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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 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.
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
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.
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.
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 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 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.
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 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:
The year 1932 saw the first text on operator theory, by
Stefan Banach, in which geometric language was used throughout.
Banach was responsible for:
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
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.
Operators in Early Analysis.
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:
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:
He also annihilated with examples the imprecise concept of infinite matrices that
had been a popular way to understand operators.
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.
Bibliography