##
**Linear operators. Part II: Spectral theory. Self-adjoint operators in Hilbert space. With the assistance of William G. Bade and Robert G. Bartle.**
*(English)*
Zbl 0128.34803

Pure and Applied Mathematics. Vol. 7. New York and London: Interscience Publishers, a division of John Wiley and Sons 1963. ix, 859-1923 (1963).

The first volume “Linear Operators. I. General theory”, comprising chapters I–VIII of the whole work, appeared in 1958 (cf. Zbl 0084.10402). The present, second volume offers the most complete account of what has been done in our century (until about 1960) in the theory of linear operators in Hilbert space as well as what was done in analysis with the use of this theory.

Chapter IX gives a short but comprehensive presentation of Gel’fand’s representation theory of commutative \(B\)- and \(B^*\)-algebras. As an application we find the Stone-Čech compactification theorem; and in the notes some facts concerning noncommutative \(B^*\)-algebras.

In Chapter X the existence of the spectral resolution for normal operators is obtained via the Gel’fand representation theory. There are paragraphs dealing with eigenvalues, minimax theorems and an integral formula for the spectral resolution. A paragraph is devoted to the representation of a normal operator as the operator of multiplication by the complex independent variable in a direct sum of \(L^2\)-spaces; such representations are called spectral and are used to construct the unitary invariants of a normal operator. In the notes and remarks one finds many consequences of the spectral theorem, multiplicity theory, invariant subspaces, unitary dilations of operators, von Neumann’s spectral sets (let us mention, however, that \(T\) is not always normal if \(\sigma(T)\) is a spectral set for \(T\)), commutators, square roots, etc.

Chapter XI is a rich collection of various pearls of analysis. It begins with the construction (based an a fixed point theorem of Kakutani) of the Haar measure on compact groups, the Peter-Weyl theorem, the Bohr compactification of the real line, and Bohr’s characterization of almost periodic functions, and the Fourier analysis on locally compact, \(\sigma\)-compact Abelian groups. Questions related to the Wiener Tauberian theorem and to A. Beurling’s problem of spectral synthesis are given in a separate section (the more recent results of P. Malliavin, J.-P. Kahane, etc., are partially quoted in the notes), which ends with the analytical characterization of Beurling’s spectral sets on the real line. A full section together with a large part of the notices at the end of the chapter contain a comprehensive exposition of Calderón-Zygmund and Marcinkiewicz theorems and their consequences on singular convolution operators, using ideas of L. Hörmander [Acta Math. 104, 93–140 (1960; Zbl 0093.11402)]. Finally there are three sections (the 6th, 8th and 9th) dedicated to the spectral study of the compact operators belonging to a von Neumann-Schatten ideal \(C_p\) \((0 < p < \infty)\), where the central place is given to the Carleman inequality, generalized from the Hilbert-Schmidt class \(C_2\) to any class \(C_p\) \((p\geq 1)\). Let us add here the reference to a pioneering paper of T. Lalesco [C. R. Acad. Sci. Paris 145, 906–907 (1907; JFM 38.0382.03)] in which such topics were firstly attacked. Among the notes and comments concluding the chapter there is a suggestive exposition of the representation theory of compact groups, Pontryagin’s duality theorem, and other aspects of the duality between groups.

Chapter XII gives the spectral resolution of self-adjoint operators, the von Neumann functional calculus with unbounded functions, the spectral representation on direct sums of \(L^2\) spaces, a detailed study of the extensions of a symmetric operator, Friedrichs’ theorem an the extension of semi-bounded symmetric operators, the Stone representation theorem (for groups of unitary operators), the polar decomposition, and finally (as applications) some classical moment problems. One has to mention the very intuitive approach (based an the analogy with the homogeneous boundary problems for ordinary differential equations) in the study of the extensions of symmetric operators with finite deficiency indices and the basic theorem (th. XII, 3.11) of W. Bade and J. Schwartz concerning the spectral representation of a self-adjoint operator by means of certain (generalized) Carleman integral operators. The different sufficient conditions for the validity of this Bade-Schwartz representation are effective and easily applicable in the eigenfunction expansions for linear differential (ordinary or elliptic) operators, avoiding the use of the nuclear or Hilbert-Schmidt imbeddings introduced with the Same purpose by Gel’fand and Kostjučenko.

Chapter XIII can be considered as the main original work included in the volume. Its contents (350 pages) constitute the best exposition, till now, of the spectral theory of formally self-adjoint operators made from the functional analytic point of view. The chapter has 10 sections.

The first one gives the elementary definitions, existence and continuity theorems, concerning the ordinary differential equation \(\tau u = 0\) in an interval \(I\), where \(\tau = \sum_{j=1}^n a_j(t) (d/dt)^j\) with \(C^\infty(I)\)-coefficients.

In section 2 the minimal and the maximal closed operator \(T_0(\tau)\) and \(T_1(\tau)\), generated by \(\tau\) in \(L^2(I)\), are defined, and the boundary values \(B\) for \(\tau\) are the continuous linear functionals on the quotient space: \[ \text{graph\;of }T_1(\tau)\Big/\overline{\text{graph\;of }T_0(\tau)}. \] A homogeneous boundary condition is in fact a relation \(B(f) = 0\) where \(f\in D(T_1(\tau))\).

Section 3 is devoted to the study of the Green function corresponding to an operator \(T\subseteq T_1(\tau)\) determined by a set of boundary conditions \(B_i(f) = 0\) \((i = 1, 2,\ldots, k)\). The cases concerning self-adjoint, or real, or second order operators are more specified. The simplicity of the new method used here (and this is true also for a great part of sections 5, 6 and 7) lies in the use of abstract functional analytic methods. For instance, the Green function is obtained by the Fréchet-Riesz theorem applied to the continuous linear functional \(f\to (\lambda I - T)^{-1} f(t)\) on \(L^2(I)\) (where \(\lambda\notin \sigma(T)\) and \(t\in I\) are fixed).

Sections 4 and 5 give the spectral representation of self-adjoint extensions \(T\) of a formally self-adjoint differential operator [i.e. \(T_0(\tau)\subseteq T\subseteq T_1(\tau)]\). The peak theorems are the Weyl-Kodaira form of the spectral representation of the matrix-valued spectral measure \(\{\rho_{ij}\}\) in the preceding theorems and the Marchenko-Coddington theorem of the unicity of \(\{\rho_{ij}\}\).

Section 6 (Qualitative theory of the deficiency indices) precedes with the introduction of the essential spectrum \(\sigma_\varepsilon(\tau)\) of a closed operator \(T\), i. e. the set of those point \(s\lambda\), for which the range of \(\lambda I - T\) is not closed. One studies especially \(\sigma_\varepsilon(\tau)= \sigma_\varepsilon(T_1(\tau))\), where \(\tau\) is formally self-adjoint, in connection with the deficiency indices \(n_{+}\), \(n_{-}\) of \(T_0(\tau)\). As application nine theorems concerning the (non) existence of boundary values at the singular end of \(I\) (for second order \(\tau)\) are obtained essentially by unitary method. A different method is used for the general case (th. XIII, 6.35, due to M. A. Naĭmark). This method is related to a new theorem (th. XIII, 6.28) which will certainly inspire many other researches since it is the first general result of the following kind (stated roughly): If \(\tau\) is stronger than \(\tau'\), then \(\tau\) and \(\tau+\tau'\) are equally strong.

In section 7 (Qualitative theory of the spectrum) the investigation of \(\sigma_\varepsilon(\tau)\) is continued. There are several theorems locating \(\sigma_\varepsilon(\tau)\) even if \(\tau\) is of arbitrary order, or comparing \(\sigma_\varepsilon(\tau)\) with \(\sigma_\varepsilon(\tau+\tau_1)\). Concerning real second order formally self-adjoint differential operators definitive connections are established between the nature of \(\sigma_\varepsilon(\tau)\) and the oscillation character of the solutions of \(\tau u = \lambda u\) with real \(\lambda\). For self-adjoint extensions \(T\) of \(T_0(\tau)\) the precise results of Hartman and Putnam are given. Finally, for a formally self-adjoint differential operator of arbitrary order with periodic coefficients (with the same period) it is shown that \(T = T_1(\tau)\) is self-adjoint, \(\sigma(T) = \sigma_\varepsilon(T)\), and \(\lambda\in\sigma(T)\) if \(\tau u = \lambda u\) has a bounded solution (on \((-\infty, \infty))\). The structure of \(\sigma(T)\) is also studied.

Section 8 (Examples) introduces some of the classical special functions related to differential equations of second order with rational coefficients.

Section 9 (modestly entitled “Exercises”) together with section 9 (Notes and remarks) contain an impressing collection of results on the subject.

The last chapter XIV (Linear partial differential equations and operators) contains some recent results about linear partial differential operators, whose proofs involve more or less linear operator techniques. After a brief discussion of the theory of distributions and the exposition of the Sobolev theorem and its Kondrashov complement (on the compactness of the imbeddings), elliptic operators are studied (section 6). One finds here the Gårding-Browder eigenfunction expansion theorem, the Gårding inequality, the compactness of the resolvent for bounded domains, and the spectral theory of formal self-adjoint elliptic operators; these last facts, concerning homogeneous boundary problems, seem to be particular but significant cases of Browder’s corresponding studies. The next section contains Friedrichs’ theorem an the existence and unicity of the Cauchy problem for symmetric hyperbolic systems, with the simplified proof due to Lax. The last section gives the resolution of the mixed problem for parabolic equations by means of the general theory of semigroups of operators and using the results about elliptic operators established in section 6. Concerning the boundary problems studied, the authors consider only homogeneous Dirichlet problems. One misses here the exposition of some basic results on linear partial differential operators (such as elementary solutions, approximation by exponential polynomial solutions, hypoellipticity, etc.). Except this lack, the chapter is very useful and its contents are well chosen.

We finish this very incomplete review of the contents of the book. Even a very competent reviewer would have difficulties in pointing out all the merits and significant features of this treatise, but it is clear that its study will from now on constitute a necessary stage for every analyst-to-be. The reader will be fascinated, whatever part of the book he studies, by the immense amount of information it offers, by the clearness and simplicity of its exposition, and by the harmony of modern and classical in the whole presentation.

{For Part III see Zbl 0243.47001.}

Chapter IX gives a short but comprehensive presentation of Gel’fand’s representation theory of commutative \(B\)- and \(B^*\)-algebras. As an application we find the Stone-Čech compactification theorem; and in the notes some facts concerning noncommutative \(B^*\)-algebras.

In Chapter X the existence of the spectral resolution for normal operators is obtained via the Gel’fand representation theory. There are paragraphs dealing with eigenvalues, minimax theorems and an integral formula for the spectral resolution. A paragraph is devoted to the representation of a normal operator as the operator of multiplication by the complex independent variable in a direct sum of \(L^2\)-spaces; such representations are called spectral and are used to construct the unitary invariants of a normal operator. In the notes and remarks one finds many consequences of the spectral theorem, multiplicity theory, invariant subspaces, unitary dilations of operators, von Neumann’s spectral sets (let us mention, however, that \(T\) is not always normal if \(\sigma(T)\) is a spectral set for \(T\)), commutators, square roots, etc.

Chapter XI is a rich collection of various pearls of analysis. It begins with the construction (based an a fixed point theorem of Kakutani) of the Haar measure on compact groups, the Peter-Weyl theorem, the Bohr compactification of the real line, and Bohr’s characterization of almost periodic functions, and the Fourier analysis on locally compact, \(\sigma\)-compact Abelian groups. Questions related to the Wiener Tauberian theorem and to A. Beurling’s problem of spectral synthesis are given in a separate section (the more recent results of P. Malliavin, J.-P. Kahane, etc., are partially quoted in the notes), which ends with the analytical characterization of Beurling’s spectral sets on the real line. A full section together with a large part of the notices at the end of the chapter contain a comprehensive exposition of Calderón-Zygmund and Marcinkiewicz theorems and their consequences on singular convolution operators, using ideas of L. Hörmander [Acta Math. 104, 93–140 (1960; Zbl 0093.11402)]. Finally there are three sections (the 6th, 8th and 9th) dedicated to the spectral study of the compact operators belonging to a von Neumann-Schatten ideal \(C_p\) \((0 < p < \infty)\), where the central place is given to the Carleman inequality, generalized from the Hilbert-Schmidt class \(C_2\) to any class \(C_p\) \((p\geq 1)\). Let us add here the reference to a pioneering paper of T. Lalesco [C. R. Acad. Sci. Paris 145, 906–907 (1907; JFM 38.0382.03)] in which such topics were firstly attacked. Among the notes and comments concluding the chapter there is a suggestive exposition of the representation theory of compact groups, Pontryagin’s duality theorem, and other aspects of the duality between groups.

Chapter XII gives the spectral resolution of self-adjoint operators, the von Neumann functional calculus with unbounded functions, the spectral representation on direct sums of \(L^2\) spaces, a detailed study of the extensions of a symmetric operator, Friedrichs’ theorem an the extension of semi-bounded symmetric operators, the Stone representation theorem (for groups of unitary operators), the polar decomposition, and finally (as applications) some classical moment problems. One has to mention the very intuitive approach (based an the analogy with the homogeneous boundary problems for ordinary differential equations) in the study of the extensions of symmetric operators with finite deficiency indices and the basic theorem (th. XII, 3.11) of W. Bade and J. Schwartz concerning the spectral representation of a self-adjoint operator by means of certain (generalized) Carleman integral operators. The different sufficient conditions for the validity of this Bade-Schwartz representation are effective and easily applicable in the eigenfunction expansions for linear differential (ordinary or elliptic) operators, avoiding the use of the nuclear or Hilbert-Schmidt imbeddings introduced with the Same purpose by Gel’fand and Kostjučenko.

Chapter XIII can be considered as the main original work included in the volume. Its contents (350 pages) constitute the best exposition, till now, of the spectral theory of formally self-adjoint operators made from the functional analytic point of view. The chapter has 10 sections.

The first one gives the elementary definitions, existence and continuity theorems, concerning the ordinary differential equation \(\tau u = 0\) in an interval \(I\), where \(\tau = \sum_{j=1}^n a_j(t) (d/dt)^j\) with \(C^\infty(I)\)-coefficients.

In section 2 the minimal and the maximal closed operator \(T_0(\tau)\) and \(T_1(\tau)\), generated by \(\tau\) in \(L^2(I)\), are defined, and the boundary values \(B\) for \(\tau\) are the continuous linear functionals on the quotient space: \[ \text{graph\;of }T_1(\tau)\Big/\overline{\text{graph\;of }T_0(\tau)}. \] A homogeneous boundary condition is in fact a relation \(B(f) = 0\) where \(f\in D(T_1(\tau))\).

Section 3 is devoted to the study of the Green function corresponding to an operator \(T\subseteq T_1(\tau)\) determined by a set of boundary conditions \(B_i(f) = 0\) \((i = 1, 2,\ldots, k)\). The cases concerning self-adjoint, or real, or second order operators are more specified. The simplicity of the new method used here (and this is true also for a great part of sections 5, 6 and 7) lies in the use of abstract functional analytic methods. For instance, the Green function is obtained by the Fréchet-Riesz theorem applied to the continuous linear functional \(f\to (\lambda I - T)^{-1} f(t)\) on \(L^2(I)\) (where \(\lambda\notin \sigma(T)\) and \(t\in I\) are fixed).

Sections 4 and 5 give the spectral representation of self-adjoint extensions \(T\) of a formally self-adjoint differential operator [i.e. \(T_0(\tau)\subseteq T\subseteq T_1(\tau)]\). The peak theorems are the Weyl-Kodaira form of the spectral representation of the matrix-valued spectral measure \(\{\rho_{ij}\}\) in the preceding theorems and the Marchenko-Coddington theorem of the unicity of \(\{\rho_{ij}\}\).

Section 6 (Qualitative theory of the deficiency indices) precedes with the introduction of the essential spectrum \(\sigma_\varepsilon(\tau)\) of a closed operator \(T\), i. e. the set of those point \(s\lambda\), for which the range of \(\lambda I - T\) is not closed. One studies especially \(\sigma_\varepsilon(\tau)= \sigma_\varepsilon(T_1(\tau))\), where \(\tau\) is formally self-adjoint, in connection with the deficiency indices \(n_{+}\), \(n_{-}\) of \(T_0(\tau)\). As application nine theorems concerning the (non) existence of boundary values at the singular end of \(I\) (for second order \(\tau)\) are obtained essentially by unitary method. A different method is used for the general case (th. XIII, 6.35, due to M. A. Naĭmark). This method is related to a new theorem (th. XIII, 6.28) which will certainly inspire many other researches since it is the first general result of the following kind (stated roughly): If \(\tau\) is stronger than \(\tau'\), then \(\tau\) and \(\tau+\tau'\) are equally strong.

In section 7 (Qualitative theory of the spectrum) the investigation of \(\sigma_\varepsilon(\tau)\) is continued. There are several theorems locating \(\sigma_\varepsilon(\tau)\) even if \(\tau\) is of arbitrary order, or comparing \(\sigma_\varepsilon(\tau)\) with \(\sigma_\varepsilon(\tau+\tau_1)\). Concerning real second order formally self-adjoint differential operators definitive connections are established between the nature of \(\sigma_\varepsilon(\tau)\) and the oscillation character of the solutions of \(\tau u = \lambda u\) with real \(\lambda\). For self-adjoint extensions \(T\) of \(T_0(\tau)\) the precise results of Hartman and Putnam are given. Finally, for a formally self-adjoint differential operator of arbitrary order with periodic coefficients (with the same period) it is shown that \(T = T_1(\tau)\) is self-adjoint, \(\sigma(T) = \sigma_\varepsilon(T)\), and \(\lambda\in\sigma(T)\) if \(\tau u = \lambda u\) has a bounded solution (on \((-\infty, \infty))\). The structure of \(\sigma(T)\) is also studied.

Section 8 (Examples) introduces some of the classical special functions related to differential equations of second order with rational coefficients.

Section 9 (modestly entitled “Exercises”) together with section 9 (Notes and remarks) contain an impressing collection of results on the subject.

The last chapter XIV (Linear partial differential equations and operators) contains some recent results about linear partial differential operators, whose proofs involve more or less linear operator techniques. After a brief discussion of the theory of distributions and the exposition of the Sobolev theorem and its Kondrashov complement (on the compactness of the imbeddings), elliptic operators are studied (section 6). One finds here the Gårding-Browder eigenfunction expansion theorem, the Gårding inequality, the compactness of the resolvent for bounded domains, and the spectral theory of formal self-adjoint elliptic operators; these last facts, concerning homogeneous boundary problems, seem to be particular but significant cases of Browder’s corresponding studies. The next section contains Friedrichs’ theorem an the existence and unicity of the Cauchy problem for symmetric hyperbolic systems, with the simplified proof due to Lax. The last section gives the resolution of the mixed problem for parabolic equations by means of the general theory of semigroups of operators and using the results about elliptic operators established in section 6. Concerning the boundary problems studied, the authors consider only homogeneous Dirichlet problems. One misses here the exposition of some basic results on linear partial differential operators (such as elementary solutions, approximation by exponential polynomial solutions, hypoellipticity, etc.). Except this lack, the chapter is very useful and its contents are well chosen.

We finish this very incomplete review of the contents of the book. Even a very competent reviewer would have difficulties in pointing out all the merits and significant features of this treatise, but it is clear that its study will from now on constitute a necessary stage for every analyst-to-be. The reader will be fascinated, whatever part of the book he studies, by the immense amount of information it offers, by the clearness and simplicity of its exposition, and by the harmony of modern and classical in the whole presentation.

{For Part III see Zbl 0243.47001.}

Reviewer: B. Sz.-Nagy and C. Foias

### MSC:

47-02 | Research exposition (monographs, survey articles) pertaining to operator theory |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

43-02 | Research exposition (monographs, survey articles) pertaining to abstract harmonic analysis |

47A10 | Spectrum, resolvent |