail. Throughout this ordeal, Chandler maintained his research interest in mathematics. He also main- tained his sense of humor. For example, a footnote in his paper on an

‘The House Un-American Activities Committee (HUAC) was one of several congressional committees that investigated persons suspected of communist affiliations. A SURVEY OF CHANDLER DAVIS 5 extremal problem [23], conceived while he was in prison but published afterward, reads

Research supported in part by the Federal Prison System. Opinions expressed in this paper are the author’s and are not necessarily those of the Bureau of Prisons.

Chandler published a very interesting account of his and others’ experiences of McCarthyism: “The Purge,” in A Century of Mathematics in America, Part I, edited by Peter Duren, American Mathematical Society, Providence, 1988. It is remarkably free of bitterness, in spite of the fact that the blacklist by American universities continued even after he got out of prison. The situation in Canada was not nearly as bad as in the U.S., and in 1962 Chandler was appointed to a professorship at the University of Toronto. He became a mainstay of our operator-theory seminars (meeting Mondays from 4 to 6 for more than 25 years) and strongly influenced many students. Chandler has been on the editorial boards of the Proceedings of the American Mathematical Society and Linear Algebra and Its Applications. He has also been a very active member of the American Mathematical Society, serving on various committees, being a member of the Council, and recently completing a term as Vice-President. Chandler remained Professor of Mathematics at Toronto until he reached the mandatory retirement age of 65 in 1992; he is now Professor Emeritus. Chandler’s retirement has not been noticed, however, because he has not really retired in any but the formal sense. He still teaches courses at all levels, supervises Ph.D. students, edits the Mathematical Intelligencer, and is very active in research. Thus Chandler has had and continues to have a full and extremely productive academic life. Chandler’s wife, , has also had a distinguished academic career. She is currently the Henry Charles Lea Professor of History at Princeton University. In addition to receiving wide acclaim for her schol- arly achievements, Natalie became known to the general public for her work as historical consultant on the award-winning film The Return of Martin Gueme and her book on the same subject. Chandler and Natalie have three children. Aaron is a very successful jazz pianist and composer; Toronto-based, he is currently most active with the Holly Cole Trio. Hannah is an anthropologist, specializing in the study of Morocco. She lives in Paris and is coeditor of the journal Mediterraneans. Simone is wo&ng on her Ph.D. in English at Berkeley. Chandler and Natalie have three grandchildren: Max, Gabriel, and Sofia.

2. DOCTORAL STUDENTS

Chandler Davis has supervised the dissertations of a number of doctoral students at the University of Toronto. Those who have finished to date 6 MAN-DUEN CHOI AND PETER ROSENTHAL

(several others are still in progress) and the titles of their theses are: John J. Benedetto The Laplace transform of generalized functions, 1964.

Hugh Barr A general theory of functional calculus, 1968. Artatrana Dash Joint spectrum, joint numerical range and joint spectral sets, 1969. Zdislav V. Kovarik Infinite-dimensional analogues of Gershgorin-type estimates, 1971. Marta A. Pojar Extensions of differentiable functional calculus for operators in Banach spaces, 1972. Man-Duen Choi Positive linear maps on C*-algebras, 1973. Mehdi Radjabalipour Operators with growth conditions, 1973. Brian McEnnis Characteristic functions and the geometry of dila- tion space, 1977.

Len Bos Near optimal location of points for Lagrange interpolation in several variables, 1981.

Bruce A. Chalmers Optimisation and approximation problems from continuous translocations, 1984.

Steve Kirkland Spectral regions for Leslie matrices, 1989. Jerzy Jarosz J-unitary dilation of semigroups, 1990. Doug Farenick The matricial spectrum and range, and C*-convex sets, 1990.

Robb Fry Approximation on Banach spaces, 1994.

3. RESEARCH WORK

Chandler has written interesting articles on academic subjects other than mathematics, such as sociobiology [57]. I n addition, he has written a number of science-fiction stories, some of which have become quite well known. Also, although Chandler’s main mathematical interests have been in linear algebra and operator theory, he has done other interesting mathematics as well. For example, he has made contributions to geometry [5, 14, 17,231, to a topic related to fractals [37], and to the philosophy of mathematics [30,48,74,78]. Given the expected audience of this journal, we concentrate on describing Chandler’s contributions to linear algebra and operator theory. A SURVEY OF CHANDLER DAVIS 7

Even these are too extensive to be covered in a single article of reasonable length, so we content ourselves with surveying a small part of his work.

4. THE EQUATION AX - XB = Y

If A and B are square matrices over C with no common eigenvalues, then, as shown by Sylvester 100 years ago, for each matrix Y of suitable size there is a unique matrix X satisfying the equation AX - XB = Y. Put differently, if ydenotes the operator that sends X into AX - XB, then 77s invertible if A and B have no common eigenvalues. This result was generalized to operators on Banach spaces by Marvin Rosenblum in 1956. The corresponding theorem holds: if A and Z3 are bounded linear operators on a Banach space, and if the spectra of A and B are disjoint, then the operator 9 defined by fi X) = AX - XB is invertible. Chandler Davis has, in collaboration with others, contributed several refinements of this result. One question he considered is the following. Infinite-dimensional linear operators can be onto even if they are not injective. This suggests that the equations AX - XB = Y may always have a solution even in some cases where the spectra of A and B intersect. It requires some terminology to describe the theorem Chandler and his collaborator obtained. The upproxi- mate defect spectrum of A, denoted 08(A), is defined to be

(A E c:A - hlisnotonto}.

The approximate point spectrum of B, denoted u,(B), is defined as

{A E C: B - hZ is not injective or does not have closed range}.

THEOREM 4.1 [49]. Suppose A and B are bounded linear operators on Hilbeti space. Then the equation AX - XB = Y has a solution Xfor every Y if and only if q(A) and a,(B) are disjoint.

This variant of Rosenblum’s theorem has found several applications (to Herrero’s study of similarity orbits, for example). The proof given in [49] involves the beginnings of a spectral mapping theorem for approximate point spectra. Chandler continued this work with another collaborator [50], producing a general spectral mapping theorem for joint approximate point spectra. There is another kind of refinement of Rosenblum’s theorem that Chan- dler and collaborators have investigated. The operator ydefined by s( X) = AX - XB is invertible if the spectra of A and B are disjoint. How large is 8 MAN-DUEN CHOI AND PETER ROSENTHAL

Iv-’ I]? It might be conjectured that ]p-ill can be bounded by an appropri- ate function of the distance between the spectra of A and B. In two joint papers, Chandler obtained useful results along these lines. First, for arbitrary A and B, there are 2 X 2 counterexamples to any reasonable conjecture [62]. But if A and B are Hermitian, or normal, there are interesting theorems. For example,

THEOREM 4.2 [62]. If A and B are normal operators and the distance between their spectra is d > 0, then there is a universal constant c2 such that II.T’II < cz/d.

It is shown in [62] that cg is at most the infimum of the 2’ norms of allA functions defined in the complex plane whose (double) Fourier transforms f satisfy

1 f(sl + is,) = ~ whenever sp + s; > 1. s, + is,

In fact, as estimated in [7O], this i&mum is at most

?r nsin t - dt -12 0 t

(which is less than 2.91). This bound on the norm of 7-l has applications to estimating angles between spectral subspaces and to spectral variation, topics which have interested Chandler for many years.

5. ANGLES BETWEEN SUBSPACES AND SPECTRAL VARIATION

Chandler Davis wrote his first paper on the effect of a perturbation on eigenvalues of a Hermitian matrix back in 1963 [24], found additional results shortly thereafter [28], and then made a detailed study in joint work with W. M. Kahan [33,36]. Estimates of norms of 7-i (described in Theorem 4.2 above) have significant applications to this topic. First, let A and B be normal operators, and let U and V be disjoint compact subsets of the plane. Let d denote the distance from U to V, and let E be the spectral projection of A corresponding to U, and F the spectral projection of B corresponding to V. Then Bhatia, Davis, and McIntosh proved the following. A SURVEY OF CHANDLER DAVIS

THEOREM 5.1 [62]. Under the above assumptions,

where cp is the universal constant of Section 4 above.

As discussed by Davis and Kahan [36], IlEFll is related to the angle between the spectral subspaces determined by E and F. This result has applications to a long-standing problem. Suppose that A and B are normal matrices with eigenvalues {cxi,. . . , a,} and {P,, . . . , P,,} respectively. Is there a universal constant c such that there is a permutation rr of{1,2,...,n] satisfying

I ai - &,I < cl1A - BII

for all i? Eq mva 1en tl y, is there a c such that there is a normal B’ which is unitarily equivalent to B, commutes with A, and satisfies I]A - B’ll < cl1 A - BII? Bhatia, Davis, and McIntosh showed that the cp of Theorem 4.2 works.

THEOREM 5.2 [62]. Zf A and B are normal matrices with eigenvalues {a , , . . . , qJ and { P1,. . . , P,,} respectively, then there exists a permutation 7~ of~l,..., n} such that

1%- P+,l < c,llA - BII for all i .

This theorem is a consequence of Theorem 5.1 [62]. It was long conjectured that c2 could be replaced by 1 in Theorem 5.2. This was proven by Weyl in the case where A and B are Hermitian; Chandler Davis and proved the same result for unitary operators [63]. Th ese and other related results are discussed in Bhatia’s book Perturbation Bourids for Matrix Eigenvalues, Longman, London, 1987. How- ever, John Holbrook has more recently found a pair of 3 X 3 normal matrices such that the constant is greater than 1.

6. NORM-PRESERVING DILATIONS

Given an operator A E B(H) and a Hilbert space K 2 H, what are all possible dilations of A to T E 9( K) with lITI = IIAll? Equivalently, given 10 MAN-DUEN CHOI AND PETER ROSENTHAL

/IA]] < 1, what are all possible B, C, D acting between appropriate spaces such that

A G

EXAMPLE 6.1. For a 2 X 2 scalar matrix

to be of norm < 1, its from must be

EXAMPLE 6.2. The 2 X 2 scalar matrix

0.6 CY [ P Y 1 is of norm at most 1 if and only if ( a 1 < 0.8, I p I < 0.8, and

y = -4?(0.75) + z[(l - 1.25@]“) ~(1 - 1.25]a]“]“a for some z of modulus at most 1.

Example 6.2 suggests that the general problem is difficult. Nevertheless, Davis, Kahan, and Weinberger gave a very explicit solution.

THEOREM 6.3 [59]. The operator

A C [ B D 1 A SURVEY OF CHANDLER DAVIS 11 has norm at most 1 if and only if

and D has the form, for some Z with ll~ll Q 1,

D = -KA*L + (1 - KK*)?Z(l - L*L)“‘, where

K = B(l - A*A)-l” and L = (1 - AA*)-“‘C.

In [59] these results are applied to approximation of integrals and to approximations of eigenvalues of Hermitian operators (see [61] for a complete exposition and history of these results). Chandler also obtained the fundamental result on a quite different dilation problem.

7. J-UNITARY DILATIONS

In 1950, Paul Halmos showed that

A (1 _ AA*)112 U= (I _ A*A)‘/” -A* 1 is unitary whenever llA/l< 1. Th us every contraction has a unitary dilation; that is, if A is an operator on a Hilbert space H of norm at most 1, then there is a larger Hilbert space K and a unitary operator U on K such that, if P is the orthogonal projection of K onto H, the restriction of PU to H is the operator A. B. Sz.-Nagy substantially strengthened this result, showing that a unitary U can be found so that the restriction of PU” to H is A” for every positive integer n. The existence of this (strong) unitary dilation for every contraction is the basis for the Sz.-Nagy-Foias theory (see Harmonic Analysis of Opera- tors on Hilbert Space by B. Sz.-Nagy and C. Foias, Akademiai Kiado, Budapest, 1970). Chandler considered analogues when A has norm greater than 1 (and may even be unbounded). 12 MAN-DUEN CHOI AND PETER ROSENTHAL

Of course, U cannot be unitary if 11AlI > 1. Chandler showed, however, that U can be J-unitary, in the sense of M. G. Krein and others. Let J be a self-adjoint unitary operator (i.e., J* = J = J- ‘> on a Hilbert space K with inner product (., . >. Then the indefinite inner product deter- mined by J is defined by

[f> d = (If> g>*

An operator U is said to be J-unitary if it is bijective and satisfies

[Uf, ug] = [f, gl forallf, g.

Thus being J-unitary is characterized by the equation U*]U = J. Chandler’s theorem is the following.

THEOREM 7.1 [39]. Let T be any closed, densely defined operator on the Hilbert space H. Then there exists a Hilbert space K 3 H and a closed, densely defined operator U on K with the following properties:

(a) there is a self-adjoint unitary operator J on K such that J restricted to H is the identity operator and [Uf, Ug] = [f, g] for all f, g in the domain of U, where [., ’ ] is the indefinite inner product determined by /; (b) Up’ is densely defined; (c) PU “1H = T” and PU-“[H = T*‘” for every positive integer n, where P is the orthogonal projection of K onto H.

The following gives an elementary example illustrating this theorem.

EXAMPLE 7.2. Regard the constant 2 as a matrix T acting on a Hilbert space of dimension 1.Then the operator

is a J-unitary dilation, with

in the sense that PU III = T (although the property does not hold for powers). A SURVEY OF CHANDLER DAVIS 13

In subsequent work with Foias, Chandler used J-unitary dilations to obtain several interesting results, including the following.

THEOREM 7.3 [42]. Zf the c h aracteristic function of the bounded linear operator T is defined and norm-bounded on the open unit disk, then T is similar to a contraction.

8. COMMUTING PAIRS OF LINEAR TRANSFORMATIONS

What makes two linear transformations commute? Is there any natural representation for a commuting pair of operators? Obviously, B, @ Z and Z @ B, commute. Also, similarity preserves com- mutativity, and so does restriction to a common invariant subspace. Thus linear operators A,, A, on a vector space V certainly commute if they have a representation

A, = T-I( B, c3 Z)T,

A, = T-l( I, 8 B,)T where Bi is a linear operator on the vector space Wj, and T is a linear isomorphism of V onto a subspace of W, 8 W, invariant under both B, @ Z and Z @ B,. Chandler has proven the following converse.

THEOREM 8.1 [38]. If A, and A, are commuting linear transformations on a vector space 7, then there exist linear transformations Bj on vector spaces 5 and a linear isomorphism T of T onto a subspace of Wl 8 WY which is invariant under both B, 8 Z and Z 8 B, such that

A, = T-‘( B, @ Z)T and

A, = T-‘( Z @ B,)T.

Chandler’s basic construction produces infinite-dimensional Wj’s even if 7 is finite-dimensional. However, he also shows that the construction can be modified so that W, and W, are finite-dimensional if 7 is. This theorem is interesting even in the 2 X 2 case. 14 MAN-DUEN CHOI AND PETER ROSENTHAL

EXAMPLE 8.2. Let

The conclusion of the theorem is satisfied by B, = A, @ I, B, = 1 8 A,, and T : C2 * C2 8 C2 defined by T,el = e, @J el, Te, = e2 8 e,, where ei = (LO> and e2 = (0,l).

9. CONCLUSION

The above survey covers only a small fraction of Chandler’s contributions; there are many others. For example, Chandler introduced the “shell” of an operator on Hilbert space [32]. Th is three-dimensional analogue of the numerical range has interesting relations to various geometric properties of operators (see [40,53,55]). One of Chandler’s earliest theorems [7] is the following very beautiful one: the algebra of all operators on separable Hilbert space is generated (as a strongly closed algebra) by three orthogonal projections (and two, as he shows, will not suffice). Chandler found a very nice proof of the Toeplitz-Hausdorff theorem on convexity of the numerical range [43]. He described the functional calculus for matrices in an explicit manner [46]. H e wrote several interesting papers on convex functions of operators [9,11,20,22,67]. He gave a very neat proof of the basic theorem on eigenvalues of compressions of Hermitian matrices [181. As the bibliography at the end of this survey indicates, Chandler has written many other papers as well. His writings have influenced a large number of mathematicians, and he has been an inspiration to those of us lucky enough to work directly with him. We expect that Chandler (like his father, who is still productive in his mid 9Os), will continue to inspire us for many years to come.

BIBLIOGRAPHY OF CHANDLER DAVIS

1 The short-cut problem, Amer. Math. Monthly 55:147-150 (1948) (MR 9-373). 2 Estimating eigenvalues, Proc. Amer. Math. Sot. 3:942-947 (1952) (MR 14-659). 3 The intersection of a linear subspace with the positive orthant, Michigan Math. J. 1:163-168 (1952) (MR 14-1055). A SURVEY OF CHANDLER DAVIS 15

4 Remarks on a previous paper, Michigan Math. J. 223-25 (1953) (MR 15-981). 5 Theory of positive linear dependence, Amer. J. Math. 76:733-746 (1954) (MR 16-211). 6 Modal operators, equivalence relations and projective algebras, Amer. J. Math. 76:747-762 (1954) (MR 16-324). 7 Generators of the ring of bounded operators, Proc. Amer. Math. Sot. 6:970-972 (1955) (MR 17-389). 8 Linear programming and computers, Comput. and Automation 4(7):10-17, 4(8):10-16 (1955) (MR 17-197). 9 ,A Schwarz inequality for convex operator functions, Proc. Amer. Math. Sot. 8:42-44 (1957) (MR 18-812). 10 Extrema of a polynomial, Amer. Math. Monthly 64:677-680 (1957). 11 All convex invariant functions of Hermitian matrices, Arch. Math. (Basel) 8:276-278 (1957) (MR 19-832). 12 A device for studying Hausdorff moments, Trans. Amer. Math. Sot. 87:144-158 (1958) (MR 19-1173). 13 Compressions to finite-dimensional subspaces, Proc. Amer. Math. Sot. 9:356-359 (1958) (MR 20 #5782). 14 Another subdivision which cannot be shelled, Proc. Amer. Math. Sot. 9:735-737 (1958) (MR #2700>. 15 Separation of two linear subspaces, Acta. Sci. Math. (keged) 19:172-187 (1958) (MR 20 #5425). 16 Various averaging operations onto subalgebras, Illinois J. Math. 3:538- 553 (1959) (MR 22 #2900>. 17 The set of non-linearity of a convex piecewise-linear function, Scripta Math. 24:219-228 (1959) (MR 21 #7479). 18 Eigenvalues of compressions, Bull. Math. Sac. Sci. Math. Phys. R. P. Roumaine 3(51):3-5 (1959) (MR 23 #2421) 19 With Ah R. Amir-Moez, Generalized Frobenius inner products, Math. Ann. 141:107-112 (1960) (MR 22 #9508). 20 Operator-valued entropy of a quantum mechanical measurement, Proc. Acad. 37:533-538 (1961) (MR 27 #599). 21 The norm of the Schur product operation, Numer. Math. 4:343-344 (1962). 22 Notions generalizing convexity for functions defined on spaces of matri- ces, in Convexity, Amer. Math. Sot. Proc. Sympos. Pure Math. 7, 1963, pp. 187-201 (MR 27 #5771). 23 An extremal problem for plane convex curves, in Convexity, Amer. Math. Sot. Proc. Sympos. Pure Math. 7, 1963, pp. 181-185 (MR 27 #4140X 24 The rotation of eigenvectors by a perturbation, J. Math. Anal. Appl. 6:159-173 (1963) (MR 26 #6799). 16 MAN-DUEN CHOI AND PETER ROSENTHAL

25 With David Carlson, A generalization of Cauchy’s double alternant, Canad. Math. Bull. 7:273-278 (1964) (MR 28 #5073). 26 With D. G. Rider, Spectral sets and numerical range, Reu. Roumaine Math. Pures A&. 10:125-131 (1965) (MR 32#359). 27 An inequality for traces of matrix functions, Cxechoslovak Math. 1. 15:37-41 (1965) (MR 31 #190>. 28 The rotation of eigenvectors by a perturbation, II, J. Math. Anal. Appl. 11:20-27 (1965) (MR 31 #5082). 29 Analyse fonctionelle et approximation des fonctions, mimeographed notes, Canadian Mathematical Congress, 1966. 30 Mathematics creative? Eviden$e 10: 124- 130 (1967). 31 Mapping properties of some CebyZev systems, in Russian, Dokl. Akad. Nauk. SSSR 175:280-283 (1967) (MR 36 #1896). 32 The shell of a Hilbert-space operator, A&. Sci. Math. 29:69-86 (1968) (MR 39 #4695). 33 With W. M. Kahan, Some new bounds on perturbations of subspaces, Bull. Amer. Math. Sot. 75:863-868 (1969) (MR 39 #7460). 34 With Heydar Radjavi and P. Rosenthal, On operator algebras and invariant subspaces, Canad. J. Math. 21:1178-1181 (1969) (MR 40 #7847) 35 On a theorem of Yamamoto, Numer. Math. 14:297-298 (1970) (MR 41 #2427). 36 With W. M. Kahan, The rotation of eigenvectors by a perturbation, III, SIAM J. Numer. Anal. 7:1-46 (1970) (MR 41 #9044). 37 With D. E. Knuth, Number representations and dragon curves, J. Recreational Math. 3:66-81, 133-149 (1970). 38 Representing a commuting pair by tensor products, Linear Algebra Appl. 3:355-357 (1970) (MR 42 #297). 39 J-unitary dilation of a general operator, Acta. Sci. Math. (Szeged) 31:75-86 (1970) (MR 41 #9032). 40 The shell of a Hilbert space operator, II, A&z. Sci. Math. (Szeged) 31:301-318 (1970) (MR 42 #8325). 41 Dilation of uniformly continuous semi-groups, Rev. Roumaine Math. Pures Appl. 15:975-983 (1970) (MR 42 #3615). 42 With C. Foias, Operators with bounded characteristic function and their J-unitary dilation, Acta. Sci. Math. 32:127-139 (1971) (MR 47 #2398). 43 The Toeplitz-Hausdorff theorem explained, Canad. Math. Bull. 14: 245-246 (1971) (MR 47 #850>. 44 A mathematical visit to North Viet Nam, Notes Canad. Math. Congress 4(3):2-7 (1971) or Notices Amer. Math. Sot. 19:128-130 (1972). 45 A mathematical visit to , Notes Canad. Math. Congress 4(4):2-6 (1972) or Notices Amer. Math. Sot. 19:167-169 (1972). A SURVEY OF CHANDLER DAVIS 17

46 Explicit functional caIculus, Linear Algebra AppZ. 6:193-199 (1973) (MR 48 #6134). 47 A combinatorial problem in best uniform approximation, in Spline Functions and Approximation Theory (A. Meir and A. Sharma, Eds.), Birkhauser, Basel, 1973, pp. 31-56. 48 Materialist mathematics, in For Dirk Struik (R. S. Cohen, J. Stachel, and M. W. Wartofsky, Eds.), Boston Stud. Philos. Sci. XV, 1974, pp. 37-66. 49 With P. Rosenthal, Solving linear operator equations, Canad. J. Math. 26:1384-1389 (1974) (MR 50 #8123). 50 With M.-D. Choi, The spectral mapping theorem for approximate point spectrum, Bull. Amer. Math. Sot. 80:317-321 (1974) (MR 48 #12104). 51 An extremal problem for extensions of a sesquihnear form, Linear Algebra AppZ. 13:91-102 (1976). 52 Geometric approach to a dilation theorem, Linear Algebra AppZ. 18: 33-43 (1977). 53 Matrix-valued shell of an operator or relation, Integral Equations Opera- tor Theory 1:334-363 (1978). 54 With M.-D. Choi, Dilations to systems of matrix units, in Spectral Theory (W. Zelazko, Ed.), Banach Centre Publ. 8, , 1982, pp. 63-75. 55 Extending the Kantorovic inequality to normal matrices, Linear Algebra AppZ. 31:173-177 (1980). 56 Beyond the minimax principle, Proc. Amer. Math. Sot. 81:401-405 (1981). 57 La sociobiologie et son explication de l’humanite, Ann. Econom. Sot. Ciwihsations 4:531-571 (1981). 58 Coeditor with B. Grunbaum and F. A. Sherk, The Geometric Vein: The Coxeter Festschtifi, Springer-Verlag, 1981. 59 With W. M. Kahan and H. F. Weinberger, Norm-preserving dilations and their applications to optimal error estimates, SIAM J. Numer. AnaZ. 19:445-469 (1982). 60 A factorization of an arbitrary m X n contractive operator-matrix, in ToepZitz Centennial (I. Gohberg, Ed.), Birkhauser, 1982, pp. 217-232. 61 Some dilation and representation theorems, in Proceedings of the Sec- ond International Symposium in West Af&a on Functional Analysis and its AppZications (T. Owusu-Ansah, Ed.), Kumasi, 1979, pp. 159-182. 62 With R. Bhatia and A. McIntosh, Perturbation of spectral subspaces and solution of linear operator equations, Linear Algebra AppZ. 52/53:45-67 (1983). 63 With R. Bhatia, A bound for the spectral variation of a unitary operator, Linear and Multilinear Algebra 15:71-76 (1984). 18 MAN-DUEN CHOI AND PETER ROSENTHAL

64 Perturbation of spectrum of normal operators and of commuting tuples, in Linear and Complex Analysis Problem Book (V. P. Havin et al., Eds.), Springer-Verlag, 1984, pp. 219-222. 65 Dilations preserving a bound on the norm of an operator, in Miniconfer- ence on Operator Theory and Partial Dzfeerential Equations (B. Jefferies and A. McIntosh, Eds.), Canberra, 1984, pp. 3-14. 66 With E. A. Azoff, On distance between unitary orbits of self-adjoint operators, Acta Sci. Math. (Szeged) 47:419-439 (1984). 67 With R. Bhatia, Concavity of certain functions of matrices, Linear and Multilinear Algebra 17:155-164 (1985). 68 With J. Dancis, An interlacing theorem for eigenvalues of self-adjoint operators, Linear Algebra Appl. 88/89:117-122 (1987). 69 Completing a matrix so as to minimize the rank, Oper. Theory Adu. Appl. 29:87-95 (1988). 70 With R. Bhatia and P. Koosis, An extremal problem in Fourier analysis with applications to operator theory, J. Funct. Anal. 82:138-150 (1989). 71 With R. Bhatia and M.-D. Choi, Comparing a matrix to its off-diagonal part, Oper. Theoy Adu. Appl. 40:151-164 (1989). 72 A Hippocratic oath for mathematicians. 2, in Mathematics, Education and Society (C. Keitel, Ed.), UNESCO, 1989, pp. 44-47. 73 With P. Ghatage, On the spectrum of the Bergman-Hilbert matrix II, Canad. Math. Bull. 33:60-64 (1990). 74 Criticisms of the usual rationale for validity in mathematics, in Physical- ism in Mathematics (A. D. Irvine, Ed.), Dordrecht, Boston, and Kluwer, London, 1990, pp. 343-356. 75 With R. Bhatia and F. Kittaneh, Some inequalities for commutators and an application to spectral variation, Aequationes Math. 41:70-78 (1991). 76 Science for good or ill, in Waging Peace ZZ (D. Krieger and F. Kelly, Eds.), Noble, 1992, pp. 71-87. 77 With R. Bhatia, More matrix forms of the arithmetic-geometric mean inequality, SIAM J. Matrix Anal. Appl. 15132-136 (1993). 78 Where did twentieth-century mathematics go wrong?, in Proceedings of the Tokyo Conference on History of Mathematics (C. Sasaki, Ed.), Birkhauser, to appear. 79 With R. Bhatia, Relations of linking and duality between symmetric gauge functions, Oper. Theory Adu. Appl., to appear. 80 With R. Bhatia, A Cauchy-Schwarz inequality for operators with applica- tions, Linear Algebra Appl., submitted for publication.

Receioed 10 F&mu y 1994; final munu.scriptnccepted 10 Murch 1994