A Short History of Operator Theory
by Evans M. Harrell II
© 2004. Unrestricted use is permitted, with proper attribution, for noncommercial
purposes.
In the first textbook on operator theory, Théorie des Opérations Linéaires,
published in Warsaw 1932, Stefan Banach states that the subject of the book is the
study of functions on spaces of infinite dimension, especially those he coyly
refers to as spaces of type B, otherwise Banach spaces (definition).
This was a good description for Banach, but tastes vary. I propose rather the
"operational" definition that operators act like matrices. And what that means
depends on who you are.
If you are an engineering student, matrices are particular symbols you manipulate
to solve linear systems. As a working engineer you may instead use Heaviside's
operational calculus, in which you are permitted to do all sorts of dangerous
manipulations of symbols for derivatives and what not, exactly as if they were
matrices, in order to solve linear problems of applied analysis. About 90% of the
time you will get the right answer, just like the student; somewhat more with
experience. And that is good enough, if the bridges you build aren't where I
drive.
In mathematics the student of elementary analysis learns that matrices are linear
functions relating finite-dimensional vector spaces, and conversely. As a working
mathematician the analyst has lost all fear of minor matters like infinity, and
will happy agree with Banach's definition.
For the students of algebra, matrices are fun objects that can be added and
multiplied, usually in flagrant disregard for the law (of commutativity). The
working algebraist still enjoys adding and multiplying, but feels that the
analyst's concern about just what the things being added and multiplied are is,
well, limiting.
In this course we'll try to please everyone, except that this is a mathematics
course, so we'll always be careful. We'll solve applied problems, we'll analyze,
and we'll add and multiply. The book by Arveson is somewhat algebraic, but the
lectures will take all three points of view.
We'll start with something completely different, namely history. It is usually
instructive to review the history of a branch of mathematics, especially in order
to understand how the subject applies and why some parts are considered
particularly interesting. Today there are excellent resources making this easy,
especially the MacTutor History of Mathematics Archive. Perhaps if Banach had had
access to the internet he wouldn't have so carelessly reduced his historical
remarks in the introduction to an unsupported repetition of Jacques Hadamard's
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assertion that it was mainly the creation of Vito Volterra. The most thorough
history of operator theory of which I am aware is Jean Dieudonné's History of
Functional Analysis, on which I draw in this account, along with some other
sources in the bibliography you may enjoy.
The concepts whose origins we should seek include: linearity, spaces of infinite
dimension, matrices, and the spectrum. (The spectrum comprises eigenvalues and, as
we shall learn, other related notions.) As with most of mathematics, these
concepts arose in applications.
I. Matrices and Abstract Algebra.
The original model for operator theory is the study of matrices. Although the
word "matrix" was only coined by James Sylvester in 1850, matrix methods have
been around for over 2000 years, as attested by the use of what we would call
Gauß elimination in a Chinese work, Nine Chapters of the Mathematical Art,
from the Han Dynasty. (Even earlier, around 300 BC, the Babylonians worked
with simultaneous linear equations.) Likewise, although Carl Friedrich Gauß
gave us the word "determinant," in the 19th Century, determinants had had
precursors for centuries, and were explicitly used since their simultaneous
discovery in 1683 by Takakazu Seki Kowa in Japan and Gottfried Leibniz in
Europe.
Eigenvalues and diagonalization were discovered in 1826 by Augustin Louis
Cauchy in the process of finding normal forms for quadratic functions. (An
early calculation equivalent to diagonalization is attributed to Johan de
Witt in 1660.) Cauchy proved the spectral theorem for self-adjoint matrices,
i.e., that every real, symmetric matrix is diagonable. The spectral theorem
as generalized by John von Neumann is today the most important result of
operator theory. In addition, Cauchy was the first to be systematic about
determinants.
All this time, what we regard as linear algebra was embedded in practical
calculations. Indeed, although today professional mathematicians intuitively
regard our subject as concerned with structures more than with particular
realizations of those structures, this idea was absent until the mid-
nineteenth century and only came to dominate well into the twentieth century.
Abstract algebra can said to have been born with William Rowan Hamilton's
discovery of quaternions in 1843, and Hermann Grassmann's introduction of
exterior algebra the following year. Grassmann was also responsible for
introducing the scalar product. Cauchy and Jean Claude de Saint-Venant also
created abstract algebraic structures at about this time. Still, these
scholars developed algebras with the idea of modeling something. For
Hamilton, quaternions were to give a better algebraic description of space
and time, and for Grassmann the goal was geometric.
In 1857 Arthur Cayley introduced the idea of an algebra of matrices, and in
1858 he showed, in modern parlance, that quaternions could be "represented"
by matrices. The goal of finding concrete realizations of abstract structures
continues to this day to be a salient feature of abstract algebra, and we
shall be concerned in this class to see how abstract operator algebras can
similarly be represented.
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In 1870, Camille Jordan published the full canonical-form analysis of
matrices, which is a prototype for the decomposition of compact operators in
the infinite-dimensional case.
The fully axiomatic treatment of linear spaces is due to Giuseppe Peano in
his 1888 book, Calcolo geometrico secondo l'Ausdehnungslehre di H. Grassmann
preceduto dalle operazioni della logica deduttiva . This is where you will
find the theorem that every operator defined on a finite-dimensional vector
space is a matrix. Peano defined the sum and product of linear operators
abstractly, and at this stage operator theory began to take shape as progress
in algebra merged with developments in analysis.
II. Operators in Early Analysis.
Leibniz was the first to think of the algebraic properties of the operations
of calculus, for example by considering higher derivatives as successive
operations we might write today as Da f(x). Reportedly he attempted to
understand the case where a might be negative or irrational.
Today, many branches of analysis are inseparable from operator theory,
notably variational calculus, transform theory, and differential equations.
Since all these subjects predated operator theory as such by a century or
two, it is no surprise that some of the earliest antecedents of operator
theory are to be found in them. Differential equations and variational
calculus were largely the creation of Leonhard Euler, Joseph-Louis Lagrange,
and the Bernoulli family. For example, we now realize that the technique of
calculating the first variation of a functional is a kind of differentiation
in a space of functions, and that a derivative in this context is a linear
operator. While the early creators of variational calculus did not avail
themselves of operators as abstractly conceived, they were implicitly using
operators.
So it is with the transforms of Pierre-Simon Laplace, Joseph Fourier and
others, which to this day remain some of the most remarkable and most studied
kinds of operators on spaces of functions. Integral operators were also
implicit in the work of the self-taught British matematician, George Green.
Fourier was a remarkable scientist (and revolutionary, civil engineer,
Egyptologist, and politician), who is perhaps less well appreciated by
mathematicians today than he should be. The folk history repeated by many
mathematicians would have you believe that the contributions attributed to
him were known earlier, and that he lacked "rigor." The latter charge,
however, is unreasonable, because current standards of mathematical rigor are
a creation of the late nineteenth century, under the influence of analysts
such as Karl Weierstraß. Fourier's standards of rigor were those of the day.
Moreover, when we read early scholars today, our understanding of the
concepts they use is often quite different from theirs. In Fourier's day, a
function was generally conceived of as a formula, and some of Fourier's
contemporaries criticized him for thinking of functions more as we do today.
Although trigonometric expansions were certainly used before Fourier, he can
be credited with many innovations, including:
The Fourier transform, which is now arguably the most important example
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of a unitary operator on Hilbert space.
The derivation and first solutions of the heat, or diffusion, equation.
The invention of the modern symbol for the definite integral.
The most pertinent of Fourier's innovations for the theory of operators, all
from his Théorie de la Chaleur, written from 1807 to 1822, are:
The first explicit use of a differential operator, when he wrote D for
the Laplacian and D2 for its square [Cajori];
the systematic expansion of functions in a basis, and
the analysis of infinite systems of equations.
The earliest significant appearance of eigenvalues in connection with
differential equations was in the theory developed by Charles François Sturm
in 1836 and Joseph Liouville in 1838. This is important because, unlike the
situation studied by Cauchy, the underlying space is infinite dimensional,
which allows phenomena that do not arise in the finite-dimensional case of
linear algebra. For example, infinite-dimensional operators can have
continuous spectrum, as became evident (though not in that language) when
George Hill presented the theory of periodic Sturm-Liouville equations in
order to study the stability of the lunar orbit. In his analysis, Hill
introduced infinite determinants.
Sturm-Liouville theory was the beginning of what we now refer to as the
spectral theory of ordinary differential operators. In the late Nineteenth
Century mathematicians were also concerned with the eigenvalues of partial
differential operators, particularly the Laplace operator. The Dirichlet
problem, named for Gustav Lejeune Dirichlet (the family name was Lejeune
Dirichlet), was to find a solution of Laplace's equation with specified
boundary conditions. Subtleties in this problem led mathematicians to a
better and more rigorous understanding of convergence of sequences of
functions and the nature of what are now termed partial differential
operators. Today we recognize this as a a question of topology, as we
familiarly treat functions as points in sets usually called function spaces,
but until the latter part of the Nineteenth Century, this notion was lacking.
Grassmann, in 1862 and Salvatore Pincherle seem to have been the first to
write functions as abstract entities f, rather than f(x), i.e. as relations
between domain and range values. The full idea of a function spaces is of the
Twentieth Century, indeed it is the central notion of Twentieth Century
analysis, and was influenced by attempts to understand the Dirichlet problem,
Fourier series and transforms, and the work of Vito Volterra and Ivar
Fredholm on integral equations.
One last Nineteenth Century influence deserving mention is the influence of
Oliver Heaviside. Heaviside was a brilliant outsider who with little formal
education made substantial contributions to the theory of electricity and
magnetism, and between 1880 and 1887 created a systematic operational
calculus, in which he boldly manipulated symbols, such as the differential
operator d/dx, in novel ways. Although he developed efficient ways to solve
differential equations, he was disdainful of mathematical rigor and had poor
relations with the scholarly community. His influence on mathematics has been
correspondingly mixed. In some respects his formal methods were ahead of
their time, anticipating Twentieth Century developments such as
pseudodifferential operators. On the other hand, the operational calculus can
be ambiguous and can interfere with the understanding of important analytical
issues. Heaviside's operational calculus has continued to have a following
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among engineers and scientists to this day, in isolation from modern
mathematics, and this situation has been a barrier to good communication
among practitioners of different disciplines.
III. Operator Theory in the First Half of the Twentieth
Century.
The subjects of operator theory and its most important subset, spectral
theory, came into focus rapidly after 1900. A major event was the appearance
of Fredholm's theory of integral equations, which arose as a new approach to
the Dirichlet problem. In a preliminary report based on his dissertation
published in 1900 and a landmark article in Acta Mathematica in 1903,
Fredholm gave a complete analysis of an important class of integral
equations, now known as Fredholm equations. Notable achievements in this work
were:
The famous Fredholm alternative theorem, which extended a non-trivial
result of linear algebra to a wide class of operators.
A careful analysis of the convergence of a sequence of operators, as
Fredholm approximated his equations with Riemann sums and passed to a
limit.
The definition of the determinant to a class of operators (greatly
extending the innovation of Hill).
The first use of the resolvent operator (although that term is due to
Hilbert).
In 1902, in his dissertation, Lebesgue defined the modern form of the
integral and introduced the most important spaces of functions, denoted in
his honor Lp.
At about this time, Hilbert founded modern spectral theory in a series of
articles inspired by Fredholm's work. (The word "spectrum" seems to have been
adopted by Hilbert from an 1897 article by Wilhelm Wirtinger.) Hilbert began
like Fredholm, with the specific idea of integral equations, and noticed that
he could obtain more precise results when the space of functions considered
was L2, the square-integrable functions, and when the integral operator was
symmetric. This was the discovery of Hilbert space and the founding of the
general study of self-adjoint operators. In 1906, Hilbert freed his analysis
from the connection with integral equations, and discovered the continuous
spectrum, which had been present but not recognized in the work of Hill.
The concept of an algebra of operators made its appearance in series of
articles culminating in a 1913 book by Frigyes Riesz, where Riesz studied the
algebra of bounded operators on the Hilbert space l2. Riesz representation,
orthogonal projectors, and spectral integrals made their first appearance in
this work. In 1916 Riesz created the theory of what he called "completely
continuous" operators, now more familiarly compact operators. Since he wrote
this in Hungarian, wide recognition came only two years later with a
translation into German. Riesz's spectral theorem for compact operators made
abstract, greatly extended, and largely supplanted Fredholm's work.
The definitive spectral theorem of self-adjoint, and more generally normal,
operators, was the simultaneous discovery of Marshall Stone and John von
Neumann in 1929-1932. Although Stone is more readable today, von Neumann's
contributions are somewhat more far-reaching. One of von Neumann's
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motivations was quantum mechanics, which had been discovered in 1926 in two
rather distinct forms by Erwin Schrödinger and Werner Heisenberg. It was von
Neumann's insight that the natural language of quantum mechanics was that of
self-adjoint operators on Hilbert space. This notion permeates modern
physics. Von Neumann introduced or transformed many concepts now at the core
of operator theory:
domains of definition
extension of operators
closure of an operator
adjoint operators
unbounded operators
He also annihilated with examples the imprecise concept of infinite matrices
that had been a popular way to understand operators.
The year 1932 saw the first text on operator theory, by Stefan Banach, in
which geometric language was used throughout. Banach was responsible for:
fixed-point theory
an understanding of contractions
the closed-graph theorem, and
weak convergence.
In a series of articles from 1935, partly with F.J. Murray, von Neumann
elaborated the theory of operator algebras, introduced by Riesz. It is this
point of view that prevails in Arveson's book. They realized that the sets of
operators that commutes with an algebra was an important tool of analysis and
classification, and made many contributions to pure algebra as well as
algebra.
The final seminal work that will be mentioned here is that of Israil
Gel'fand, who in a 1941 article in Matematicheskii Sbornik extended thei
spectral theorem to elements of normed algebras, and in the process
introduced
the spectral radius formula,
C* algebras (though not with that name), and
the character of an algebra
Since Gel'fand's time operator theory has become an enormous branch of pure
and applied mathematics, and further developments are beyond the scope of a
brief historical sketch.
Bibliography
1. A.D. Alexandrov, A.N. Kolmogorov, and M.A. Lavrent'ev, eds.,
Mathematics. Its Content, Methods, and Meaning, in three volumes.
Cambridge, Mass.: MIT Press, 1963 (original publication by Akademiya
Nauk, 1956).
2. Florian Cajori, A History of Mathematical Notations, New York: Dover,
1993.
3. Jean Dieudonné, History of Functional Analysis, Amsterdam, New York, and
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Oxford: North-Holland, 1981
4. Felix Klein's Vorlesungen über die Entwicklung der Mathematik im 19.
Jahrhundert, New York: Chelsea, 1967.
5. The MacTutor History of Mathematics Archive
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