N.K. Nikolkij (Ed.) Functional Analysis q’ Linear Functional Analysis L . Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest Encyclopaedia of Mathematical Sciences Volume 19 Editor-in-Chief: R.V. Gamkrelidze List of Editors, Authors and Translators Editor-in-Chief R.V. Gamkrelidze, Academy of Sciences of the USSR, Steklov Mathematical Institute, ul. Vavilova 42, 117966 Moscow, Institute for Scientific Informa- tion (VINITI), ul. Usievicha 20a, 125219 Moscow, USSR Consulting Editor N.K. Nikolkij, Steklov Mathematical Institute, Fontanka 27, St. Peterburg D-l 11, USSR Author Yu.1. Lyubich, Department of Mathematics, State University of New York at Stony Brook, Stony Brook, NY 11794-3651, USA Translator I. Tweddle, Department of Mathematics, University of Strathclyde, Livingstone Tower, 26 Richmond Street, Glasgow Gl lXH, Scotland Linear Functional Analysis Yu.1. Lyubich Translated from the Russian by I. Tweddle Contents Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Chapter 1. Classical Concrete Problems .......................... 6 0 1. Elementary Analysis ........................................ 6 1.1. Differentiation ........................................ 6 1.2. Solution of Non-linear Equations ........................ 7 1.3. Extremal Problems .................................... 10 1.4. Linear Functionals and Operators ....................... 10 1.5. Integration ........................................... 11 1.6. Differential Equations .................................. 16 $2. The Fourier Method and Related Questions .................... 19 2.1. The Vibrations of a String .............................. 19 2.2. Heat Conduction ...................................... 21 2.3. The Classical Theory of Fourier Series .................... 21 2.4. General Orthogonal Series .............................. 27 2.5. Orthogonal Polynomials ............................... 28 2.6. The Power Moment Problem ........................... 29 2.7. Jacobian Matrices ..................................... 34 2.8. The Trigonometric Moment Problem .................... 35 2.9. The Fourier Integral ................................... 36 2.10. The Laplace Transform ................................ 39 2.11. The Sturm-Liouville Problem ........................... 45 2.12. The Schrodinger Operator on the Semiaxis ................ 47 2.13. Almost-Periodic Functions ............................. 51 0 3. Theory of Approximation ................................... 53 3.1. Chebyshev Approximations ............................. 53 3.2. Chebyshev and Markov Systems ......................... 56 3.3. The Chebyshev-Markov Problem ........................ 58 2 Contents 3.4. The L-problem of Moments ............................. 60 3.5. Interpolation and Quadrature Processes .................. 61 3.6. Approximation in the Complex Plane .................... 65 3.7. Quasianalytic Classes .................................. 69 9 4. Integral Equations ......................................... 70 4.1. Green Function ...................................... 70 f 4.2. Fredholm and Volterra Equations ....................... 74 ) 4.3. FredholmTheory ..................................... 76’ - 4.4. Hilbert-Schmidt Theory ................................ 77 4.5. Equations with Difference Kernels ....................... 78 4.6. The Riemann-Hilbert Problem .......................... 82 Chapter 2. Foundations and Methods ............................ 85 Q1 . Infinite-Dimensional Linear Algebra .......................... 85 1.1. Bases and Dimension .................................. 85 1.2. Homomorphisms and Linear Functionals ................. 88 1.3. The Algebraic Theory of the Index ....................... 93 1.4. Systems of Linear Equations ............................ 95 1.5. Algebraic Operators ................................... 97 1.6. General Principles of Summation of Series ................ 100 1.7. Commutative Algebra ................................. 101 4 2. Convex Analysis ........................................... 107 2.1. Convex Sets .......................................... 107 2.2. Convex Functionals ................................... 109 2.3. Seminorms and Norms ................................. 111 2.4. The Hahn-Banach Theorem ............................ 112 2.5. Separating Hyperplanes ................................ 113 2.6. Non-negative Linear Functionals ........................ 115 2.7. Ordered Linear Spaces ................................. 117 9 3. Linear Topology ........................................... 119 3.1. Linear Topological Spaces .............................. 119 3.2. Continuous Linear Functionals .......................... 127 3.3. Complete Systems and Topological Bases ................. 132 3.4. Extreme Points of Compact Convex Sets .................. 138 3.5. Integration of Vector-Functions and Measures ............. 140 3.6. w*-Topologies ........................................ 144 3.7. Theory of Duality ..................................... 147 3.8. Continuous Homomorphisms ........................... 153 3.9. Linearisation of Mappings .............................. 163 0 4. Theory of Operators ........................................ 165 4.1. Compact Operators ................................... 165 4.2. The Fixed Point Principle .............................. 174 4.3. Actions and Representations of Semigroups ............... 178 4.4. The Spectrum and Resolvent of a Linear Operator ......... 181 . / Contents 4.5. One-Parameter Semigroups ............................. 4.6. Conjugation and Closure ............................... 4.7. Spectra and Extensions of Symmetric Operators ........... 4.8. Spectral Theory of Selfadjoint Operators ............... /. .. 197 4.9. Spectral Operators .................................... 206 4.10. Spectral Subspaces .................................... 207 4.11. Eigenvectors of Conservative and Dissipative Operators ..... 211 4.12. Spectral Sets and Numerical Ranges ..................... 215 4.13. Complete Compact Operators .......................... 216 4.14. Triangular Decompositions ............................. 220 4.15. Functional Models .................................... 222 4.16. Indefinite Metric ...................................... 227 4.17. BanachAlgebras ...................................... 232 Q5 . Function Spaces ........................................... 242 5.1. Introductory Examples ................................ 242 5.2. Generalised Functions ................................. 244 5.3. Families of Function Spaces ............................ 252 5.4. Operators on Function Spaces .......................... 255 Commentary on the Bibliography ................................ 261 Bibliography ................................................. 261 Author Index ................................................. 273 Subject Index ................................................. 277 Up to a certain time the attention of mathematicians was concentrated on the study of individual objects, for example, specific elementary functions or curves defined by special equations. With the creation of the method of Fourier series, which allowed mathematicians to work with rbitrary’ functions, the individual approach was replaced by the lass’ approach, in which a particular function is considered only as an element of some unction space M ore or less simultane- ously the development of geometry and algebra led to the general concept of a linear space, while in analysis the basic forms of convergence for series of functions were identified: uniform, mean square, pointwise and so on. It turns out, moreover, that a specific type of convergence is associated with each linear function space, for example, uniform convergence in the case of the space of continuous functions on a closed interval. It was only comparatively recently that in this connection the general idea of a linear topological space (LTS)’ was formed; here the algebraic structure is compatible with the topological structure in the sense that the basic operations (addition and multiplication by a scalar) are continuous. Included in this scheme are spaces which, historically, had appeared earlier, namely FrCchet spaces (metric with a complete translation invariant metric), Banach spaces (complete normable) and finally the class which is the most special of all but at the same time the most important for applications, Hilbert spaces, whose topology and geometry are defined in a manner which goes back essentially to Euclid - the assignment of a scalar product of vectors. Using contemporary formal language we can say that LTSs form a cate- gory in which continuous homomorphisms, or continuous linear operators (we usually apply the last term to homomorphisms of a space into itself), serve as morphisms. Specific classical examples of linear operators are differentiation and integration or, in a more general form, differential and integral operators. As a rule, integral operators are continuous but this cannot be said of differential operators. Thanks to the construction of a sufficiently general theory of linear operators it became possible to include the latter case. The story was repeated at the operator level. At first mathematicians studied individual operators but later on it turned out to be useful and necessary to pass to classes. First and foremost in this connection, multiplication of operators (as a rule non-commutative) went out and algebras of operators appeared. Moreover many natural function spaces are also algebras (commutative, of course: typical multiplication is the usual one, i.e. pointwise, or its Fourier equivalent - convo- lution on an Abehan group). With regard to topology all these situations are covered by the concept of a topological algebra but with a considerable excess of generality. A satisfactory approach is achieved in the narrower setting of 1 Translator note. I will use LTSs for the plural and write an LTS rather than the correct a LTS for the indefinite form since the former reads more smoothly. Other abbreviations of similar type will be treated in the same way. < Preface 5 Banach algebras which have proved to be extremely fruitful in harmonic analysis, in representation theory, in approximation theory and so on. The development of functional analysis ran its course under the powerful inlhrence of theoretical and mathematical physics. Here we may mention, for example, spectral theory, which evolved from wave mechanics, the technique of generalised functions (or distributions), whose construction and widespread introduction was preceded by the systematic practice of using the b-function in quantum mechanics, ergodic theory, whose fundamental problems were posed by statistical physics, the investigation of operator algebras in connection with applications to quantum field theory and statistical physics and so on. At the present time the ideas, terminology and methods of functional analysis have penetrated deeply not only into natural science but also into such applied disciplines as numerical mathematics and mathematical economics. In the introductory volume presented below the classical sources of functional analysis are traced, its basic core is described (with a sufficient degree of generali- ty but at the same time with a series of concrete examples and applications) and its principal branches are outlined. An expanded account of a series of specific sections will be given in the subsequent volumes, while certain questions closely connected with linear functional analysis have already been elucidated in previ- ous volumes of the present series. We only touch fragmentarily upon non-linear aspects. The general plan of the volume was discussed with R.V. Gamkrelidze and N.K. Nikolkij and individual topics with A.M. Vershik, E.A. Gorin, M.Yu. Lyubich, A.S. Markus, L.A. Pastur and V.A. Tkachenko. Valuable information on certain questions which are elucidated in the volume was kindly provided to the author by V.M. Borok, Yu.A. Brudnyj, V.M. Kadets, M.I. Kadets, V.Eh. Katsnelon and Yu.1. Lyubarskij. The author offers profound thanks to all the named individuals. He is also most grateful to Dr. Ian Tweddle for his translation of the work into English. Chapter 1 Classical Concrete Problems $1. Elementary Analysis The overwhelming majority of functional relations encountered in mathemat- ics and its applications are non-linear n the large’ but linear nt he small For example the linear theory of elasticity is founded on the assumption that stresses depend linearly on deformations when the latter are small. The method of mathematical analysis, which is firmly established in the works of Newton and Leibniz, consists of two basic procedures: local linearisation of the functions being studied (differentiation) and recon- struction of the global relation according to its local structure (integration. Functional analysis was conceived in the womb of classical mathematical analysis but it took shape as an independent discipline only at the end of the 19th and the beginning of the 20th centuries. 1.1. Differentiation. If we speak about classical analysis using modern lan- guage, we can say that one of its fundamental objects is a dlfirentiable real or complex function of n real variables, i.e. a mapping f: G --f @ (G a domain in Euclidean space W), having the property that at each point x E G the increment f(x + h) -f(x) as a function of the displacement vector h can be approximated by a linear function to within order o( lhl) as Ihl+ 0: f(x + 4 -f(x) = Mx),~) + 4lW. (1) The uniquely determined vector g(x) in this situation is the gradient Vf(x), whose af coordinates are the partial derivatives ax (1 < k < n). The principal linear part of (1) is by definition the dzfirential df of tke function f at the point x. In classical notation h - dx and df = (VfW, dx) = $I $x,. Thus the function f is locally linearised: f(x + d4 =fW + kil g dx,. The function f is said to be smooth if its partial derivatives exist everywhere in G and are continuous. Any smooth function is differentiable. If a function has continuous partial derivatives of all forms up to order r inclusive, then we say 1 Not only direct integration but also indirect processes such as the solution ofdifferential equations. . 5 1. Elementary Analysis I that it belongs to the class2 C’(G). The class C”(G) is defined as the intersection of all the P(G). The enumerated classes can also be considered on closed domains, for example on the interval [a, b] c R (and even on arbitrary subsets x of W). If the function f belongs to C’(G) (I c co) then at any point x E G we have TaylorS formula f(X + dx) =f(x) + mtl i i,+..TinE, dxilarnfaxin dxflxi + o(ldxlY 1 n i.e. to within order o(ldxI the function f is a polynomial of rth degree in dx. The inner sum in Taylor formula is the mth differential of the function f at the point x. For a function f E CG) we can formally write down its Taylor series by passing in Taylors’ formula to I = co (and of course rejecting the error term). If the Taylor series of a function f E Cm(G) converges to it in some neighbourhood of the point x E G, then f is said to be analytic at the point x. A function which is analytic at all points of the domain G is said to be analytic on G. A function which is analytic on the whole space is said to be an entire function. All that has been said above carries over immediately to complex functions of n complex variables. In the 18th century the latter situation had already begun to separate out little by little into a distinct branch of mathematical analysis (complex analysis) in view of the fact that complex analytic functions of one or several complex variables have very distinctive properties. In particular, from the definition of an analytic function (complex or real) one easily obtains the uniqueness theorem: an analytic function which is flat at some point (i.e. such that all its derivatives are equal to zero at this point) is identically zero. The standard example of an infinitely differentiable non-zero function which is flat (at x = 0) is f(x) = {z’ I: f ;; In what follows, either it will be stated or it will be clear from the context whether the functions under consideration and their independent variables are real or complex. 1.2. Solution of Non-linear Equations. This is one of the first applications of linearisation. Let us consider the equation f(x) = 0, where f is a real function which is smooth on a certain interval [a, b] c R. The existence of a root is guaranteed if, for example, f(u) < 0 and f(b) > 0. The uniqueness of the root is guaranteed if the derivative fx) > 0 for all x. Assuming these conditions are satisfied, let us denote the root by X. If some approximation x,, to the root x is known then, using (l), we obtain f(x0) +fxo)@ - x0) = 0, *For convenience we also put Co(G) = C(G), where C(G) is the class of continuous functions on G.