Semigroups of Linear Operators and Applications: Second Edition
320
Semigroups of Linear Operators and Applications: Second Edition
320eBook
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Overview
Author Jerome A. Goldstein emphasizes motivation and heuristics as well as applications. Each of the two sections concludes with further applications and historical notes. Challenging exercises appear throughout the text, which includes a substantial bibliography. This edition has been updated with supplementary transcripts of five lectures given by the author during a 1989 workshop at Blaubeuren, Germany.
Product Details
| ISBN-13: | 9780486822228 |
|---|---|
| Publisher: | Dover Publications |
| Publication date: | 05/17/2017 |
| Series: | Dover Books on Mathematics |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 320 |
| File size: | 29 MB |
| Note: | This product may take a few minutes to download. |
About the Author
Read an Excerpt
Semigroups of Linear Operators & Applications
By Jerome A. Goldstein
Dover Publications, Inc.
Copyright © 2017 Jerome A. GoldsteinAll rights reserved.
ISBN: 978-0-486-82222-8
Contents
Chapter 0. A Heuristic Survey of the Theory and Applications of Semigroups of Operators, 3,Chapter 1. Semigroups of Linear Operators, 13,
Chapter 2. Linear Cauchy Problems, 83,
References, 181,
Transcripts of Five Lectures Given by the Author at a 1989 Workshop in Blaubeuren, Germany, 232,
Participants from the 1989 Blaubeuren Workshop, 234,
Index of Symbols, 296,
Author Index, 299,
Subject Index, 306,
CHAPTER 1
A Heuristic Survey of the Theory and Applications of Semigroups of Operators
The evolution of a physical system in time is usually described by an initial value problem for a differential equation. (The differential equations can be ordinary or partial, and mixed initial value-boundary value problems are included.) The general setup is as follows. Let u(t) describe the state of some physical system at time t. Suppose that the time rate of change of u(t) is given by some function A of the state of the system u(t). The initial data u(0) = f is also given. Thus
[MATHEMATICAL EXPRESSION OMITTED] (0.1)
(For short du/dt = Au, u(0) = f.)
First of all we must make sense out of
[MATHEMATICAL EXPRESSION OMITTED]
The function u takes values in a set X. In order for u(t + h) - u(t) to make sense, X is taken to be a vector space. In order that limits make sense in X, X is taken to be a Banach space. (More generally, X could be a topological vector space or a differentiable manifold. But the desire to present a clean and complete theory with lots of applications in a reasonable number of pages led us to omit any setting more general than a Banach space.)
A is an operator (i.e. a function) from its domain D(A) in X to X. The equation du/dt = Au is interpreted to mean that u(t) belongs to D(A) and that
[MATHEMATICAL EXPRESSION OMITTED]
where • denotes the norm in X.
Here are three examples.
Example 1. Let Ω be a bounded domain in n-dimensional Eudidean space Rn and let [partial derivative]Ω denote the (nice) boundary of Ω. Let Δ = Σni = 1 [partial derivative]2/[partial derivative]x2i denote the Laplacian. Consider the following classical mixed initial-boundary value problem for the heat equation. We seek a function w = w(t,x), defined for 0 ≤ t < ∞, x [member of] [bar.Ω] = Ω [union] [partial derivative]Ω, such that
[MATHEMATICAL EXPRESSION OMITTED] (0.2)
(For consistency we should have f(x) = 0 for x [member of] [partial derivative]Ω.) Write u(t) = w(t, •), regarded as a function of x, and take X to be a space of functions on Ω, e.g. LP(Ω) for some p ≥ 1 or C([bar.Ω]), the continuous functions on the closure of Ω. The derivatives du/dt and [partial derivative]w/[partial derivative]t are both limits of the difference quotient h-1 [w(t + h,x) — w(t,x), the first limit being in the sense of the norm of X and the second limit being a pointwise one. Even so, we can formally identify [partial derivative]w/[partial derivative]t with du/dt. Clearly the functions denoted by f in both (0.1) and (0.2) can be identified with each other. To define A we take X = C([bar.Ω]) for definiteness. Let D(A) = {v [member of] C([bar.Ω]): v is twice differentiable, Δv [member of] C([bar.Ω]), and v(x) = 0 for each x [member of] [partial derivative]Ω}. Define Av = Δv for v [member of] D(A). Equations (0.2) are thus written in the form (0.1). Note that the boundary condition of (0.2) is absorbed into the domain of definition of the operator A and into the requirement that u(t) [member of] D(A) for all t ≥ 0.
Example 2. Consider the initial value problem for the wave equation
[MATHEMATICAL EXPRESSION OMITTED] (0.3)
For SC we take a space of pairs of functions on Rn. We set
[MATHEMATICAL EXPRESSION OMITTED]
i.e.
[MATHEMATICAL EXPRESSION OMITTED]
Then formally (0.3) becomes (0.1).
Example 3. Consider the initial value problem for the one-dimensional Hamilton-Jacobi equation
[MATHEMATICAL EXPRESSION OMITTED] (0.4)
We take X to be a space of functions on R and set u(t) = w(t,•), Av = -F(dv/dx). Then (0.4) formally becomes (0.1). Note that the operator A of this example is nonlinear, in contrast to the two preceding linear examples.
We return to the notion of a physical system which we imagine being housed in our (imaginary) experimental laboratory. In a well-posed physical experiment something happens, only one thing happens, and repeating the experiment with only small changes in the initial conditions or physical parameters produces only small changes in the outcome of the experiment. This suggests that if the initial value problem (0.1) is to correspond to a well-posed physical experiment, then we must establish an existence theorem, a uniqueness theorem, and a (stability) theorem which says that the solution depends continuously on the ingredients of the problem, namely the initial condition f and the operator A.
Suppose (0.1) is well-posed in the above (informal) sense. Let T(t) map the solution u(s) at time s to the solution u(t + s) at time t + s. The assumption that A does not depend on time implies that T(t) is independent of s; the physical meaning of this is that the underlying physical mechanism does not depend on time.
The solution u(t + τ) at time t + τ can be computed as T(t + τ)f or, alternatively, we can solve for u(τ) = T(τ)f, take this as initial data, and t units of time later the solution becomes u(t + τ) = T(t)(T(τ)f). The uniqueness of the solution implies the semigroup property
T(t + τ) = T(t)T(τ) t, τ > 0.
Also, T(0) = I = the identity operator (this means that the initial condition is assumed), t ->T(t)f is differentiable on [0, ∞ [[and (d/dt)T(t)f = AT(t)f so that u(t) = T(t)f solves (0.1)], and each T(t) is a continuous operator on X. (This reflects the continuous dependence of u(t) on f.) The initial data f should belong to the domain of A, which is assumed to be dense in X. Finally, each T(t) is linear if A is linear.
We are thus led to the notion of a strongly continuous one-parameter semigroup of bounded linear operators on a Banach space X. Such a semigroup is called a (C0) semigroup; this terminology, introduced by Hille, has become standard. The definition is as follows. A family T = {T(t):0 ≤ t< ∞} of linear operators from X to X is called a (C0) semigroup if
(i) T(t) < ∞ (i.e. sup{ T(t)f : f [member of] X, f ≤ 1} < ∞) for each t ≥ 0,
(ii) T(t + s)f = T(t)T(s)f for all f [member of] X and all t,s ≥ 0,
(iii) T(0)f = f for all f [member of] X,
(iv) t ->T(t)f is continuous for t ≥ 0 for each f [member of] X.
T is called a (C0) contraction semigroup if, in addition,
(v) T(t)f ≤ f for all t ≥ 0 and all f [member of] X, i. e. T(t) ≤ 1 for each t ≥ 0.
Roughly speaking, for most purposes it is enough to consider only (C0) contraction semigroups. (This will be fully explained in Section 2 of Chapter I.)
Let T be a (C0) semigroup. Define its generator (or infinitesimal generator) A by the equation
[MATHEMATICAL EXPRESSION OMITTED]
where f is in the domain of A iff this limit exists. Formally, the semigroup property suggests that T(t) = "etA" where A = (d/dt)T(t)|t = 0. This also suggests that the solution of (0.1) is given by u(t) = T(t)f, where T is the semigroup generated by A. The following result is basic.
Theorem I (well-posedness theorem). The initial value problem (0.1) (with A linear) is "well-posed" iff A is the generator of a (C0) semigroup T. In this case the unique solution of (0.1) is given by u(t) = T(t)f for f in the domain of A.
See Chapter II, Theorem 1.2 and Exercise 1.5.4 for precise versions of this.
The obvious question that arises at this point is: which operators A generate (C0) semigroups? For simplicity we work with a (C0) contraction semigroup T. If A is the generator of T, think of T(t) as etA. The formula
[MATHEMATICAL EXPRESSION OMITTED]
which is valid when A is a number and λ > Re(A), suggests the operator version
[MATHEMATICAL EXPRESSION OMITTED] (0.5)
which turns out to be valid for all λ > 0 and all f [member of] X; here I is the identity operator on X. The f is there to make the integrand nice, namely, continuous and bounded in norm by the integrable function f e-λt. The estimate
[MATHEMATICAL EXPRESSION OMITTED]
suggests that:
[MATHEMATICAL EXPRESSION OMITTED] (0.6)
Theorem II (Hille-Yosida generation theorem). A linear operator A generates a (C0) contraction semigroup iff the domain of A is dense in X and (0.6) holds.
One can recover the semigroup T from the generator A by inverting the Laplace transform (0.5) or by other methods, such as the formula
[MATHEMATICAL EXPRESSION OMITTED]
note that (I - αA)-1 = λ(λI - A)-1 where λ = 1/α.
The important implications in Theorems I and II are: (i) A densely defined operator A satisfying (0.6) generates a (C0) semigroup, (ii) If A generates a (C0) semigroup T, then the initial value problem (0.1) is well-posed and is governed by T. In other words, we solve (0.1) by solving equations of the form λh - Ah = g and getting a solution h satisfying the estimate h ≤ g λ; this should be true for all g [member of] X and λ > 0.
The semigroup method has certain limitations. To illustrate this, consider the heat equation initial value problem
[MATHEMATICAL EXPRESSION OMITTED]
One can show that this is governed by a (C0) contraction semigroup on LP(R), 1 ≥ p< ∞ or on the space of bounded uniformly continuous functions on R. It is well-known that another solution of this problem is given by
[MATHEMATICAL EXPRESSION OMITTED]
where u is the semigroup solution and [MATHEMATICAL EXPRESSION OMITTED]. Here v(t,x) does not grow more slowly then [MATHEMATICAL EXPRESSION OMITTED] as x -> ± ∞ for fixed t > 0. Thus, while the semigroup approach gives the unique solution in a certain Banach space context, there may be other solutions (if one widens one's notion of solution).
The next two results are samples from the theory of (C0) semigroups.
Theorem III (perturbation theorem). If A generates a (C0) semigroup and if B is a bounded linear operator on X, then A + B generates a (C0) semigroup.
Theorem IV (approximation theorem). For n = 0, 1,2, ... let An generate a (C0) contraction semigroup Tn. Then
[MATHEMATICAL EXPRESSION OMITTED]
holds for all λ > 0 and all f [member of] iff
[MATHEMATICAL EXPRESSION OMITTED]
holds for all t > 0 and all f [member of] X. Sufficient for this is that D(A0) [subset] D(An) and
[MATHEMATICAL EXPRESSION OMITTED]
holds for all f in the domain of A0.
The perturbation theorem says, roughly, that once we know how to solve (0.1), we can alter A in certain way and solve the resulting problem. The approximation theorem says that the solution of (0.1) depends continuously on A.
Let A be an operator like the one in Example 1, e.g. Au(x) = [ZIGMA]ni, j=1aij(x) [partial derivative]2u/[partial derivative]xi]partial derivative]xj and u(x) = 0 for x [member of] [partial derivative]Ω whenever u is in the domain of A. After one shows that A generates a (C0) semigroup, a variant of Theorem III can be applied with Bu(x) = a0(x)u + Σni = 1 = ai(x) [partial derivative]u/[partial derivative]xi. Thus when we add certain variable coefficient differential operators to constant coefficient operators which are generators, the resulting sums are generators. Theorem IV says that for A, B as above, the unique solution of du/dt = (A + B)u, u(0) = f depends continuously on the coefficients ai, ajk (0 ≥ i ≥ n, 1 ≥ j,k ≥ n).
We emphasize that solving (0.1) by showing that A generates a (C0) semigroup produces a number of bonuses besides existence, uniqueness, and continuous dependence on the data f — namely continuous dependence on A. Moreover, we can perturb A by a large class of allowable perturbations and still get a semigroup generator; thus (0.1) is automatically imbedded in a large class of well-posed problems. Many problems of the form (0.1) can be solved by a variety of techniques, but few other methods have the advantages just described.
The theory of (C0) semigroups has many applications to problems that are not concerned with the solution of differential equations. A consequence of the Hille-Yosida theorem (Theorem II) is Stone's theorem: Let A be a densely defined operator on a complex Hilbert space. Then A and -A both generate (C0) contraction semigroups iff A generates a (C0) group of unitary operators iff iA is self-adjoint.
As a consequence of this we deduce the spectral theorem, i.e. the structure theorem for self-adjoint operators. One version of this theorem concludes that every self-adjoint operator on a complex Hilbert space is (unitarily equivalent to) an operator on an L2 space given by multiplication by a measurable real-valued function. Other results follow. For instance, perturbation theorems for self-adjoint operators follow from corresponding results for generators.
Concerning applications to quantum physics, we use semigroup theory to explain and justify the Feynman path integral formula for the wave function of nonrelativistic quantum mechanics, and we show that the Klein-Gordon equation has the correct nonrelativistic limit as the speed of light becomes infinite. We also discuss scattering theory (in some detail), asymptotic equipartition of energy, and the ergodic theorem.
Semigroups of operators are intimately connected with probability theory, particularly with Markov processes. We sketch a number of aspects of this relationship, including potential theory. There are other connections with probability theory as well. We illustrate this by proving the central limit theorem as a consequence of the approximation theorem.
(Continues...)
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