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Q 3. Linear
established by means of Parseval
Topology
135
equality
All this is proved in exactly the same way as for classical Fourier series in L2(  R, n). As a corollary we obtain the result that all separable Hilbert spaces are isometrically isomorphic to one another. In the example considered the coordinate functionals are continuous and the situation is exactly the same for any Schauder basis (eJ$ in a Banach space E (see Section 3.8). Moreover the systems (ei); c E and (C)g c E* are biorthogonal:
Zf (A,), EE is a sequence of real numbers which satisfies sup II,  nl < a, ”
then the corresponding system of exponentials logical basis in L2(  71,n).
(e n ,,z is an unconditional
topo
It is not possible to improve the constant $: if L, = n  $sgn n the system even fails to be complete (Levinson, 1936). On the subject of unconditional convergence of vector series C&, xi we note that in a Banach space this is equivalent to 1) convergence of all its arts’ CEO Xir32) convergence Of all series I& EiXi (Ei = f 1). In a complete LCS E a sufficient condition for unconditional convergence of a series I& xi is that all series ~~ op(xi) converge (p runs through a family of basic seminorms). It is natural to call such convergence absolute. In a Banach space absolute convergence of a series means that csO llxi II < co. In any infinitedimensional Banach space there exists a series which converges unconditionally but not absolutely 23 It is even possible (Pelczy6ski, 1969).
to construct
a reflexive
separable
Banach
space having
no unconditional
basis
136
Chapter 2. Foundations and Methods
(DvoretzkiRogers,
1950) which cannot happen in the finitedimensional case space the set of sums obtained from all possible rearrangements which preserve convergence is an alline manifold (Steinitz, 1916). In any infinitedimensional Banach space there is a counterexample to this (V.M. Kadets, 1986). The concept of an unconditional topological basis can be generalised to the case of a nonseparable LTS E by applying this term to a system of vectors (ei)isI with respect to which there exists for each x E E a unique unconditionally convergent series (theorem of Riemann). For a convergent series in a finitedimensional
x = C &ei iel
(the set of coordinates 5i different from zero is no more than countable). Completeness and linear independence are clear in this case also. The coordinate space is now a subspace E^c @(I, K) which is invariant with respect to rearrangments (it is symmetric) and moreover each e E E^has no more than countable support. Example. Let E be any Hilbert space and let B = (ei)isI be a complete orthonormal system (its existence can be shown with the help of Zorn lemma). This orthonormal basis is an unconditional topological basis. The unique expansion of any vector x is its Fourier series X =
C iEI
(X9
e&i,
and the coordinate space coincides with lf = L Z, v) (v is the measure on I such that u( {i}) = 1 for all i E I); an isometry between E and lf is established by means of Parseval equality
In particular,
the system of exponentials (eiar)2c u on the whole axis co < basis in the space B2, the completion of the preHilbert space AP of almostperiodic functions. Thus B2 is isometrically isomorphic to 1;. In any LTS E the least power of all the complete systems is called its topological dimension and is denoted by dimt E. In the finitedimensional case it is equal to its algebraic dimension, while in the infinitedimensional case it is the least power of all the dense subsets. In particular, E has countable topological dimension if and only if it is separable. A system of vectors (ei)iol in a linear metric space E with metric d is said to be 6  distal if t < co is an orthonormal
d(ei, er) > 6 > 0
(i # k).
A system is said to be distal if it is bdistal for some 6. The power of a distal system does not exceed dimt E if E is infinitedimensional. In fact in each ball d(x, ei) < J/2 there is a point xi from the dense subset and xi # xlr (i # k). For an infinitedimensional space E the power of any complete distal system is equal to dimt E.
37
5 3. Linear Topology
In a Hilbert space E any orthonormal system (ei)i=l is distal: d(e,, e = ,/2 dimt E. (i # k); therefore the power of any orthonormal basis in E is equal e isometriConsequently, all Hilbert spaces of the same topological dimension cally isomorphic. Y Example. The topological dimension of the space Bz is equal to the power of the continuum. Let E be an infinitedimensional normed linear space. By Zorn lemma there exists for arbitrary 6 (0 c 6 < 1) a maximal bdistal normalised system (ei)i.,. It is complete since otherwise it would be possible by Riesz lemma to extend it in such a way that the &distal property is preserved. Therefore its power is equal to dimt E. This result remains valid if in place of (ei)ip, we construct by transfinite induction a system (uj)jpJ with the following properties: 1) the index set J is wellordered, 2) llujll = 1 0 E J); 3) d(Uj, Lj) 2 6 where Lj is the closed linear hull of the subsystem (I&<~. If we now choose fi E E* such that llfi II = 1, Ifi > 6/2 and fil=, = 0, then the system (f,)j,, will be 6/2  distal. Consequently, dimt E* 2 dimt E. If E* is separable then E is also separable. If E is reflexive then dimt E* = dimt E. The approach to the theory of topological dimension of normed spaces which has been described was proposed by M.G. Krejn, M.A. Krasnosel kij and D.P. Mil an (1948) as a basis for the investigation of those properties of subspaces which are stable under small perturbations. It is natural to measure the size of a perturbation by means of the gap 8(L, M) = max
sup d(x, M), sup d(y, L) ,
( XSS‘
>
YSSM
where S,, S, are the unit spheres of the subspaces L, M. Theorem 1. 1f 8(L, M) = 8 < 4 then dimt L = dimt M. Proof. 24 Suppose that (Ui)iel is a maximal &distal normalised system in L with 28 < 6 < 1. We take E < 612  8 (E > 0) and choose vi E M such that
d(ui, Vi) < 8 + E. Then the system (Vi)iel will be [S  2(8 + s)]distal. quently, dimt L < dimt M. 0 Theorem 2. Zf 8(L, M) = 0 K 4 and the subspaces dimt(E/L) = dimt(E/M).
Conse
L, M are closed then
Proof. Suppose that dimt(E/L) > dimt(E/M). We take in E/L a maximal bdistal normalised system ([Ui])ipr, where [Ui] is the class of the vector ui E E, IIUiII < 1 + E (E > 0). Let US consider the class (Ui) modulo M. The system is not distal. Therefore for any q > 0 we can find i,, i2 E Z such that (< ))iol d((u,,), (Ui,)) < q and then there exists z E M such that d(ui,  ui2) Z) < q. Consequently 24For the infinitedimensional case. The finitedimensional an application of topological methods.
case requires a separate treatment with
138
Chapter
2. Foundations
and Methods
d(  l uiz7 L,  d( , ui29 z,, ’ llzll
6v 2(1 + E) + r/’
Letting 6 + 1, E + 0, q + 0 we obtain 8(L, M) 2 $, contrary to the given condition. 0 Remark 1. In the case of a Hilbert require 8(L, M) < 1.
space it is sufficient in both theorems to
Remark 2. The modified gap 8(L, M) = max
sup d(x, S,), sup d(y, S,) ( XeSL YCSM >
is a complete metric on the set of subspaces of a Banach space (I. Ts. Gokhberg  A.S. Markus, 1959), whereas the original gap does not satisfy the triangle inequality. Moreover they are topologically equivalent because of the inequality 1 < 8(L, M)/B(L, M) < 2. 3.4. Extreme Points of Compact Convex Sets. For any convex set X in a linear space E a point x E X is called an extreme (or extremal) point if it does not belong to any interval [x,, xZ] c X whose end points are both different from x. This is equivalent to the assertion that if x = (u + u)/2 where u, u E X then u = u. The set of extreme points of a convex set X is denoted by extr X. Its role is contained in the following fundamental result. KrejnMil an Theorem. Let E be an LTS and suppose that E* is total. Then any compact covex set Q c E coincides with the closed convex hull of the set extr Q.
In this connection we note that the set of extreme points of a compact convex set need not be closed and hence it can fail to be compact. Example. Let us consider in R3 any circle C and any interval [x1, x2] not lying in the plane of the circle but having a common point c with it different from x1, x2. For the compact set Q = Co(C u [x1, xJ) the set of extreme points is (C\{C))” {x19 x2). Nevertheless, if E is finitedimensional any compact convex set Q c E coincides with the convex hull of its set of extreme points. This assertion had already been proved by Minkowski (by induction on the dimension). In any LTS the closure of a convex set is convex, from which it is clear that the closure of the convex hull of any set X is the smallest closed convex set containing X. We call this the closed convex hulP of the set X. The convex hull of a set which is actually closed may fail to be closed. Example. The convex hull of the set of points (<, r2) (r > 0) in R2 consists of zero and the points (5, q) for which q > r2 and < > 0. Its closure contains in addition the semiaxis (0, q) (q > 0).
25The
closed absolutely
convex
hull is defined
similarly.
5 3. Linear Topology
139
In a finitedimensional space the convex hull of any compact set is closed and consequently it is compact (its boundedness is obvious). In an infinitedimensional space (even if it is a Hilbert space) the convex hull of a compact set may fail to be closed. In certain infinitedimensional spaces there exist compact sets whose closed convex hulls are not compact. This is essentially the effect of incompleteness of such a space. In any complete LCS the closed convex hull of a compact set is compact. For a locally convex FrCchet space a similar assertion holds in the weak topology: the weak 26 closure of the convex hull of a weakly compact set is weakly compact (KrejnShmul an theorem). If the closed convex hull of a compact set Q in an LTS E with total E* is compact then all of its extreme points are contained in Q (D.P. Mil an, 1947). We now give a proof of the KrejnMil an theorem. First let us assume that E is an LCS (without loss of generality we can also assume that it is real). We put K = Co(extr Q). If K # Q then any point z E Q\K can be separated from K by a closed hyperplane {x: f(x) = a} such that f(x) < a (x E K), f(z) > a. We put B= maxXGaf(x). Clearly /I > a and so the hyperplane H = { y: f(y) = /I} does not intersect K or, consequently, extr Q. This situation leads to a contradiction since we are able to find an extreme point of the compact set Q in H. For this purpose let us call a closed alline manifold A extreme if A n Q # 0 and whenever an interior point of an interval [a, b] c Q belongs to A then the entire interval is contained in A. For example, the hyperplane H is extreme. Using Zorn lemma it is easy to show that there is a minimal extreme manifold M contained in H. It is a singleton set since otherwise there exists a continuous linear functional g which is not constant on M and then the smaller extreme manifold {y: y E M, g(y) = r}, where y = max xGMnp g(x), is contained in M. But any singleton extreme manifold must simply consist of an extreme point of the compact set Q. Now if we only know that E* is total, from what has been proved the theorem is true for the weak topology, i.e. Q = e where e is the weak closure of the set C = Co(extr Q). Denoting the strong closure by c we have C c c c e. But c is compact, consequently weakly compact and so also weakly closed. Therefore c = e, i.e. Q = c 0 It is not possible to dispense with the condition that E* be total: there exists an LTS in which a certain nonempty compact convex set has no extreme point (Roberts, 1977). The condition of compactness in the KrejnMil an theorem is also essential; for example, it is not possible to replace it by closedness and boundedness. Example. If S is a connected compact space the closed unit ball 1)cp11< 1 in the Banach space of real continuous functions on S has no extreme points other than the constants f 1. 26It is also the closure in the strong (original) topology since the space is locally convex.
Chapter 2. Foundations and Methods
In this connection we stress that the closed unit ball of an infinitedimensional Banach space is not compact, since otherwise the space would be locally compact and consequently finitedimensional. One of the many useful corollaries of the KrejnMil an theorem is that under its conditions the maximum of any continuous convex functional on a compact convex set is attained on its set of extreme points. 3.5. Integration of VectorFunctions and Measures. The initial development of this important analytic tool was carried out in the 1930s. We give the Gel andPettis definition of the integral of a vectorfunction (1938). Let S be a set with a measure p, E an LTS with total E* and x(s) (s E S) a uectorfunction, i.e. a function with values in E. Let x(s) be integrable in the sense that all scalar functions of the form f@(s)) (f E E*) are summable with respect to the measure p. Then the linear functional
is called the integral of x(s) with respect to the measure p. Thus, in general, J E (E*) #, however the most interesting case is when J(f) = f(F) where X E E. If such an x exists it is unique; it can be identified with J and we put correspondingly
y=ss44dp. Theorem. Suppose that ,u(S) < CO and x(s) is an integrable vectorfunction. If the closed convex hull X of the set of values x(S) is compact then the integralof x(s) with respect to the measure ,u is a vector X E E (further, X E X if p(S) = 1). Proof. We may assume that p(S) = 1 and that E is a real space. For any collection {S,}; c E* we consider the continuous mapping Fx = (J(x)); from X into R Its image is a compact convex set. Moreover (.T(f;:)); E Im F since otherwise there exists a vector (ai); such that
i.e. for the functional
g = ~~=r aif the inequality
dp >I;g(x) a g(x(s)) !J s&(s)) S
is satisfied, which is absurd. If we now consider the closed hyperplanes H, = {z: f(z) = J(f)} (f E E*)in E, we see that the family of compact sets H, n X has the finite intersection property and so its intersection X, is nonempty. The vector X E X, is what is required. 0 Integrability with respect to a generalised measure o is defined as integrability with respect to each of its components and, correspondingly, the preceding
141
Q3. Linear Topology
theorem can be formulated componentwise ” . In particular it is true if the generalised measure o is absolutely continuous with respect to some measure /A (infinite in general), o(A) =
(P E L’(S, PI, A = 9, ~(4 & sA while the function x(s) satisties the previous conditions. In the most developed situation we have to integrate a function x(s) with values in a Banach space E which is defined and weakly continuous on a locally compact topological space S provided with a generalised measure o. In this situation the norm of the function is measurable since 11x(s)11= suplf(x(s))l (f E E*, II f I[ = 1) and ail the f@(s)) are continuous. If
~(4 = then x(s) is integrable
s
IWII b4 < ~0
(in this case it is natural
to say that it is absolutely
integrable2’) and the integral x(s) dw E E**
XC s S
is a vector in E; moreover llX\l < v(x). In fact for any E > 0 we can choose a compact set Q, c S such that
Il~(~)IIW4 < E.
s s\Q. The set x(QE) is weakly compact. By the KrejnShmul an closed convex hull is also weakly compact. Consequently x, E
theorem its weakly
x(s) dw E E.
s Q. Further IIX  X,IlE*+ =
sup
fsE** 11/11=1IS S\Qc
f(x(s))
do < E.
Since E is closed in E** we deduce that X E E. Let S be any space with a measure p. Following Bochner (1933) we can define the Banach space L S, p; E) of absolutely integrable (summable) vectorfunctions with values in the Banach space E as the L completion of the space of simple 27Or with respect to (w(. 28If S is compact then sup, If(x(s)l < cc for all f E E*, hence by the BanachSteinhaus theorem If(x(s))l < Cllfll and consequently Ilx(s)ll < C, i.e. any weakly continuous function on a compact set with values in a Banach space is bounded in norm. Moreover it is absolutely integrable with respect to any generalised measure with finite total variation.
142
Chapter 2. Foundations and Methods
functions, i.e. finite combinations of the form29 xkuk&), where the uk are arbitrary vectors in E and the xk are characteristic functions of measurable sets Sk such that ,u(S,) < co and Si n Sk = 0 (i # k). Moreover if x E L’(S, p; E) the function 11x(s)11is measurable and
llxll =
ss
IWII &
On L’(S, p; E) the Bochner integrul of x with respect to the measure ,u is defined as the strong limit of integrals of simple functions and
The Lebesgue theory of integration extends to this situation with no significant changes. All functions in L1(S, p; E) are GelfandPettis integrable and the values of the integrals in both senses coincide in this case. If x E L’(S, 1~;E) the set function
on the aalgebra associated with the measure p is countably additive and in this sense it is a vector measure. For any vector measure m(A) its ouriution is the measure
Id (4 = sup $r II&%) II < 00, where (Ak}; runs through all finite measurable partitions original example
ImA
=
sA
of the set A. In the
IIx(s)II G.
The total variation (or norm) of a vector measure m is the quantity IIml1 = Iml (S). A vector measure m is said to be absolutely continuous if l(rnll < co and m(A) =
~(4 dImI,
sA where x E L’(S, Iml; E). We say that a real Banach space E has the RudonNikodjm property (RN) if every vector measure with finite norm is absolutely continuous3’. The RN property has a purely geometric characterisation which is closely connected with the KrejnMil an theorem. r9Any summable function is the limit almost everywhere of some sequence of simple functions. Therefore its set of values is separable (except possibly for values taken on a set of measure zero). 30The classical RadonNikodjm theorem asserts that if there are two measures p, v (on a common oalgebra) and p(A) = 0 + v(A) = 0 then v(A) = sA
where p is some function (the density or RadonNikodjm
P dp, deriuatiue).
$3. Linear Topology
143
Let M c E be a closed convex bounded set. A point z,, E M is said to be strongly exposed if there exists a functional f E E* such that f(z) < f(zo) for all z E M (z # zo) and the diameter of the ap’f(z) > f(zO)  E(z E M, E > 0) tends to zero as E + 0. PhelpsRieffel Theorem. A necessary and suficient condition for (I Banach space to have property RN is that each of its closed bounded convex subsets be the closed contiex hull of its set of strongly exposed points. Since all strongly exposed points are clearly extreme3 ‘, in an RNspace E each closed bounded convex set is the closed convex hull of its set of extreme points32 (Lindenstrauss, 1966). The converse assertion holds if E is a conjugate space (HuffMorris, 1976). Each separable conjugate Banach space has the property RN (DunfordPettis, 1940). We now use the tools of integration to give another formulation of the KrejnMil an theorem. Let E be an LTS with total E*, let Q c E be a compact convex set and put S = extr Q. Let us consider on the compact set S a normalised measure p and the function x(s) = s. The centre of muss xp =
s dp IS belongs to Q and the KrejnMil an theorem implies that any point X E Q is the centre of mass of some normalised measure p5 on S. A significant refinement of this result is due to Choquet (1956): if the compact set Q is metrizable then pz (X E Q) can be chosen to be concentrated33 on extr Q and not on its closure S (in this situation extr Q is a G, set). Such a pz is called a Choquet measure for the point X. It can be regarded as a solution of a singular abstract moment problem. This way of looking at it embraces in a single form all those classical moment problems in which the required measure is finite (the general case needs an additional limiting process). For example, in the power moment problem on the whole axis we have a convex set Q of positive (in the sense of the corresponding Hankel matrix) sequences (m& (m. = l), its extreme points are {t"}$ (t is a real parameter) and the integral representation al m, = tk da(t) (k=0,1,2,...) s0 corresponds to Choquet theorem. The converse is false. However, as we see, strongly exposed points play the same role as extreme points in a series of cases. We note here that any weakly compact convex set in a Banach space is the closed convex hull of its set of strongly exposed points (AmirLindenstrauss, 1968). In a uniformly convex Banach space every point of the unit sphere is a strongly exposed point of the closed unit ball. 32Thus, for example, if S is a connected compact space with card S > 1 then C(S) is not an RNspace. 33We say that a measure v on a set X is concentrated on a subset M c X if v(X\M) = 0. If X is a topological space the smallest closed subset on which the measure v is concentrated is called its support and is denoted by supp v.
144
Chapter
2. Foundations
and Methods
The integration of vectorfunctions also plays a significant role in the extension of complex analysis to the vector situation, which in turn is absolutely essential for constructing spectral theory. A vectorfunction x(c) defined on some domain G of the complex plane @ and taking values in an LTS E with total E* is said to be analytic on G if all scalar functions of the form f(x(c)) (f E E*) are analytic on G34. Theorem. If x(c) is an analytic function on a domain G with values in a Banach space E then at each point [,, E G it has a strong expansion as a power series
(K  lol < 6 = d(to,@\@I,
(17)
in which the vector coeflicients are uniquely determined. Proof. We have
f&(c)) = kzotk(f
)(c  cdk
(fEE*),
where
= &
f(W) s I<[ol=p(C
4 ro)k+l
(0 < p < 6).
Since x(c) is weakly continuous on the circle I[  [,, 1 = p, we have tk( f) = f (x,), wherex,EE(k=0,1,2 ,... ). Moreover /lxkII < M(p)pek, where M(p) = sup16501=P[lx(c) 11.Consequently, the power series (17) converges absolutely on the disk Ic  co 1 < p for any p < 6, i.e. finally, on the disk I[  co1 < 6. Its sum s(c)  x(c) since f(s(c)) = f(x([)) for all f E E*. 0 The further development of the theory of analytic vectorfunctions with values in a Banach space presents no problem. Concerning this we note only the CauchyHadamard formula for the radius of convergence of a power series with vector coefficients ck (k = 0, 1,2, . . .):
The sum of the power series is analytic inside the disk of radius r. 3.6. w*Topologies. Let us consider an arbitrary linear space E and its algebraic conjugate space E’. Since E’ c @(E, K) the topology of pointwise convergence is defined on ES. It is called the w*topology on E#. The LCS EX is w*closed in @(E, K) and consequently it is w*complete. If E is an LTS, its conjugate space E* is contained in E’ and so the w*topology is also defined on E*. This is the weakest topology in which all the linear 34A vectorfunction
which
is analytic
on the whole
plane is said to be entire.
145
$3. Linear Topology
functionals of the form R(f) = f(x) (X E E, f E E*) are continuous (these are the canonical images of the elements x E E in the space (I?*)#). Each w*continuous linear functional E* has the stated form. A fundamental result of a general character concerning the w*topology is the BanachAlaoglu Theorem. Let E be an LTS and U c E a neighbourhood of zero. Then the set U” consisting of those linear functionals on E which satisfy the condition35 supXScr If(x)1 G 1 (and so are continuous) is w*compact. Proof We put A = {I: 111< l> and consider the set of functions @(U, A) with its product topology. This is compact by Tikhonov theorem. The mapping Rf = f Iu of the set U” c E* provided with the w*topology into @(U, A) is a homeomorphism between U” and RU But, as is easily verified, RU” is closed. Consequently it is compact. 0
From the BanachAlaoglu Corollary
theorem and the KrejnMil an
1. U” coincides with the w*closed
theorem follows
convex hull of the set extr U
Now let E be a normed space. Then we have on E* at least three natural locally convex topologies: the strong topology defined by the norm of the linear functionals (with respect to which E* is a Banach space); the weak topology defined by the family of strongly continuous, i.e. bounded (for the norm), linear functionals; and the w*topology 36. Applying the BanachAlaoglu theorem to the open unit ball D t E we obtain the closed unit ball of E* as Do. Thus we have established Corollary 2. Zf E is a normed space the closed unit ball in the Banach space E* is w *compact. By the KrejnMil an theorem the ball Do is the w*closed convex hull of its set of extreme points. Hence it is clear, incidentally, that a Banach space need not be the conjugate of some normed space; for example the spaces C[O, 11, L’ [0, l] are not conjugate spaces (even up to isomorphism). The w*compactness of the closed unit ball in the space of measures C(S)* (S compact) is the content of a theorem of Helly. The spaces of continuous functions on compact sets have the special property of universality with respect to the class of all Banach spaces. This also follows from the BanachAlaoglu theorem. Corollary 3. Any Banach space E is isometrically isomorphic to a closed subspace of the space of continuous functions on some compact set. 35For any nonempty X c E the set X0 = {f: f E E*, supxEX If(x)l < 1) is called the polar of the set X. The polars of a set and of its absolutely convex hull clearly coincide. It is also clear that X0 is absolutely convex and w*closed. The term olar’ is also used in a relative sense, i.e. for the intersection of X0 with subspaces of EY (in particular, with E*). The notation X0 is retained in this situation. If X is a subspace then X0 is its annihilator: X0 = XI. 36This is also often called weak in view of the fact that it is weaker than the norm topology. In order to avoid confusion we will not do this (sometimes we call the w*topology the weakened topology).
146
Chapter 2. Foundations and Methods
Proof. The canonical isometry u: E + E** associates with each x E E the function (ox)(f) = f(x) which is w*continuous on the ball Do. The norm of this function in C(DO), i.e. maxfED If(x is equal to IIxII. 0
If E is separable the w*topology on the unit ball Do c E* is metrizable. Now, as is wellknown, any compact metric space is the image of the Cantor set 6 c [0, l] under some continuous mapping. Thus in the case under consideration C(D”) is isometrically isomorphic to a subspace of C(G) and this in turn is isometrically isomorphic to a subspace of C[O, 11. Thus we have the BanachMazur Theorem. Any separable Banach space is isometrically to a closed subspace of C[O, 11.
isometric
We give a further application of the metrizability of the w*topology on Do in the separable case. Let E be a separable Banach space with a closed cone K. Then there exists a continuous linear functional f > 0 (M.G. Krejn, M.A. Rutman, 1948). For the proof we note that the compact set Do, being metrizable, is separable. Consequently there is also a countable dense subset {f.}? of the intersection Do n K*. It just remains to put, for example, f = c,, 2~ ,. Remark. Separability of the space E in the last assertion is essential. Any Banach space E coincides with the subspace of E** consisting of all w*continuous linear functionals on E*. Thus if E is reflexive (and only in this case) each continuous linear functional on E* is w *continuous. The weak topology on E is induced by the w*topology of the space E**. With respect to this topology E is dense in E**, since for any < E E** and any finite collection (fi); c E* the system of linear equations J(x) = <(A) (1 < i < n) has a solution x E E. The unit ball D c E is also w*dense in the unit ball D** c E** which contains it, since otherwise there exists an element 5 E D** which is separated from D by a w *continuous linear functional; this is a functional f E E* such that r(f) # 0 and flD = 0  a contradiction. Theorem. For reflexivity of a Banach space E it is necessary and sufficient that the ball llxll < 1 be weakly compact. Necessity.
This follows from Corollary
2 of the BanachAlaoglu
theorem.
Sufficiency. The image of the closed ball in E** is w*compact and consequently w*closed. On the other hand it is w*dense in the closed ball of the space E**. These balls therefore coincide and then we also have E = E**. 0 Corollary. A necessary and sufficient condition for each closed bounded convex set in a Banach space E to be weakly compact is that E be reflexive.
This result is extremely useful in the theory of extremal problems, in particular in the theory of approximation. Reverting to the latter, let us consider a nonempty closed subset Yin any metric space E. For any point x0 an element y, E Y is called a best approximation in Y if
8 3. Linear Topology
4x0,
147
yo)= inf 4x0,Y) (sf(x,,
Y)).3’
YEY
If E is a reflexive Banach space then for any nonempty closed convex set Y c E and any point x0 E E a best approximation y, E Yexists. In fact it is sufficient to look for y, in the intersection Yeof the set Y with the ball d(x,, y) < d(x,, Y) + 1. Now Ye is weakly compact and the functional d(x,, y) = [lx0  yll is weakly lower semicontinuous. Therefore d(x,, y) attains its inlimum on Y,. The condition of reflexivity is necessary because of James criterion. We note in passing that the theorem on the uniqueness of the best approximation for all x0 E E and all convex Y c E holds if and only if E is strictly convex. Thus in a uniformly convex Banach space E any nonempty closed convex set Y c E (in particular, any closed subspace) is a Chebysheu set in the sense that a best approximation y, E Y exists and is unique for each point x0 E E. N.V. Efimov and S.B. Stechkin have investigated the general problem of the convexity of Chebyshev sets (1961). Example. Let us consider the problem of the best approximation in a Hilbert space E38. Let L c E be a closed subspace, let x0 E E and let y, be the best approximation in L for x,; put z0 = x0  y,. Then for any y E L the function CPM = lb0 + ~~11~= llzol12 + 22 reh
Y) + ~zll~l12
(r E w
has a minimum at the point r = 0, hence re(z,, y) = 0. Consequently, (z,,, y) = 0 (in the real case this is immediate, while in the complex case it comes from substituting iy for y in the result). Thus y, E L and x0  y, is orthogonal to the subspace L (i.e. to every vector in L). These properties characterise y, geometrically as the orthogonal projection of the vector x0 on the subspace L. The value de of the deviation of x0 from L is calculated by Pythagoras theorem: d, = Jcll%ll”
 llYol12).
3.7. Theory of Duality. Let E and F be two linear spaces and let (x, y) (x E E, y E F) be a bilinear functional such that Vx # 0 3y with (x, y) # 0 and Vy # 0 3x with (x, y) # 0. Then we say that there is a duality between E and F or that (E, F) is a dual pair of spaces. Example 1. Let E be any linear space, let F = E put (x, f) = f(x). Clearly this is a dual pair. Example 2. If E is an LTS (E, E*) is a dual pair with (x, If (E, F) is a dual pair there is total on E. Thus F defines
its algebraic conjugate, and
and E* is total (for example, if E is an LCS) then f) = f(x). is a natural embedding of F in E# under which F a locally convex topology on E which is denoted
(x,,, y,,) is called the deviation of the point x,, from the set Y “*i.e. we are concerned with the methodof least squares.
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by o(E, F). The space E*, the conjugate of E in this topology, coincides with F. Now we can consider the w*topology on F = E*. Clearly it coincides with the topology o(F, E) for the dual pair (F, E) and consequently F* = E. In this way a complete symmetry between the linear topology space?’ E and E* is obtained. This symmetry also appears in the following property of reciprocity of polars. Theorem. Let (E, F) be a dual pair, X c E (X # 0) and Y = X0, the polar of X in F. Then Y”, the polar of Y in E, coincides with the a(E, F)closed absolutely convex hull 2 of the set X. In particular, if X is absolutely convex and o(E, F)closed then X = Y” (i.e. X0’ = X). Proof. Since 1(x, y) 1 < 1 (x E X, y E Y) we have X c Y” and hence 8 c Y”. Let x0 I$ 2. By the theorem on the separating hyperplane, there exists y E F such that 1(x, y)( < 1 (x E 2) and 1(x0, y)l > 1. But then y E Y and consequently x0 4 Y”. It follows that 2 = Y”. IJ Corollary 1. Let (E, F) be a dual pair, L a subspace of E and N = L’, the annihilator of L in F. Then the annihilator NL of N in E coincides with the a(E, F)closure of L. In particular, if L is a(E, F)closed then L = N1, i.e. (Ll), = L.
We note further that, clearly, for any family of sets {Xi} in E
Hence on replacing Xi by Xi” we obtain from the previous theorem Corollary 2. For any family of a(E, F)closed absolutely convex sets Xi c E the polar of their intersection coincides with the a(F, E)closed absolutely convex hull of the union of their polars X,!‘.
If (E, F) is a dual pair we say about a locally convex topology on E for which E* = F that it is consistent with the given duality. The weakest among such topologies is a(E, F). It turns out that the class of topologies which are consistent with the given duality can be described and, moreover, among these topologies there is a finest. MackeyArens Theorem. Let (E, F) be a dual pair. In order that a locally convex topology on E be consistent with the given duality it is necessary and sufficient that it be a topology of uniform convergence on some family of absolutely convex subsets of the space F which are compact in the topology a(F, E). Necessity. The polar U” of any neighbourhood of zero U c E is compact in the topology o(F, E) by the BanachAlaoglu theorem, since by assumption F = E* and o(F, E) is the w*topology. The initial topology T on E coincides
391n this connection, if E is an LTS to start with, its initial topology is to be replaced by the weak topology; moreover E* is preserved and it is provided with the w*topology. f
;
0 3. Linear
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149
with the topology of uniform convergence on the family {UO} where U runs through the family T of absolutely convex (open) rneighbourhoods of zero in E. In fact, if U E T and XXEE, sup I(x,y)I < 1 , YEW 1 I then U c I$, since 1(x, y) I < 1 (x E U, y E U’), U” is compact and the function cp,(y) = 1(x, y)l on U” is continuous and, consequently, attains its maximum. On the other hand if WE T we can find U E T such that its a(E, E*)closure 0 is contained in W. Thus 0 = Uoo I> VU and so VU c W. l’,=
Sufl~ciciency. The topology p being considered on E is stronger than a(E, F), hence F c E*. If f E E* then If(x)1 < 1 on a pneighbourhood
u, = {x: x EE, I<x, Y> I < 1 (Y E Q>> (Q is an absolutely convex o(F, E)compact for x E Q”, i.e. f E Qoo (the second polar is in the space E* 3 F and therefore closed and hence f E F. This shows that E* = F.
subset of F). Consequently If(x)1 < 1 taken in E*). Since Q is also compact in E* we have Qoo = Q. Thus f E Q 0
Corollary. If (E, F) is a dual pair the strongest topology on E which is consistent with the given duality is the topology of uniform convergence on all absolutely convex a(F, E)compact subsets of F.
This topology is called the Mackey topology and is denoted by z(E, F). An LCS E whose topology coincides with 7(E, E*) is called a Mackey space. Any barrelled (and also any metrizable)
LCS is a Mackey space.
Remark. If (E, F) is a dual pair, the closure of any convex set X c E is the same for all topologies which are consistent with the given duality since, as a result of the theorem on the separating hyperplane, the operation of closure on the class of convex sets is determined by the set of continuous linear functionals. Taking this as his starting point, D.A. Rajkov (1962) constructed a purely algebraic theory of duality and obtained an algebraic variant of the KrejnMil an theorem. Many important questions in the theory of duality for LTSs are connected with the consideration of bounded sets. Theorem of Mackey. Zf (E, F) is a dual pair the same sets are bounded for all topologies on E which are consistent with the given duality. Proof. By the MackeyArens theorem it is sufficient to show that each o(E, F)bounded set X c E is z(E, F)bounded. Moreover we need consider only the case where X is countable: X = {x.}?. If it is not r(E, F)bounded there exists {a,,}? c R, such that lim,,, a, = 0 but {a,,x y does not tend to zero in the topology z(E, F). Then we can find E > 0 and an absolutely convex o(F, E)compact set Q c F such that maxy.o I( a,x,, y)l > Efor an infinite set of suffices Y (we can suppose that this holds for all n). There is therefore a sequence
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~Y,>Y’ = Qsuchthat<x,,Y.> + cc. At the same time, sup” 1(x,, y) ( < co for each y E F because X is o(E, F)bounded. This situation contradicts the BanachSteinhaus theorem. 0 Corollary.
In any LCS weak boundedness of a set is equivalent to its bounded
ness.
We introduce for any LT.3 E the uniform topology /I(E*, E) on E* which is defined to be the topology of uniform convergence on all bounded subsets X c E. For a normed space E this topology coincides with the norm topology of E*, i.e. its strong topology. Clearly &E*, E) is always finer than the w*topology4’ and the strengthening of the topology can lead to an enlargement of the conjugate space. The space of uniformly continuous linear functionals on E* is called the second conjugate of E and is denoted by E **. The standard canonical homomorphism u: E + E** is injective if E* is total. In this case we have an embedding E c E** in place of the equality under the weak topology. The space E is said to be (topologically) reflexive if E = E** and the initial topology on E coincides with /J(E**, E*). In other words, reflexivity of E means that u is an isomorphism of the LTSs E and E**. Moreover E is necessarily an LCS. If E is reflexive, it is clear that E* is also reflexive under its uniform topology. Theorem. In order that an LCS E be reflexive it is necessary and suflicient that it be barrelled and each weakly closed bounded set X c E be weakly compact. For the proof we note that, in terms of duality, barrels in an LCS E can be characterised as the polars of w*bounded4’ subsets Y c E*. Therefore any barrel in E contains a neighbourhood of zero for the topology on E induced by /?(E**, E*). It is clear from this that if E is reflexive it must be barrelled. The second necessary condition follows from the BanachAlaoglu theorem. Conversely, if the stated conditions are satisfied, we have t?(E*, E) = T(E*, E) and hence by the MackeyArens theorem, /?(E*, E) is consistent with the duality, i.e. E = E**. Moreover /?(E, E*) is stronger than the initial topology on E for the reasons set out above. On the other hand, any /?(E, E*)neighbourhood of the form u,,, = {x: IfWl < E UE Y,}, where Y c E* is a bounded set, contains a barrel, namely the polar of the bounded set 2/e Y and so if E is barrelled there is a neighbourhood of zero for the initial topology contained in Ur,,. q Thus weak compactness of the bounded weakly closed sets is a criterion for reflexivity of a barrelled LCS (in particular, a locally convex Frechet space). A barrelled LCS in which all the closed bounded sets are compact is called a Monte1 space. All finitedimensional spaces are of this type according to the classi@However the topology j?(E*, I$) may not be consistent with the duality. 41 They can also be assumed to be absolutely convex.
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151
cal BorelLebesgue theorem. Among Banach spaces only the finitedimensional ones are Monte1 spaces, but infinitedimensional Monte1 space are found among Frechet spaces, for example42, C” [0, 11. The space ,4(G) of functions which are analytic on the domain G c C” provided with the topology of uniform convergence on compact sets is a Monte1 space ( WeierstrassMantel theorem). It follows from the general criterion for reflexivity that every Monte1 space is reflexive. Moreover the conjugate of a Monte1 space is a Monte1 space under its strong topology. The determination of weak compactness of subsets can be difficult, because the weak topology is not in general metrizable. The following result is useful in this situation. EberleinShmul an Theorem. Zf E is a complete LCS and the subset X c E is such that each sequence (xn}y c X has a weak limit point, then its weak closure X is weakly compact.
In the proof of this theorem we use the following criterion for a@*, E)continuity of a linear functional on E* which goes back to Banach. Theorem. Let E be a complete LCS. Zf a linear functional on E* is w*continuous on the polars of the neighbourhoods of zero in E, then it is w*continuous (and consequently it is generated by an element x E E). Proof. Let us consider the linear space L of all linear functionals on E* which satisfy the condition of the theorem. Clearly L I> E and (L, E*) is a dual pair. We can introduce on L the topology p of uniform convergence on the polars of the neighbourhoods of zero in E since the functionals in L are bounded on these sets. This locally convex topology induces the initial topology on E and, since E is complete, it is therefore closed in L. By the BanachAlaoglu theorem the polars of the neighbourhoods of zero in E are compact in the topology o(E*, E). On each such set P the topology a(E*, L) is on the one hand stronger than a(E*, E), since L 3 E, while on the other hand it is weaker, since it is, by construction, the weakest topology under which all the glP (g E L) are continuous. Consequently p is a topology of uniform convergence on a certain family of absolutely convex o(E*, L)compact sets. By the MackeyArens theorem it is consistent with the duality of the pair (L, E*), i.e. L* = E*. But then if f E L* and f IE = 0 we have f = 0. Thus L = E. 0
Turning to the proof of the EberleinShmul an theorem, we stress that the nontriviality of the assertion lies in the fact that E need not be weakly complete.43 However the space (E*)” is complete in the w*topology and therefore Y, the closure of X in (E*)#, is compact. It remains to show that Y c E, i.e. to establish for any functional y E Y its w *continuity on E* which, by the preceding theorem, reduces to the w*continuity of the restriction ~1,~ on the polar of any 42 The topology
of C” [0, l] is defined lIdIn = max ObkSn
43 For example,
the Banach
space C(X),
by the countable max OSx+l where
Iv l
family
of norms
(n=0,1,2
X is an infinite
compact
,... ). set, is such a space.
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neighbourhood of zero U c E. It suffices to establish the absence of this for some E > 0 we can construct and {f y c U” such that If,(z)1 > E for all m, lf,(z Ifm(xn)l < +E for m > n. This leads to a contradiction on taking a weak limit w*limit point f of {f ?. 0
continuity at zero. In inductively ix.}? c X x,)1 c 3.s for m < n and point x of {x.}? and a
We note further the following Shmul anDieudonne criterion: in a complete LCS a necessary and sufficient condition for the weak compactness of a weakly closed set X is that the intersection of any sequence V, 13 V, 1 9.. of closed convex sets having nonempty intersection with X be nonempty. A very general consideration of the problem of weak compactness was carried out by Grothendieck (1952) and further results were obtained by Ptak (1954). To the latter is also due the credit for discovering the deep connection between the theory of duality and topological properties of homomorphisms, which are wellknown as Banach theorems on the closed graph, the inverse homomorphism and the open mapping. We will discuss these fundamental results later on but for the time being we will be concerned with corresponding aspects of duality. First of all we formulate a criterion for completeness of an LCS in terms of the conjugate space. Theorem. In order that an LCS E be complete, it is necessary and sufficient that a hyperplane in E* be w*closed whenever its intersections with the polars of the neighbourhoods of zero in E are weclosed.
For the proof we associate with an arbitrary LCS E the space E^ of linear functionals f on E* for which all the sets f O) n U” are w*closed (U runs through the neighbourhoods of zero in E) or, equivalently, whose restrictions f IuO are w*continuous. We embed E in ,!? in the canonical way and introduce on i the topology of uniform convergence on the polars U The preceding theorem follows from the following assertion. Lemma. For any LCS E the space .@is its completion, i.e. 1) E is complete, 2) the topology of E is induced by the topology of i?, 3) E is dense in E^. Proof The completeness of E^follows from the w*compactness of each polar U” and the completeness of the space of continuous functions C(U . The coincidence of the induced and the initial topologies on E is trivial. Finally the topology on E^ is consistent for the dual pair (fi, E*) by the MackeyArens theorem, i.e. @)* = E*. Moreover if E is not dense in E^there exists f E (I?)* such that f # 0, f IE = 0, which is a contradiction since f E E*. 0 Remark. The criterion for completeness of an LCS can be reformulated as follows: E is complete o each linear functional on E* which is w*continuous on the polars of the neighbourhoods of zero is w*continuous. An LCS E is said to be fully complete (or to be a Ptdk space) if each subspace of E*, whose intersections with the polars of the neighbourhoods of zero in E
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153
are w*closed, is w*closed. It follows from the previous result that a fully complete space is complete. The converse is false, however the following holds. BanachKrejnShmul an
Theorem.
Each locally convex Frtchet
space is fully
complete.
3.8. Continuous Homomorphisms. Let E, and E, be LTSs. Continuity of a homomorphism h: E, + E, is equivalent to its continuity at zero, for which in turn it is sufficient (and for locally bounded E, also necessary) that h be bounded in the sense that the image of some neighbourhood of zero is bounded. Example. In the space Cm[O, l] differentiation d/dt is continuous but not bounded. For any continuous homomorphism h the images of the bounded sets are bounded, since if (x,)? is a bounded sequence and a, + 0, then tl,x, + 0 and hence a,(hx,) = h(a,x,) + 0. If E, is a normed space and the homomorphism h: E, *E, maps bounded sets into bounded sets then h is bounded and so continuous. Example. Let E be an LCS and j? the same space but now provided with its weak topology, which we will asume to be different from the initial topology (as a rule this is generally the case). Then the identity homomorphism E”+ E is not continuous (and so not bounded) but it takes bounded sets into bounded sets. In the case of normed spaces E,, E, boundedness of a homomorphism h is equivalent to finiteness of its norm
llhll = sup IIWI. IIXII=1 In this case the set of bounded (i.e. continuous) homomorphisms of E, + E, is itself a normed space (even a Banach space if E, is a Banach space). For any LTSs E,, E, the set of continuous homomorphisms of E, + E2 is denoted by Homc(E,, E2). It is a linear space44 and can be turned into an LTS by the following standard procedure: a covering 52 of E, by bounded sets is chosen, after which a base of neighbourhoods of zero in Homc(E,, E,) is given by the sets NV,, = {h: hX c V}, where X E 0 and V is an arbitrary neighbourhood of zero in E,. This is the Qtopology or the topology of uni$orm convergence on the family a. It becomes stronger as the family Sz is extended but, at the same time, it is not changed if we adjoin to the sets of !2 all their subsets, finite unions and linear combinations; it is also unaffected if all the sets in Q are replaced by their closures and if E, and E, are LCSs, all the sets in Sz can be replaced by their closed absolutely convex hulls. If E, is an LCS and { pi)i. I is a family of basic seminorms the Qtopology on Homc(E,, E,) is defined by the family of seminorms 7Cx,i(h) = supx E x p i (hx) (X E Sz),i.e. Homc(E,, E,) turns out to be an LCS. 441t is even an algebra (associative, with unity) in the case E, = E,. We use the notation: Homc(E, E) = X(E).
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Among the Gtopologies on Homc(E,, E,) the weakest is the topology of pointwise convergence and the strongest is the topology of uniform convergence on all the bounded subsets of E, . The latter is termed briefly the uniform topology. In the case Homc(E, K) = E* this is just the topology /?(E*, E). For normed spaces E,, E, the uniform topology on Homc(E,, E,) is generated by a norm4’. The topology of pointwise convergence on Homc(E,, E,) is sometimes called the strong topology46, since the weak topology defined in the case where Ez is total by the neighbourhoods ofzero {h: If( < E} (x E E,, f E Ez, E > 0)is also used (this is clearly locally convex). The topology of uniform convergence on compact sets is also contained in the class of Gtopologies. In this case we have the wellknown criterion for precompactness 47 of a family of continuous (not necessarily linear) mappings of E, + E, (E, metrizable), namely equicontinuity of the restrictions to each compact set Q c E, and pointwise precompactness (ArzelriAscoli theorem). The following assertion follows in turn from the BanachSteinhaus theorem (and also bears the same name4*). Theorem. Zf E, is a barrelled space and E, is any LCS then each pointwise bounded family F c Homc(E,, E,) is uniformly continuous and consequently uniformly bounded on some neighbourhood of zero.4g Proof. Let {pi}i.J be a family of basic seminorms. Let us consider the continuous seminorms p,(hx) (h E F) on E,. According to the condition xi(x) = sup,pi(hx) < co. Consequently the seminorms 7Ziare continuous and hence the uniform continuity of the family F follows immediately. 0 Remark. If an LCS E, is such that the BanachSteinhaus theorem holds for E, = K (i.e. in E:) then E, is barrelled. In fact any barrel T c E, is the polar of a w*bounded and therefore equicontinous set F c E:. But then on some neighbourhood of zero U c E, we will have 1f (x)1 < 1 for all f E F and hence
UCT
451f E is a normed space, the norm on U(E) satisfies the usual properties: I/h,h, 11< llhl 11Ilh,II, 11111 = 1. Thus the continuous linear operators on E form a Banach algebra f!(E) which is noncommutative if dim E > 1. We note here that in any Banach algebra the operation of multiplication is continuous. 461n the case of E* it coincides with the w*topology. set X in an LTS E is said to be precompoct if it is relatively compact in the completion of the space E (a set is said to be relatively compact if its closure is compact). The property of precompactness is weaker than relative compactness (in a complete space they are clearly equivalent), but it is convenient to work with it in view of Hausdorff criterion: X is precompact o for any neighbourhood of zero U there exists a finite Unet, i.e. a covering of the set X by neighbourhoods of the form x, + U (1 < i < m). It is also called the principle of uniform boundedness. 4QIf E, is a normed space uniform boundedness on some neighbourhood of zero is equivalent to uniform continuity. If both spaces E,, E, are normed then uniform continuity of a family F is equivalent to boundedness in norm: sup,,, jlhll < co.
5 3. Linear
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Topology
For any LTSs E,, E, and any h E Homc(E,, E,) the conjugate5’ homomorphism h*: Ef + ET is defined by the formula @*f)(x) = f&x) (x E E,,fe Ef). It is important to know for what topologies on the dual space h* will be continuous. Lemma. Let ET and E,* be provided respectively where hS2, c 52,. Then h* is continuous.
with Q, and On,topologies
Verification of this basic result presents no problem. In particular, h* is continuous if ET, E,* are provided with their w*topologies or if both have their uniform topologies. In the latter situation if both spaces are normed then
Ilh*II = IIW We note in passing the duality relations of the homomorphism h E Homc(E,, E,) annihilators considered below are relative necessarily dual pairs). With no restrictions on the LTSs E, and
between the images and the kernels and of h* E Hom(Ez, ET). All the to the pairs (E,, E:), (E,, Ez) (not E, we have the formula
Ker h* = (Im h)’ and the inclusions Im h* c (Ker h)‘,
Ker h c (Im h*)l,
(the closure of Im h* is taken in the w*topology Im h* = (Ker h)‘,
Ker h = (Im h*)l,
Im h c (Ker h*)l on51 ET). If Ez is total we have Im h = (Ker h*)l.
In all three cases equality is established by considering a continuous linear functional which annihilates the smaller space. In many applications we have to deal with discontinuous homomorphisms, however the weaker property of closedness of the graph (h E Hom(E,, Ed) I‘,,= {k WL,el = El x E, usually holds; this is called closedness of the homomorphism itself. Example. Let E, = C O, 11, E, = C[O, 11, where the topology on C[O, l] is the usual one and the topology on C O, l] is that induced by C[O, 11. Differentiation d/dt: E, + E, is discontinuous but closed. Closed Graph Theorem (CGT). Let E,, E, be LCSs with E, barrelled and E, fully complete. Then if the homomorphism h: E, + E, is closed it is continuous.
ore precisely, the topological conjugate. Clearly h* = h#),; and moreover note the following formulae: 1) (hi + h,)* = h: + hz; 2) (ah)* = ah*; 3) (hg)* h E Homc(E,, E,) has a continuous inverse then h* is invertible and (h*)’ = not be continuous. If E, is a reflexive Banach space the w*topology on ET coincides with the normed space ET and the weak closure of any subspace of ET coincides with
Im h* c ET. We also = g*h*; 4) l* = 1. If (hl)* but (h*)’ may weak topology of the its strong closure.
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and Methods
In the proof of the CGT we make use of the following closedness of a homomorphism.
general criterion for
Lemma 1. Let E,, E, be LCSs and let h E Hom(E,, E,). A necessary and sufficient condition for closedness of h is that the space A,, of those f E Ez for which h#f E E: be dense in Ef under the w*topology5’. Proof. .For any LTSs E,, E, the conjugate space of E, x E, is naturally identified with E: x Ef in accordance with the formula cp(x, y) = g(x) + f(y) (g E E:, f E Ez). Moreover for any h E Hom(E,, E,) the annihilator of its graph I, is the subspace Ih’ of functionals of the form f(y  hx) where f E A,,. Now let Et, E, be LCSs and suppose that h is closed. If y E E,, y # 0 then, since (0, y) 4 I,, there is a functional f E A,, such that f(y) # 0. Conversely, if for each y E E,, y # 0 there exists a functional f, E A,, such that f,(y) # 0, then r,, = n, Ker f,, from which the closedness of the graph I, follows immediately. 0
We also require the following Lemma 2. Zf E,, E, are LCSs and E, is barrelled, then for any h E Hom(E,, E,) the intersections of the set A,, c Et with the polars of the neighbourhoods of zero V c E, are w*closed. Proof. If the neighbourhood V c E, is absolutely convex, the closure of its inverse image is a barrel in E, and therefore it contains a neighbourhood U of zero in E,; the w*closedness of the set A,, n V” follows from its representation by means of the system of inequalities 1f(hx)l < 1 (x E U), 1f(y)1 < 1 (y E V). 0 Proof of the CGT. By Lemma 1 the set A,, c Ef is w*dense and by Lemma 2 it is w*closed since the space E, is fully complete. Consequently A,, = Ez, i.e. h is weakly continuous. If V is an absolutely convex neighbourhood in E,, its closure v is weakly closed and therefore the inverse image h’ v c E, is a barrel. But then there exists a neighbourhood of zero U in E, such that lJ c hl c i.e. hUcE,:O
The following result is one of the most important
consequences of the CGT.
Inverse Homomorphism Theorem (IHT). Let E,, E, be LCSs with E, barrelled and E, fully complete. Zf the homomorphism g: E, + E, is bijective and continuous then it is a topological isomorphism, i.e. gl is continuous.
In fact the inverse of a closed (and so of a continuous) homomorphism is closed. In a certain sense the next fundamental theorem is the dual assertion for the CGT. Open Mapping Theorem (OMT). Let E,, E, be LCSs with E, barrelled and E, fully complete and suppose that the homomorphism g: E, + E, is surjective and continuous. Then g is an open mapping.
or
any LCS E a subspace
N c E* is w*dense
if and only if it is total.
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157
Proof. The space Ker g is closed 53, therefore we can pass to the factor space G = EJKer g and consider the bijective homomorphism y: G + E, associated with g by the relationship g = yj where j: E, * G is the canonical homomorphism. Full completeness is inherited by factor spaces and y is continuous (because g is continuous and j is open). According to the IHT, y’ is continuous, i.e. y is open, but then g is also open. 0 As Banach had already shown, all three theorems are true for Frechet spaces without the assumption of local convexity. Their applications are numerous and varied (the IHT is used particularly often). The first application which we note relates to complemented subspaces. A closed subspace L in an LTS E is said to be complemented if it has a closed complement s4 M. For examp 1e, a n y closed subspace of finite codimension, in particular any closed hyperplane, is of this type. In an LCS every tinitedimensional subspace is complemented. If E is a Hilbert space, any closed subspace L c E is complemented, the orthogonal complement Ll = {y: (x, y) = 0 (x E L)}
being a closed complement for L. The decomposition L i L’ = E (in this case we also write55 L @L’ = E) is obtained by associating with each x E E its orthogonal projection x’ on L (moreover x  x’ is the orthogonal projection of the vector x on L*). Lindenstrauss and Tzafriri (1971) showed that if every closed subspace of a Banach space E is complemented then E is isomorphic to a Hilbert space. If L is a complemented subspace in an LTS E and M is its topological complement then the projection P onto L along M is closed. Thus if E is a barrelled fully complete LCS or a Frechet space, P is continuous. If P is a continuous projection on a Banach space E, P # 0, then llPl[ 2 1. If lIPI/ = 1 or P = 0 it is said to be an orthoprojection since in a Hilbert space this corresponds exactly to the orthogonal decomposition E = Im P @ Ker P. A bounded operator A on a Banach space is said to be a contraction56 if 11 AlI < 1 (a uniform contraction if IIA 11< 1). It is precisely in this case that distance is not increased: d(Ax, Ay) < d(x, y) (respectively d(Ax, Ay) < qd(x, y) where q < l), i.e. A is a contraction (uniform contraction) in the metric sense. We now apply the IHT to a problem connected with metrization of an LTS. Two invariant metrics d,, d, on a linear space E are said to be equivalent if they 53For any closed homomorphism 9. 541t is called a topological complement. In this connection we say that E decomposes into the topological direct sum of the subspace L and M. This concept extends naturally to any finite and then by means of closure also to any infinite collection of subspaces. A system of subspaces is said to be complete if the closure of its sum is the whole space E. e also use the term orthogonal sum. This term extends immediately to the case of an arbitrary family of pairwise orthogonal subspaces (if the family is infinite it means the closure of the algebraic orthogonal sum). 56If a contraction A is invertible and A’ is also a contraction then A is an isometry. Isometries of a Hilbert space onto itself are called unitary operators.
158
Chapter 2. Foundations and Methods
define the same topology. We say further that d, is weaker than d2 (or subordinate to d,) if the corresponding topologies are in this relationship. If the inequality d,(x, y) < Cd&, y) holds for some constant C > 0 then d, is weaker than dl. Theorem. Zf the space E is complete under both metrics d, and d2 and also d, is weaker than d, then the given metrics are equivalent. Proof. The identity mapping acts continuously from the d,topology d,topology. According to the IHT we have a homeomorphism. 0
into the
Two norms II.II 1 and II * IIz on a linear space E are said to be equivalent if the corresponding metrics are equivalent. The relation of subordination is defined correspondingly and moreover a necessary and sufficient condition for this is that the inequality IIx II I < C IIx IIz be satisfied for some positive constant C. Thus for equivalence of the norms it is necessary and sufficient that a llxll z < I/XII 1 < b llxll z (x E E) where a, b are positive constants. Any two norms on a linitedimensional space are equivalent. From the preceding theorem we obtain Corollary. Zf a space E is a Banach space with respect to each of two norms, one of which is subordinate to the other, then these norms are equivalent.
Equivalent renorming is a powerful method for investigating topological properties of Banach spaces. We mention one example. The following easily established result holds in a Hilbert space: if x, + x weakly and IIx,(I + llxll then x, + x. It is very convenient to have this Hproperty in a Banach space. It does hold in a uniformly convex space (V.L. Shmul an, 1939) but in the general case it is lacking. M.I. Kadets (1957) showed that in any separable Banach space it is possible to introduce an equivalent norm possessing the Hproperty. He used this result in the solution of a problem of Banach on the homeomorphism of all infinitedimensional separable Banach spaces. We note that in any separable Banach space it is possible to introduce an equivalent strictly convex norm (Clarkson, 1936) and also an equivalent smooth norm (Day, 1955). The separability requirement is essential. In 1941 Day showed that a reflexive Banach space need not have a uniformly convex equivalent norm. The possibility of introducing a uniformly convex norm is equivalent to the superreflexivity of the space. This deep result was obtained by Enflo (1972). The concept of superreflexivity, which is defined in the following way, is also deep. Let E, E, be Banach spaces. We say that E, is finitely representable in E if for each finitedimensional subspace L c E, and each a > 1 there exists an embedding i: L + E which is aclose to an isometry:
al lbll G IIWI G allxll
(x E L).
The space E is said to be superreflexive if each space which is finitely representable in it is reflexive. Superreflexivity of E* is equivalent to superreflexivity of E (James, 1972). As a corollary to Enflo theorem the complete duality between uniformly convex and uniformly smooth Banach spaces shows that the possibility
0 3. Linear
Topology
159
of introducing a uniformly smooth norm is also equivalent to superreflexivity. Further in a superreflexive space E there is an equivalent norm which is simultaneously uniformly convex and unformly smooth. Returning after this digression to other applications of the IHT, let ,us consider a separable Banach space E with a Schauder basis B = (ei):. Let E be the corresponding coordinate space. By introducing the norm IlPll = SUP i$ tiei (t = (5iE) n II II on E we turn it into a Banach space. The expansion in terms of B can be regarded as a homomorphism g: E + E which is clearly continuous ( 11911= 1) and bijective. According to the IHT, h = gi is also continuouss7 and then
15.1< zllell < 2. ’ ’ Ileill
lIeill
IIXII.
Thus in a Banach space the coordinates of a vector with respect to a Schauder basis are continuous linear functionals (theorem of Banach). A Schauder basis (e,); in a separable Hilbert space E is called a Riesz basis if the set i? coincides with 12. In this case “:P
J
i& lti12 < C0
(t E G
and, by the BanachSteinhaus theorem, II [ Ill2 < C II [ 11~.Since i is complete under both norms they are equivalent, i.e. 1: E^+ l2 is an isomorphism of the Banach spaces. Choosing an orthonormal basis (ai): in E we consider the isometry of l2 + E which is defined so that Si H Ui (i = 0, 1,2, . . .). Then the composite isomorphism E + E^+ l2 + E carries the Riesz basis (eJ$ onto the orthonormal basis (ui)$. Conversely, the image of an orthonormal basis under the action of any isomorphism of E + E is a Riesz basis, since under topological isomorphisms a Schauder basis carries over to a Schauder basis and the coordinate spaces are preserved. Thus we have Theorem of Bari. The Riesz bases in a separable Hilbert space E are the images of the orthonormal bases under all possible isomorphisms of E + E.
The Riesz bases therefore form an equivalence class with respect to the action of the group of automorphisms of the Hilbert space. Corollary. Any Riesz basis (ei)$ is unconditional sense that infi llei 11> 0, sup, (Iei II < CO.
and almost normalised
Toeplitz (1926) showed that any unconditional basis in a Hilbert space is a Riesz basis.
almost normalised
hus
9 is an isomorphism
of the Banach
spaces B, E.
in the
Schauder
160
Chapter 2. Foundations and Methods
Thus all unconditional almost normalised Schauder bases in a Hilbert space (which can be regarded as 1’) are equivalent. The situation is similar in I’, in only in the stated spaces (LindenstraussZippin, CO 58 and (up to isomorphism) 1969). Now we make a simple but useful remark relating to the theory of bases. Let (ei)g be a complete minimal system of vectors in a Banach space E. Then to each x E E there corresponds the formal expansion x N boo ci(x)ei, where (ri)~ c E* is the system which is biorthogonal to (ei)$. The partial sum S, = &o li(‘)ei (n = 0, 1,2, . . .) is a bounded projection onto Lin{ei};. If(ei)$ is a Schauder basis, the sequence (S,): converges strongly (to the identity operator) and consequently it is uniformly bounded (by the BanachSteinhaus theorem): sup IIS I < co*
(18)
”
Conversely, it follows from (18) that (ei)g is a Schauder basis since if x E Lin(e,}$ there exists a number v = v(x) such that S,x = x for all n > v(x), i.e. (S,,x); converges to x on a dense subspace and therefore also on the whole of E (because of (18)). We now mention two examples which illustrate the effectiveness of the IHT in concrete analysis. The first of these is due to Banach (1931). Example 1. Let us consider the system of linear equations
(1% Any sequence x = (&.)y such that the series (19) converges with sum equal to vi for each i = 1,2, 3, . . . is called a solution of the system. It turns out that, if the system (19) has a unique solution for all (rli)~, then the rows of the matrix (4~) have finite support, i.e. (19) is a Toeplitz system. For the proof we consider the space R of all x = (&,)y for which all the series (19) converge and introduce on R a locally convex topology by means of the seminorms Pitx)
=
suP $1 ” I
(i = 1, 2, 3, . . .).
%ktk I
This is a Frtchet space. The system (19) defines a continuous homomorphism g: R + s, where s is the space of all sequences with the topology of pointwise convergence (s is also a locally convex FrCchet space). According to the condition g is bijective. Consequently, h = gl is continuous. Since the topology on R is stronger than the pointwise topology, the linear functionals <,(hy) on s are continuous. But then (Y
=
(qj)?,
nk
<
ao).
s* cO is the space of sequences converging to zero with the sup norm. We note that co*z I
0 3. Linear Topology
161
The functionals <,(hy) (y E s) are linearly independent since this is true of &(x) (x E R, k = 1,2,3, . . .). Consequently, the rows of the matrix (/Jr) are linearly independent. By the theorem of Toeplitz the system of equations
has solution y = (qj)y for any (c&‘. But then cl, = &(hy) (k = 1,2, 3, . . .) and the series
kzl f converge. Because of the arbitrariness have finite support.
(i = 1,2, 3, . . .)
of (lk)y, the rows of the matrix (ai& must
Example 2. Let (A4.k be a sequence of positive numbers defining a quasianalytic class of functions on [0, 11. Let us consider the Banach space B(A4,) of infinitely differentiable functions cp(t) (0 < t < 1) for which 1 llcpll = sup max I# (t)l < co. n M. o
sup I(P n M,,
O)1 <*
and cpis uniquely determined by (cp O))~be cause of the quasianalyticity of the class. It is natural to ask if there exists for an arbitrary numerical sequence a = (a& satisfying the condition llall = sup H < cc n Mn a function cpE B(M,) such that cp O) = a, (n = 0, 1,2, . . .). The answer turns out to be negative except for the analytic case M, = O(n!). For the proof we consider the Banach space b(M,) of sequences satisfying condition (20) and the homomorphism g: B(M,) , b(M,) defined by the formula (gcp), = cp O). Clearly g is continuous ( I(g II = l), injective and, in the case where there is a positive answer to the question posed, bijective. But then h = gl is continuous. Let a, = o(M,) (n + co). Then a = 2 anan, n=O
where the series converges in b(M,). Consequently
(W(t) =n=O f an;, . where the series converges in B(M,)
and so in particular
at the point t = 1.
162
Chapter 2. Foundations and Methods
Moreover
Setting a, = &,M,
we obtain Mk < Ck!.
The last example of the application of the IHT which we present here is concerned ith the regularisation of Noetherian operators. Let E be any LTS in which the IHT holds and let T be a Noetherian operator on E whose image Im T is closed5’ (its kernel Ker T is also closed since it is finitedimensional). The regularisors for T are defined by the formulae (see Section 1.3) R, = (1  .4)FT , $ = ? , where A is a projection onto Ker T, P is a projection onto Im T and T = TIKerA. The projections A, P can be chosen to be continuous and FT’ is continuous in accordance with the IHT. Therefore both regularisors turn out to be continuous. A sufficient condition for the preservation of the Noetherian property, normal resolvability and the value of the index under the change from T to T + V, where I/ E 2(E), is that the operators 1 + R,V and 1 + VR, be invertible. An effective formulation is obtained from the following general result: in any Banach algebra the elements of the form e  x with I(xIJ < 1 are invertible.60 Thus in the Banach space E it is sufficient to impose on the perturbation V the condition of smallness of norm (Atkinson, I. Ts. Gokhberg, 1951), namely
IIVII < max(llRrII
IIW9.
Invertibility of the operator 1  U where II UII C 1 is a simple but very useful result. In terms of the equation x = Ux + y it implies unique solvability for all y (this also follows from the contraction mapping principle). Among its many applications there is, for example, the theorem on stability of buses: if (ei): is a Schauder basis in a Banach space E and (ci)g is a sequence of coordinate functionals then each system of vectors (ni); satisfying the condition
iz IICII * llei  411< 1 is a Schauder basis in E (M.G. Krejn, D.P. Mil an, the identities
X = 2 liei,
UX
=
igo
rifei
M.A. Rutman, 
1940). In fact
4)
i=O
define a linear operator U with norm less than 1, therefore S = 1  U is an automorphism; but Se, = ui (i = 0, 1, 2, . . .). n operator T with closed image is said to be normally resolvable. This term is connected with the equation TX = y. If E* is total, Im T = (Ker 7’*)I. In this case normal resolvability is equivalent to the identity Im T = (Ker T*),. 601n fact, the Neumann series ~~=oxk converges absolutely (since jlx’(I < Ilxll’) and its sum is (e  x)i. We note that it follows from this result that the set of invertible elements of a Banach algebra is open.
163
5 3. Linear Topology
Here is a further example. WienerPaley theorem. Zf (ei)g is an orthonormal system (ui)$ which satisfies the inequality
Iliz %(eiI
%)I2
G
0 i$
Jail2
basis in a Hilbert space, any
(0<8<
1).
for all possible systems of numbers (ai)” is a Riesz basis. 3.9. Linearisation of Mappings. Let E,, E2 be Banach spaces and let G c E, be an open set. A mapping F: G + E, is said to be (Frtchet) dtfferentiable at the point x E G if there exists a bounded homomorphism g: E, + E2 such that
IIW + d  Fx  wll = 4II4I)
@E&, II4 +O).
The homomorphism g is uniquely defined. It is called the (Frdchet) derivative of the mapping F at the point x and correspondingly it is denoted by F’(x) (further it can be written in the traditional way: dF(x) = F’(x) dx). From standard results of elementary analysis we note the formula of finite increments in the form 1 Fx,  Fx, =
F’(x, + z(x2  x1))(x2
 x1) dz.
s0 The following are conditions which are sufficient for its validity: 1) [x1, x2] c G; 2) F is differentiable at all points of the interval [xi, x2]; 3) the derivative F’(x, + t(x2  x1)) is a weakly continuous function of r (0 < z < 1). They are clearly satisfied in a convex domain G for a differentiable mapping whose derivative is weakly continuous. If moreover q = SUP,,~ IIF’(x)I) < co then the mapping satisfies the Lipschitz condition lb,
 Fx, II G 4 lb2  XI II
(~1,
x2
E ‘3.
Consequently it is uniformly continuous and extends by continuity>0 a mapping F: G + E,. Let E, = E, and suppose that FG c G, q < 1. Then F is a uniform contraction and consequently there exists in G a unique fixed point Z; for any x0 E G the iterates FnxO converge to X with bound O(q”). For any differentiable mapping F: G + E, the mapping x H F’(x) is defined from the set G into the Banach space Homc(E,, E,). If this mapping is differentiable at some point x E G then by definition its derivative F”(x) (the second derivative of F) belongs to the space Homc(E,, Homc(E,, E,)). Subsequent derivatives are defined in a similar way. Linearisation of the mapping @: G + E, allows us to construct the Newton process x,+1
=
x,

[@‘(x,)] x,
(n=0,1,2,...)
for solving the equation @x = 0. The convergence of this process and of certain of its modifications was investigated in detail by L.V. Kantorovich (1948). In all cases, if 1) the root x is simple in the sense that the operator Q’(X) has a bounded
164
Chapter 2. Foundations and Methods
inverse, 2) the second derivative W(Y) exists and is continuous in the uniform topology on some ball D(X, 6) = (x: 11x Xl1 < S}, then for all x0 in a sufficiently small ball 0, the Newton process will be defined and will converge to the root X In fact, the formula Fx = x  [@(x)1 x defines a differentiable
mapping F: 0, + E, :
F x) dx = [F x)] F x)
dx)[F x)]lFx.
F x) is continuous in the uniform topology and F F) = 0. Therefore supxE,,. 11F x) 11< 1 for sufficiently small E. A further standard area for applications of linearisation is the local theory of differential equations. The language of functional analysis provides a single means for considering evolutionary systems, both finitedimensional (systems of ordinary differential equations) and infinitedimensional (partial differential equations, integrodifferential equations). An bstract’ evolutionary system in a Banach space E has the form i = @(x, t), where x = x(t) is a vectorfunction with values in E defined on some interval Z of the real axis (we will assume further that it is on the semiaxis t > 0) and having continuous derivative i(t), and #: G x Z + E where G c E is, as before, some open set (for simplicity we will take G = E). If the mapping @ is continuous and in each ball D c E it satisfies a Lipschitz condition in x uniformly with respect to t on each finite interval, then the Cauchy problem which consists of finding a solution of the equation satisfying the initial condition x(O) = x0 has a unique solution on some interval 0 < t < E. The proof follows the classical scheme of Picard:
x(t) =XIJ +sf@(x(4,4
i.e. the required solution is a fixed point of the mapping defined by the right hand side on the space of continuous vectorfunctions on [0, E]. For sufficiently small Ethis mapping is a uniform contraction on a neighbourhood of the constant x0. The requirement of continuity of the mapping @ is not satisfied (in the Banach space setting) if, for example, @ is a differential operator. Therefore the scheme which has been described is not suitable for partial differential equations. Even for the investigation of the linear equation 1= Ax where the operator A is unbounded we require special methods which are connected with spectral theory (see Section 4.5).
$4. Theory
of Operators
165
0 4. Theory of Operators 4.1. Compact Operators. A homomorphism h: E, + E, of any LTSs is said to be compact or completely continuous if there exists in E, a neighbourhood of zero whose image is relatively cpmpact.’ Any compact homomorphism is clearly bounded and so continuous. The property of compactness is valuable because it is the basis of a satisfactory generalisation of Fredholm theory. Such a generalisation was effected for normed spaces by F. Riesz (191Q for LCSs by Leray (1950) and for arbitrary LTSs by Williamson (1954). The classical Fredholm theory is easily covered by these generalisations since linear integral operators are compact in the corresponding function spaces under the usual assumptions (for example, operators on C(S) (S compact) with continuous kernel or operators on L2(X, p) with HilbertSchmidt kernel are of this type). The compact homomorphisms form in Homc(E,, E,) a subspace Ci(E,, E,) which is invariant with respect to the natural action of !i!(E,) x !i?(E,): h H ahb (a E i?(E,), b E !i?(E,)). Thus (E(E, E) = CC(E) is a twosided ideal in the algebra L?(E). If E, is complete c(E,, E,) is closed in the uniform topology. The subspace B(E,, E,) of continuous, finite rank homomorphisms and consequently its uniform closure g(E,, E,) are contained in 6(E,, E,). There is considerable interest in the question of when (I;(E,, E,) and g(E,, E,) coincide, i.e. of the possibility of uniform approximation of an arbitrary compact homomorphism by continuous, finite rank homomorphisms. In fact Fredholm constructed his theory of integral equations in this way by approximating their kernels by degenerate ones, i.e. by kernels of the form
Let E, be a barrelled LCS with a Schauder basis (ei)g and let h be a compact homomorphism of any LTS E, into E,. Let us consider the partial sums
S,X= k tiei
(x=2&i).
i=O
The family of linear operators S, (n = 0, 1,2, . . .) is equicontinuous (by the BanachSteinhaus theorem). Therefore if U is a neighbourhood of zero in E, such that hU is relatively compact, (S,h),” converges to h uniformly on U and consequently this convergence is uniform on each bounded set X c E,. But all the S,h are clearly of finite rank and continuous. Thus in the case under consideration S(E,, E,) = CC(E,, E2). We say that an LTS E has the approximation property if the identity operator (and hence also any continuous operator) is a limit point of the set g(E, E) = g(E) 1 A continuous mapping X P Y of any HausdorlT topological spaces is said to be compact a covering of the space X by open sets whose images in Y are relatively compact.
if there is
166
Chapter
2. Foundations
and Methods
of finite rank operators under the topology of uniform convergence on relatively compact sets. We saw above that any barrelled space with a Schauder basis has the approximation property. We can show as above that if E, has the approximation property then g(E,, E,) = 6(E,, E2) for any LTS E,. The converse of this assertion holds in the category of Banach spaces. In the example constructed by Enflo of a separable Banach space with no Schauder basis even the approximation property is lacking. Remark. Let El, E, be reflexive Banach spaces, at least one of which has the approximation property. If the Banach space Homc(E,, E2) is reflexive (and only in this case), every continuous homomorphism of E, + Ez is compact, i.e. Homc(E,, E,) = CC(E,, E,) (Holub, 1973). Passing directly to Fredholm theory in any LTS E, we denote by A an arbitrary compact linear operator on E and fix a neighbourhood of zero lJ for which the image AU is relatively compact. A theory will be constructed for the operator T = 1  A, one of its basic results being that T is a Fredholm operator (i.e. it is Noetherian and ind T = 0) and it is normally resolvable (i.e. Im T is closed). We note at once that the subspace Ker T is finitedimensional since AIKcrT = 1 and consequently Ker T is locally compact. Lemma 1. The operator T is normally resolvable. Proof. Using the compactness of A, it is easy to verify that if X c u and X is closed then TX is closed. Therefore the ylinder’ V = T‘(Tu) = u + Ker T is closed while at the same time the cylinder V, = T‘(TU) = U + Ker T is open. Consequently 2 = V\ V, is a closed set. Its nflation’ 2 = {x: x = az (a E 2, a 2 l)} together with the cylinder V covers the whole space E. But TV = To is closed. It remains to establish the closedness of the set Tz = fi. This follows from the fact that the set T(u n v,) is closed and does not contain zero. 0 Corollary.
The subspaces E, = Im T” (n = 1,2, 3, . . .) are closed.
The Fredholm Alternative. Either the operator T’ is continuous) or Ker T # 0.
T is invertible (and in this case
Proof. Suppose Ker T = 0. Using the compactness of Tit is easy to verify that the homomorphism S which is inverse to E 5 Im T is continuous Let us suppose that Im T # E and take x 4 T. We consider the ndimensional subspace L, = Lin{Tkx};’ (n > 1, Lo = 0). Since AIL, is injective and maps L, n 0 into a compact set we have that L, n 0 is compact. Because of the closedness of Im T and the continuity of S we can construct a neighbourhood of zero U’ such that (L, + U n Im T c TU. Now we take a balanced neighbourhood of zero u” such that u” c U n u’ and choose in L, n 0” (n 2 1) any point x, & L,, + U Clearly x, # x, (n # m). We have x, = TY,~ + a&, where y,, E L,,. Since Ty,, E L, + t?” then TY,~ E TU and hence y,, E U because T is injective. Consequently the set ’ Under
additional
assumptions
about
E this is guaranteed
by the inverse
homomorphism
theorem.
5 4. Theory of Operators
167
{Ay,} is relatively compact. But then it must be finite since Ay,  Ay, z x,+,(mod
L,)(n > m)
and hence Ay,  Ay, $ U”. As a result the set (x contradiction. 0
turns out to be finite  a
Now let us consider the root subspaces W, = Ker T” (n = 1,2,3,. . .). Since the intersections W, n 0 are compact, if W, # W,, (n = 2,3, . . .) we can choose for each n a point x, E (W, A u)\(W,, + U). Then Ax,  Ax, $ U (n > m) in spite of the relative compactness of {Ax,}. Consequently there exists a number v such that W, = W,,, = . . * (W, # WV,). The maximal root subspace W, is tinitedimensional. The operator T maps the invariant subspace E, = Im T’ into itself and moreover E,+1 is its image. But Ker T n E, = 0. By the Fredhohn alternative E,+1 = E, and then E, = E, (n > v), in particular E,, = E,. Therefore for any x E E there exists y E E such that T = T , i.e. x  T E W,. Thus we have established Lemma 2. The following
direct sum decomposition
holds:
E = Ker T’ i Im T Corollary.
The operator
T is a Fredholm operator.
Proof. The codimension of the subspace Im T r> Im T’ is clearly finite and equal to the codimension of Im( T lKerTY) (since T Ii,,, rV is bijective). As a result it is equal to dim(Ker T) = def T. The conjugate operator A* of a compact operator A on a Banach space is compact (theorem of Schauder), however this is false in general even in an LCS. Nevertheless, in any LTS def T* = def T since Ker T* = (Im T)* and Im T is a closed subspace with codimension equal to def T. Since T” = 1  A,, where A, is a compact operator, and (T * = (T*) we have def(T*)” = def T” (n = 1, 2, 3, . . .). The property of compactness of an operator on an infinitedimensional space is a special kind of smallness. This is evident in particular in the result that the index is stable under compact perturbations (Atkinson, I.Ts. Gokhberg, 1951). Theorem. Let S be a normally resolvable Noetherian operator on an LTS E for which the IHT holds and let V: E + E be a compact operator. Then S + V is Noetherian, normally resolvable and
ind(S + V) = ind S. Proof. The regularisors R,, R, for S are continuous.
In the usual notation
we
have R,(S+
V)=l+(R,VA),
(S + V)R, = 1 + (VR,  B).
Since the parts added to the identity operator are compact we have that R,(S + V) and (S + V)R, are normally resolvable Fredholm operators, from which all the assertions follow. 0
168
Chapter
2. Foundations
and Methods
One of the most important areas of application of this theorem is the theory of singular integral equations 3. The kernel of a singular integral operator is frequently the sum of a standard singular kernel (for example, the Cauchy kernel ([  z)‘) and a regular4 kernel which defines a compact operator. In this case the calculation of the index is reduced to that for the standard kernel. Compactness of integral operators with regular kernels is brought about by the existence of finite rank approximations. Moreover the properties of an operator are better the more rapidly it can be approximated by finite rank operators. This observation leads us to define the approximation numbers (anumbers) of a compact homomorphism h: E, + E, of normed spaces as its deviations (n = 0, 1,2, . ..) a,+,@) = g*; Ilh  911 n from the subspaces ?j, = &(E,, E,) = {g: g E 7j(E,, E,), rk g < n]. Clearly, llhll = a,(h) 2 a,(h) 2 . . . and a,(h) + 0 if and only if h E $j(E,, E,). Putting G,(E,, E,) = ‘&JX,, E,) we define further the class G,(E1, E2) (0 c p < co) by the requirement
The spaces GJE,, E,) (0 < p < co) are invariant with respect to the action of f?(E,) x f?(E,); consequently G,(E) = G,(E, E) is a twosided ideal of the algebra WI. For any h E GJE,, E2) the quantity
N,(h) = (.fl b.(h)lp~ defines a quasinorm. Thus G,(E,, E,) is a complete LTS. If E, and E, are Hilbert spaces then N, is a norm. In this case the class G,(E,, E,), whose elements are called HilbertSchmidt homomorphisms, is of special interest. Suppose that E, and E, are separable and let (ej”);, (e~“)~ be orthonormal bases in E,, E, respectively. In order that h E Homc(E,, E,) be a HilbertSchmidt homomorphism it is necessary and sufficient that i$ Ilhei lZ =
I(he$
ej2 12< co,
i, j=O
since this quantity is also N2 (the HilbertSchmidt norm). Any integral operator on L X, p) with a HilbertSchmidt kernel is a HilbertSchmidt operator, for example, the operator on l2 defined by the matrix (aji)Ti=o such that
,I, 3 In this context S.G. Mikhlin in a Hilbert space. 4 Or eakly singular
(1947) discovered
lclji12< 00. the stability
ofthe
index under
compact
perturbations
44.
Theory
of
Operators
169
One of the numerous applications of the class $ in a Hilbert space E is concerned with the theory of bases. Let (I@ be an orthonormal basis in E and (ui): an olinearly independent system which is quadratically close to the basis (ei); in the sense that
Then it is a Riesz basis and it is called a Bari basis (in honour of N.K. Bari who established the result in question in 1951). The proof is based on the fact that the operator defined on E by the equations Aei = e,  ui (i = 0, 1,2, . . .) is a HilbertSchmidt operator and so is compact. If T = 1  A then Tei = Ui (i = 0, 1,2, . . .). But Ker T = 0 because of the olinear independence of the system (Ui)~ . Therefore T is an automorphism of the Hilbert space E. Remark. M.G. Krejn (1957) characterised a Bari basis (Ui)gmby means of a certain collection of intrinsic properties, namely, 1) completeness, 2) olinear independence, 3) quadratic closeness of the Gram matrix ((Ui, Uj))Tj=o to the identity, i.e.
i go I(%, uj)  4j12 < C0* Let E,, E2 be Banach spaces. A homomorphism nuclear if it can be expanded in a series of the form
h: E, + E2 is said to be
(21)
where fk E ET, uk E E,, llfkll = 1, )IuJ = 1 and & (~1 < co; thus series (21) converges in the uniform operator topology so that all nuclear homomorphisms are compact. The class of nuclear homomorphisms %(E,, E,) in Homc(E,, E2) is an L?(E,) x g(E,)invariant subspace (in f?(E) it is a twosided ideal 3(E)). The nuclear norm in YI(E,, E,) is defined as v(h) = inf ck lcxkl over all representations of the form (21). Clearly llhll < v(h). With respect to the norm just introduced ‘%(E 1, E2) is a Banach space. Pietsch (1963) showed that all homomorphisms of the class 6, are nuclear and moreover for any p > 0 there exists a nuclear operator on 11 which does not belong to (Sp. In a Hilbert space the class of nuclear operators coincides with 6,. If A: E + E is a nuclear operator on a Banach space E we associate with each of its representations in the form (21) (f. E E*, u, E E) the number (22)
which we may call the truce5 of the operator A and denote by tr A whenever this 5 More precisely, not equivalent.
the tensor
truce.
There
are several
other
definitions
of trace which
in general
are
hapter 2. Foundations and Methods
170
number is independent of the representation under consideratio#. Grothendieck (1955) proved that the existence of the trace for all nuclear operators on E is equivalent to the approximation property; moreover in this case the trace is a continuous linear functional on %(E) (with respect to the nuclear norm v) and its norm is equal to one. This applies in particular to a Banach space with a Schauder basis. On Enflo space there exists a nuclear operator which does not have a trace, i.e. it is such that the quantity (22) depends on the representation of the operator in the form (21). In any Banach space all operators of class 6, have a trace (Pietsch, 1963). If a Banach space E has a Schauder basis (ei); then corresponding to each operator A E L?(E) there is a matrix a = (aji)~ii=, which is defined by the expansion Ae, = 2 ajiej
(i = 0, 1, 2, . . .).
j=O
We can consider its trace tr a = g ajj j=O
whenever this series converges. If A is a nuclear operator tr a exists and is equal to tr A and so the trace of the matrix of the operator does not depend on the choice of basis’ (G.M. Litvinov, 1979). Consequently, if the basis is unconditional,
(A.S. Markus, V.I. Matsaev, 1971). Let A be a nuclear operator on a Banach space E which has the approximation property and suppose that v(A) < 1. Then we can define the determinant det(1 + A) = exp{tr[ln(l
+ A)]},
wheres ln(1 + A)d; zl ( l)n . The prior condition v(A) < 1 can be removed by the following means: whatever the nuclear operator A, the function D(n) = det(1 + LA) of the complex variable 6 Clearly tr A is a linear functional on its domain of definition. n this situation we can refer to the matrix trace of the operator A. n general, if cp(l) is an analytic function on the disk (II < r, for any operator A E Q(E) such that l[All < r we put m p 0) d4 = 1 n!A *=0 This is one of the principal versions of jimctional caZculus in Q(E).
8 4. Theory
of Operators
171
1 is analytic o\n the disk 111< [v(A)]‘. It can be shown that D can be continued analytically to the whole plane, i.e. it is an entire function. It remains to take as definition: det(1 + A) = D(1). Remark.
If
A = k=l2 gk(.bkT where rk E E, gk E E* and & llgkll ’llVk[l < Co, then det(1 + AA) = 1 + F c,A”, n=l
det(gjJujs)):,,,l (n = 1, 2, 3, . . .) (KochGrothendieck where cn=&,<...<jn mula  a generalisation of a wellknown formula from linear algebra).
for
Example. det(1 + g()u) = 1 + g(u). The theorem on multiplication of determinants extends to the situation just described. As a result of this it follows that det(1 + A) # 0 is a criterion for invertibility of an element 1 + A (A E s(E)) in the algebra %(E) with the identity adjoined. Continuous linear functionals on the Banach spaces 6,(E) where E is a Hilbert space can be described in terms of the trace, namely f(A) = tr(AF) where F E 6,(E)
 +  = 1 , and the correspondence thus established between : > 6,(E)* and 6,(E) is in fact an isometric isomorphism. For p = 2, i.e. in the space of HilbertSchmidt operators, we can introduce a scalar product c
(A, B) = tr(AB*), with respect to which &(E) (with the previous norm) is a Hilbert space. In this particular case the previous formula follows from the theorem of Riesz. The concept of a nuclear homomorphism carries over to the category of LCSs if we replace the conditions on fk, uk by the following: the family {fk} is equicontinuous, the family {&} is bounded and its closed absolutely convex null U provides a norm on 0 = Lin U with respect to which 6 is a Banach space. All nuclear homomorphisms are compact. The theory of the trace outlined above extends to nuclear operators on an LCS (G.M. Litvinov, 1979). Starting from the previous definitions Grothendieck (1955) introduced the important class of nuclear spaces as those9 spaces for which every continuous homomorphism into a Banach space is nuclear. The class of nuclear spaces is closed with respect to completions and passage to subspaces and factor spaces (and certain other
his
is one of several
equivalent
definitions.
172
Chapter
2. Foundations
and Methods
natural operations). Nuclear spaces are encountered frequently in analysis. In particular the usual spaces of infinitely differentiable and analytic functions and many natural spaces of sequences are nuclear. It is appropriate to note that all nuclear barrelled complete LCSs are Monte1 spaces (consequently they are reflexive). Thus if a space is nuclear and Banach, then it must be finitedimensional. If E,, E, are LCSs a homomorphism h E Homc(E,, .I&) is said to be absolutely summing if each unconditionally convergent series c,,x” in E, is mapped to an absolutely convergent series 1. hx, in E,. If E,, E, are Hilbert spaces the absolutely summing homomorphisms are precisely the HilbertSchmidt homomorphisms (Grothendieck, 1956). The same author also established the following deep result. Theorem. Zf h: E, + E 2, g: E, + E, are absolutely of Banach spaces then gh: E, + E, is nuclear.
summing homomorphisms
From this we obtain immediately the DvoretskiRogers theorem on the existence in any infinitedimensional Banach space E of a series of vectors which converges unconditionally but not absolutely. In fact, in the contrary case, the operator 1: E + E is absolutely summing, therefore nuclear (1 = 12) and consequently compact. Closely connected with the DvoretskiRogers theorem is Grothendieck criterion for nuclearity of a space: in order that a locally convex Frechet space be nuclear it is necessary and sufficient that all unconditionally convergent series of vectors in it converge absolutely. Associating with any LCS E the space I1 (E) of unconditionally convergent series and the space 1’{E} of absolutely convergent series, Pietsch (1963) extended this criterion to the general case. He provided these spaces with natural locally convex topologies such that the embedding i: l E) + l E) is continuous and for nuclearity of the space E it is necessary and sufficient that i be a topological isomorphism. The problem of the existence of a Schauder basis in a nuclear space has a rather different character than in a Banach space, since all nuclear locally convex Frechet spaces have the approximation property. However there also exists in this class a space with no Schauder basis” (N.M. Zobin, B.S. Mityagin, 1974). If there is a Schauder basis in a space of this class, then it is absolute (A.S. Dynin, B.S. Mityagin, 1960). A nuclear LCS has certain special structure. For its description we make use of (without discussion) the general topological concept of the projective limit of a directed family of topological spaces. A projective limit of LTSs (LCSs) is an LTS (LCS) which is complete if all the spaces in the given family are complete. Any complete LCS is isomorphic to a projective limit of Banach spaces. Any complete nuclear LCS E is isomorphic to a projective limit of a certain family of Hilbert spaces. If in addition E is metrizable, the family concerned can be chosen
lo We note that any nuclear
locally
convex
Frkchet
space is separable.
$4. Theory of Operators
173
to be a sequence (E,),” (with the natural ordering); in this sense E is countablyHilbert.
Let us pursue further the question of topologising the tensor product E, 60 E, of two LCSs E,, E,, which is closely connected with the preceding material. The algebraic object E, @ E, is obtained by factorising the space of formal linear combinations’ ’ c Ak(Uk x Vk) bk E El> vk E E2) k
by the linear hull of the set of all combinations
of the following types:
u x (VI + VJ  u x Vl  u x v,;
(Ul x u2) x v  u1 x v  u2 x v;
u x uv  a(u x v).
au x u  u(u x v);
The canonical image of the element u x v E E, x E, in E, @ E, is denoted by u @ v. If E is a further linear space then for any bilinear mapping 8: E 1 x E, + E there exists a unique linear mapping 4: E, @ E, + E such that the following diagram is commutative (j is the canonical mapping):
For any two seminorms p (I), pt2’ on E,, E, respectively we define their tensor product by
(p
@p
where the infimum
(w)= inf T p (uk)p
uk)
is taken over all decompositions
(w
E E,
C3 E,),
of the form
w=~u,@v,. k
This is a seminorm
on E, 6 E, which has the crossproperty: (p
@ pq(u
@ 0) = p (u)p
v).
If E, , E, are LCSs their topologies are given by families of seminorms {p! , respectively. Then the family {p!‘) @ pr)} defines a locally convex topology on E, 8 E, which is the strongest under which the canonical mapping j is continuous’2. The completion of the space E, ($3E, with respect to this topology is denoted by E, @ E, and is called the topological tensor product of the spaces {pr)}
El,
E,.
Theorem of Grothendieck. If E,, E, are metrizable w E E 1 @ E, can be represented in the form
LCSs then each element
I1 i.e. the space of functions of finite support C&E, x E,, K). here are other natural topologies on E, @IE, but we will not discuss them here.
174
Chapter 2. Foundations and Methods
w = k=l2 ak(ukC3v,), where u& E E,, lim&+, u& = 0, u&E E,, lim&,,
II& = 0 and c;=i
lakl < 03.
In this sense all elements of E, 6 E, are nuclear. If E, , E, are normed spaces then E, & E2 is a Banach space with norm defined by the ,tensor product of the norms given in E, and E,. Particular interest attaches to the tensor product of the form E* @ E. In this case we have the natural homomorphism Tfk
z (
@
=
$fk( k
>
from E* @ E to S(E). In the Banach space case the topological tensor product E* @ E is also a Banach space. Since the inequality v(zw) G IIw 1)(v is the nuclear norm) holds for all w E E* @ E, then r can be extended by continuity to a homomorphism 2 E* &I E + %(E) which is clearly surjective. Consequently ‘%(E) z (E* @ E)/Ker z This topological
isomorphism
is an isometry.
4.2. The Fixed Point Principle. A famous theorem of Brauer asserts that if X is a compact convex set in a finitedimensional linear space then any continuous mapping F: X + X has a fixed point13. It was extended to inlinitedimensional normed spaces by Schauder (1930) and then to LCSs by A.N. Tikhonov (1935). Theorem. Let E be an LCS, X c E a nonempty closed convex set and F: X + X a continuous mapping such that the set Y = FX is relatively compact. Then there exists a fixed point: x = Fx. Proof. If x # Fx for all x E X, there exists a neighbourhood of zero U such that x  Fx 4 U (x E X). Let V be an absolutely convex neighbourhood of zero such that V + V c U. We can construct a finite Vchain (x& + V}; on the set K The convex hull P c X of the set {xk}T is compact and lies in a subspace of dimension
$4. Theory
of Operators
175
this leads to the contradiction y  Fy = Py  Fy = I$1 Tj(PYi,  FYij) + 2 rjtFYij  Fy) ’ ’ + ’ c ’ j=l
for some
Zj
> 0
(Cj
Zj
= 1). 0
The fixed point principle has many important will go into the details of some of these here.
and interesting applications.
We
Example 1. Let us consider a Banach space E which is ordered by a closed cone K. Let T: E + E be a compact (in general nonlinear) mapping which is monotone in the usual sense, i.e. x < y + TX < Ty, and positively homogeneous: T(ax) = aTx (a > 0, x E E). If under these conditions there exists a vector u 2 0 ( /lull = 1) such that TV > cu for some c > 0, then there exists an eigenvector e 2 0 (e # 0) corresponding to some positive eigenvalue15 112 c. This theorem of Rutman is a generalisation of the classical PerronFrobenius theorem concerning a matrix with nonnegative elements16. For the proof l7 we consider the closed cone K, = {x: x 2 E1IxI1u} (0 < E K 4) in E and a functional f E K* such that f(u) > 0, Ilf II = 1. If x E K, then TX > Ellxll cu. We extract from K, a closed convex subset X, by imposing the additional conditions: IIx (I < 1, f(x) > ef (0). It is nonempty: if x E K, and llxll = 1 then x E X, (in particular, u E X,). If x E X, then llxll 3 &f(u) and consequently TX # 0 for x E X,. The formula TX + 2eIITxllu FEx = IITx + 2.~1)TxIlull
defines a continuous mapping of the set X, into a relatively compact subset of it (because of the compactness of T). By the fixed point principle there exists x, E X, such that FExE = x,, hence IIx,II = 1 and TX, = &x,  p,u, where 1, 3 c, & > 0. Using the compactness of T once again, we choose E, JO such that that sequence ( Tx,,),“=~ converges. Then (x,,)$i will also converge to some x > 0 ( (Ix II = 1) and (Q, will converge to some 1 3 c. We obtain in the limit TX = Ax. Example 2. Let us consider the problem of the existence of a nontrivial closed invariant subspace for a continuous linear operator A on an LTS E. Using the algebraic construction (see Section 1.2) with subsequent closure does not generally give the required result since the possibility that Lin{Akx}, , = E for all x E E is not excluded (however in this case E is separable). In 1984 Read constructed a counterexample in a certain nonreflexive Banach space. However, if A is a compact operator on an arbitrary Banach space, it has a nontrivial closed invariant subspace (AronszajnSmith, 1954).
IsThe concepts of eigenvalue and eigenvector for nonlinear mappings are defined exactly as for linear mappings: Te = r2e. r6The corresponding generalisation for integral operators was obtained by Yentsch (1912). he fact that the PerronFrobenius theorem can be obtained by the method of fixed points (in fact from Brauer theorem) was noted by P.S. Aleksandrov and Hopf (1935).
176
Chapter 2. Foundations and Methods
Does there exist such a subspace for a pair of commuting compact operators? Using the fixed point principle, V.I. Lomonosov (1973) obtained the following general result”. Theorem. Let A be a nonzero compact operator on an infinitedimensional LCS E. Then the algebra (A)’ of all continuous operators which commute with A is (topologically) reducible, i.e. there is a nontrivial closed subspace which is invariant for all” TE (A)‘.
The following lemma, which also has an important constitutes a fundamental step of the proof.
independent
significance,
Lemma. Let 52 be an irreducible subalgebra of the algebra of continuous operators on an LCS E which contains a compact operator A # 0. Then there is a compact operator B E 9l which has a fixed point x # 0. Proof. For any y E E, y # 0, the subspace L, = {z: z = Ty (T E ‘%)}is different from the zero subspace, invariant for all S E I and, consequently, dense. We choose a point x0 such that Ax, # 0 and a convex neighbourhood of zero U such . that the set Q = Ax, + AU 1s compact and does not contain zero (clearly, Q is convex). Then for each y E Q there is an operator TYE Isuch that T,y  x,, E U. By continuity Tyv  x0 E U for all v E Q close to y. There exists therefore a finite collection {K}; c such that for each point v E Q at least one of the inclusions TV  x0 E U holds, i.e. at least one of the inequalities q( TV  x,,) < 1 (1 < i < n) holds, where 4 is the gauge of the neighbourhood U. We put
1(r) = max(1  z, 0) 4i(v) =
wiV
 x0))
,z 4(4W
The mapping
(1 < i < n).
 x0))
n @V
=
2 i=l
4i(V)KV
(0 E
Q)
is continuous and so @Q is compact; moreover @Q c x0 + U since if TV 4 x0 + U then $i(v) = 0. The mapping F = @A has a fixed point x in x0 + U. It is fixed for the compact operator B=
~ ~i(AX)T,AE~ i=l
and x # 0 since 0 $ x0 + U. 0 I8 For Banach spaces; however the extension to LCSs does not require significant changes. 19(A)’ is called the centralizer (in general algebraic terminology) or the cmnmutant (in operator theory terminology) of the operator A. The centralism (commutant) M’ of any set M c UT(E) is defined as the intersection of all (A)’ (A E M). If !!I c f?(E) is a subalgebra then QI n %’ is the centre of the algebra 9X. In any case it contains the scalar operators A = 11 (I. E K). 201t is said to be hyperinvariant for A.
$4. Theory of Operators
111
It is suhicient to prove the theorem of Lomonsov in the complex case. If (A)’ is irreducible, by the Lemma there is a compact B E (A)’ for which L = Ker(1  B) # 0. Since L is finite dimensional and invariant for A then Al, has an eigenvalue ,4.The eigenspace Ker(A  Al) is invariant for TE (A)’ and consequently it coincides with E, i.e. A = Al, hence A = 0 since it is compact and E is infinite dimensional. 0 Remark. The infinitedimensionality of E is not used until the last step. Therefore if we require that A is not scalar 21, the assertion of the theorem will also be true in the finite dimensional case but of course this case allows a purely algebraic treatment: a classical theorem of Burnside asserts that any irreducible subaglebra of the algebra End E with finite dimensional E coincides with End E. Therefore if (A)’ is irreducible, then A commutes with all operators in E. But then, by Schur lemma, A is scalar. Corollary 1. Zf T is a continuous operator on an LCS22 E and the algebra (T) contains a compact operator A # 0, then T has a nontrivial closed invariant subspace. Corollary 223. Let 2I be an irreducible algebra of operators on an LCS E which contains a compact operator A # 0. Then its centraliser ’ consists of the scalar operators. Example 3 (MarkovKakutani theorem). If E is an LCS, X c E is a compact convex set and (Ai}ieI is any commuting family of continuous affine mappings of X + X, then there exists a common fixed point x E X for all Ai. In fact, for each Ai its set of fixed points Xi is nonempty, convex, compact and invariant for all Aj. By induction these properties carry over to finite intersections the intersection of all the Xi is nonempty. xil n . * * n Xin. Consequently One of the applications of the MarkovKakutani theorem is concerned with invariant means. Let us define this important concept. Let S be any semigroup (to be specific we take it to be multiplicative). Any nonnegative linear functional m on the space B(S) which is invariant with respect to left shijts (m[dJ = m[f, where bt(s) = &ts)) and normalised in the sense that m[l] = 1 is called a leftinvariant mean on S. Clearly, the value of ml41 is contained between inf 4 and sup 4 from which it follows that the functional m is continuous (llrnll = 1)24. If a leftinvariant mean exists” on S then S is said to be leftamenable. If S is a group and m is a leftinvariant mean, by putting 4*(s) = 4(sl) and m*[b] = m[r5*] we obtain a rightinvariant mean. Thus for groups (and clearly also for Abelian semigroups) we can speak simply of amenaIn the real case we have to require in addition that A is not a root of any quadratic equation with real coefficients and negative discriminant. 22dimE>10verCand>20verR. his is a generalisation of Schur lemma to the intinitedimensional case. 24Conversely ifp E B(S)* and satisfies the conditions y(l) = 1,llp)l < 1, then p 2 0. In fact if cpE B(s), llrpll = 1 and cp > 0, then 111 cplj d 1, hence ~(1  cp) < 1, i.e. p((p) > 0. *’ Moreover it need not be unique.
178
Chapter 2. Foundations and Methods
bility. Invariant means were brought into use by von Neumann (1929) who established the amenability of abelian groups and gave a simple example of a nonamenable group: the free group with two generators. Theorem.
Any Abelian semigroup is amenable.
Proof. In the space B(S)* the set A = {f:f(l)
= 1, U11 < l> = {f:f(l)
= l,f>
O>
is w*compact by the BanachAlaoglu theorem, convex and invariant with respect to the mappings which are the conjugates of shifts: (A,f)q5 =f(&. Clearly all the A, are affine and w*continuous on A and At1Af2 = A,,,l, i.e. the family {A,} commutes since the semigroup is abelian. By the MarkovKakutani theorem there is a common fixed point m for all the A,. This is also an invariant mean. 0 An invariant mean on the additive semigroup of natural numbers N is called a Banach limit. This is a nonnegative linear functional on the space of bounded sequences which is equal to 1 at (1, 1, . . . ) and invariant with respect to the shift (a,) + (a,,,). Its value at a convergent sequence is the same as the limit. A Banach limit provides a regular method for the summation of series with bounded partial sums. 4.3. Actions and Representations of Semigroups. Let S be a topological semigroupz6 and X a Hausdorff topological space. An action of S on X (or in X) is a family of continuous mappings T(s): X + X (s E S) such that T(st) = T(s)T(t) and the orbit T(s)x is continuous in s for each fixed x E X. If there is an identity e in S then T(e) is idempotent. If T(e) = 1, the action is said to be monoida12’. This condition is usually assumed to be satisfied if S is a group2* and consequently in this case all the T(s) are bijective. The actions of a semigroup S on a space X are also called (topological) dynamical systems on X; in this context X is called the phase space, the points x E X are the states of the system and the variable s E S is time. The following is the classical situation: S = R (pow), S = R, (semiflow2’), S = N (iterations of a mapping F: X + X). One of the central problems of topological dynamics is the existence of an invariant measure, i.e. a measure p on X such that
for all 4 E L1 (X, p).
sX
WW
& =
sX
4(x) dp
Z6i.e. a semigroup provided with a HausdorlI topology with respect to which multiplication is continuous. * semigroup with an identity is sometimes called a monoid. 281n a topological group not only multiplication but also inversion s I+ sl is continuous (by definition). * or the sake of brevity we sometimes also call them flows.
6 4. Theory
of Operators
179
BogolyubovKrylov Theorem 30. If the phase space X of a dynamical system is compact and the time semigroup S is leftamenable (in particular, Abelian), there exists an invariant measure p on X with ,u(X) = 1. Proof. Let m be a leftinvariant mean on S. We take any point x0 E X and consider the linear functional f (4) = m[q5(T(*)xo)] on C(X). Since it is nonnegative, by the theorem of Riesz it can be represented in the form jx 4 d,u. This measure p has the required properties. 0
Any semigroup S acts on itself (X = S) by left shifts: T(s)x = sx. Therefore if S is a compact abelian semigroup there is a measure on it which is invariant with respect to shifts. For groups a considerably more general result holds. Theorem of Haar.
On any locally compact group there is a left inoariant measure
(left Haar measure). In the same way there also exists a right Haar measure. Each of them is unique up to a positive multiple (von Neumann, 1936) but a left Haar measure need not be a right Haar measure. Groups for which left and right Haar measures are the same are called unimodular. In particular, all compact and all discrete groups are of this type. Haar measure is a very valuable tool for constructing the theory of representations of locally compact groups, in particular for the construction of general harmonic analysis. By a representation of a topological semigroup S in an LTS E we mean a linear action T on E, i.e. a strongly continuous homomorphism T: S + f?(E). Thus all the T(s) are continuous linear operators on E and T(s)x is a continuous function of s for each x E E; T(s1s2) = T(s,)T(s,). If T(s) depends continuously on s in the uniform topology then the representation T is said to be uniformly continuous.
A representation is said to be bounded if all its orbits are bounded. For a Banach space E this is equivalent to boundedness in norm: sups 11T(s)11< co. If lIT(s < 1 (s E S) the representation is said to be contracting and if all the T(s) are isometries it is called isometric. Any contracting representation of a group is isometric. Any bounded representation T of a semigroup S in a Banach space is contracting in the equivalent norm
IM = max IM sup s IITWII > . ( A representation of a group G in a Hilbert operators T(g) (g E G) are unitary.
space is said to be unitary if all the
Theorem. Let S be a rightamenable topological representation of it in a Hilbert space E. Zf 3o In a somewhat
generalised
form.
semigroup and T a bounded
180
Chapter 2. Foundations and Methods
infs IIW4
2 ~llxll
(a = constant > 0),
then there exists on E an equivalent Hilbert norm with respect to which T is unitary. Proof. We consider on S the bounded function 4Xx,Y(s) = (T(s)x, T(s)y) (x, y E E) and put (x, y) = m[&J. Then (T(s)x, T(s)y) = (x, y) (s E S) and a2 llx112 < 6, x> G b2 11412(B = sups IIWI). In many situations this theorem allows us to restrict attention to unitary representations. The following are classical examples: 1) finitedimensional representations of finite groups, whose theory was developed at the end of the 19th and the beginning of the 20th centuries (Frobenius, Molin, Bumside, Schur); 2) representations of compact groups in Hilbert space (the theory of unitary representations of compact groups was essentially constructed in the work of Peter and H. Weyl, 1927). If T is a finitedimensional representation of a semigroup S, the function x(s) = tr T(s) (s E S) is called its character. This term reflects the fundamental classical result that, for example, unitary representations with identical characters are indistinguishable from the geometric point of view; more precisely, they are equivalent in the sense defined below. Let T,, T2 be representations of a topological semigroup S in LTSs E,, E, respectively. Any continuous homomorphism V: E, + E, which satisfies the relation T,(s)V = VT,(s) (s E S) is called an intertwining operator for the pair of representations T,, T2. For example, V = 0 is such an operator. If among the intertwining operators there is a topological isomorphism U (so that T,(s) = UT,(s)Uwe say that the representations T,, T, are equivalent. If the space is a Hilbert space and U is an isometry we use the term unitary equivalence. In a series of situations generalisations of the concept of trace allow us to consider characters of infinitedimensional representations and to apply them for classification purposes. On the other hand, the characters of onedimensional representations are very important and interesting (especially in the Abelian case). They are the continuous homomorphisms of a semigroup S into the multiplicative group of the field C and are called (onedimensional) characters of the semigroup S. For example, the exponentials are the characters of the additive semigroup R, (the character which is identically zero is usually excluded from consideration). Now let G be a locally compact group and p a right Haar measure. The group G acts by right shifts on the Hilbert space L2(G, cl). This unitary representation R of the group G is said to be regular” and is the starting point of harmonic analysis on a group. If G = Iw or G = T (the unit circle) then p is Lebesgue measure and harmonic analysis on G is Fourier analysis. We now pass to a
31This term is retained for representations by means of shifts in any function spaces on a group which are invariant with respect to shifts.
$4. Theory
of Operators
181
consideration of the basic aspects of the spectral theory of operators with which harmonic analysis has a deep connection. 4.4. The Spectrum and Resolvent of a Linear Operator. Many applications require that spectral theory covers linear operators which are not necessarily defined on the whole of the underlying LTS E and are not necessarily continuous. The domain of definition D(A) of an operator A on E can be any subspace of E and in general it is neither closed nor dense, i.e. A is simply a homomorphism of D(A) + E. Such a generality has significant implications for the algebra of linear operators. For example D(A + B) = D(A) n D(B) and it can happen that D(A + B) = 0 for dense D(A), D(B). We note here that D(U) = BID(A), the Bpreimage of the subspace D(A). In accordance with the usual formal rules, just one change in the domain of definition changes the operator. If D(A) c D(J) and A aA, = A then A scalled an extension of the operator A and we may write A c A (in this connection the operator A is said to be a restriction or part of the operator A . Let A, B be operators such that BA c AB and B is defined on the whole space E. Then we say that they commute even if BA # AB, since if BA = AB and D(AB) = E then D(A) = E, a restriction which is unacceptable for many applications. Example. Suppose that D(A) = D(B) = E and A, B commute (AB = BA). If A is injective, the operator A’ (D (A’ )  Im A) commutes with B in the sense defined above. The theory of operators is concerned in the first place with their properties which are invariant with respect to automorhisms of the underlying space. From this point of view the operator UAW’ (U is a topological automorphism) is just as suitable as A; in this connection, just as for representations, we say that they are equivalent (the term similar operators is also used). In the finitedimensional case the problem of classifying linear operators up to similarity reduces to the theory of elementary divisors. In the infinitedimensional case the problem has on the whole a transcendental nature. However knowledge of certain, if not all, invariants already allows the possibility of penetrating deeply into the internal structure of an operator. The spectrum is the most important invariant of this sort. As the ground field in spectral theory it is expedient both from the internal point of view and in connection with applications to settle on the field C, which has a unique combination of properties: algebraic closure, local compactness, connectedness. A complex number I is said to be a regular point of a linear operator A on an LTS E if A  11 maps D(A) bijectively onto E and the resoluent R, = R,(A) = (A  II)’ is a continuous operator on E + D(A) (in this connection it is convenient to regard R,(A) as an operator on E + E by using the inclusion of D(A) in E; in any event Im R,(A) = D(A)). The set reg A of regular points of an operator A is called the resoluent set and its complement spec A is the spectrum.
182
Chapter 2. Foundations and Methods
The spectrum of an operator contains the set of all eigenvalues and sometimes reduces to this, for example, if A: E + E is a continuous algebraic32 operator. Example. Let P be a projection different from 0, 1. Then the set of its eigenvalues is (0, l} and if P is continuous spec P = (0, l} (otherwise, spec P = C). An eigenvalue 1 is said to be normal if corresponding to it there exists a linitedimensional root subspace with an invariant topological complement L and A # spec(A IL). For any linear operator A and all 3, E @we put A1 E A,(A) = Im(A  Al). The eigenvalues il are characterised by the fact that the homomorphism (A  11): D(A) + A,(A) is not invertible. For all other A the inverse homomorphism is defined. As before we can call it the resoloent of the operator A, denote it by R, = R,(A) and regard it as an operator on E with domain of definition A,(A); Im R,(A) = D(A). The eigenvalues of an operator A are contained in the approximate spectrum spec,A which is defined in the following way: I E spec,A if there is a neighbourhood of zero V such that for any neighbourhood of zero U a vector x $ V can be found for which Ax  kx E U. The set spec,A = spec A\spec,A is called the residual spectrum. The union rzg A = reg A u spec,A is characterised by the fact that the resolvent R,: A1 + E is a continuous operator (but AI # E for I E spec,A in contrast to the situation for regular points). Points 1 E rzg A (i.e. L $ spec,A) are said to be quasiregular. If an operator A on a complete LTS E is closed33, the subspace A,(A) is closed for all 1 E rzg A. In this case the factor space E/A, is called the defect space and its topological dimension n, is called the defect number of the operator A at the point A. The approximate spectrum of an operator A on a normed space E is the set of those ;I E C for each of which there exists a quasieigensequence of vectors (x.)p :
l&n IIX ) > 0. lim (Ax,  kc,) = 0, n+co n+4, We can assume moreover that llxnll = 1 (n = 1,2, 3, . . . ). The quasiregular points 1 can be characterised by an inequality of the form
IlAx  WI 2 c~II4l
(x E D(A), cI. > 0).
Thesetr~gAisthereforeopen(ifI~l
and D.P.
Theorem. Let A be a closed operator on a Banach space E. Then the defect number n, is constant in a sufficiently small neighbourhood of any quasiregular point. 32The requirement of continuity is essential; the definition of the spectrum in an LT.5 takes topology into account. 33This is guaranteed if reg A # (21.
183
5 4. Theory of Operators
For the proof it is sufficient to note that if y = Ax  J.x = (Ax  px) + (A  ,u)&y and (IyJI = 1 then d(y, A,u) < I,u  Ilc;‘. Therefore we have the bound for the gap 1 1 > <8(A,, A,,) 6 1~  L( max L 2 cl l  1p  A1 if 1~  Lj < *cl. But then np = dimt(E/A,)
= dimt(E/A,)
= n,. 0
Corollary 1. The defect number n, is constant on each connected component of the set rzg A.
In particular, if rzg A is connected, then either rzg A = reg A or rzg A = spec, A. Corollary
2. The resolvent set reg A and the residual spectrum spec,A are open.
In fact, these sets are separated out from rrg A by the respective conditions n, = 0 and n, # 0. Corollary
3. The spectrum of an operator A is closed.
However it can turn out to be empty. Example. For the differentiation operator on L O, 1) the natural (maximal) domain of definition consists of all absolutely continuous functions with derivative in L*(O, 1). Let us restrict the domain of definition by imposing the condition 4(O) = 0. We obtain a closed operator with no spectrum since its resolvent
’e ll/(t)dt s0 is a bounded operator on the whole of L*(O, 1) for ah I E @. We note further that any nonempty closed set S c @ is the spectrum of some closed linear operator. For this we consider the Hilbert space L*(S, ,u), where p is a measure on S which is positive on all open subsets X of S. Let us denote by A the operator obtained by multiplying by the independent variable [: (A&(c) = &(C). Its domain of definition consists of those q4E L*(S, p) for which c+(l) belongs to the same L*. The operator A is closed and if S is compact (and only in this case) then A is even bounded. It is easy to see that spec A = spec,A = S. The points with positive measure are the eigenvalues in this example. For any operator A the resolvent R, is an analytic function of 1 on reg A since the power series (R,+)(s) =
converges in the uniform is34 R,.
2 R;+ L  ,u)” n=O operator topology for II  ~1 < IIRpll’ and its sum I’
34This argument shows once again that reg A is open. Analyticity of the resolvent is the basis for the application of the methods of complex analysis in spectral theory.
184
Chapter 2. Foundations and Methods
Suppose that the operator A is bounded and D(A) = E. The Laurent series (23) converges in the domain )A1> l/All under the uniform operator topology and its sum is R,. Thus spec A lies in the disk 111< 11All. We see that the spectrum of a bounded operator is compti.,On the other hand spec A # 0 since otherwise R, is an entire function which tends to zero as 111+ co and so, by Liouville theorem, Rt F 0 which is impossible for an inverse operator. The smallest p = p(A) > 0 for which spec A lies in the disk 111< p is called the spectral radius of the operator A. Clearly p(A) < /[All but we also have GelIfand formula3 5 p(A) = lim l IjA = inf ~IIA I. n+co n The existence of the limit and its equality to the inlimum is guaranteed by the theorem of Fekete that if a sequence (01,) of real numbers is subadditive, i.e. a ,,, < a, + a,,,, then 5 tends to the infimum of the sequence as n + co. Further, n the outer radius of convergence r of the series (23) is equal to the stated limit and r > p(A) since the domain of convergence of the series is contained in reg A. On the other hand, r < p(A) since the resolvent is analytic on the domain IA) > p(A). If p(A) = 0, i.e. spec A = (O}, the operator A is said to be quasinilpotent. In particular, any bounded nilpotent operator is of this type, however the class of quasinilpotent operators is more extensive than this. Example.
On Lp(O, 1) (1 < p < co) the integration W(s)
is not nilpotent
but it is quasinilpotent.
1 (W)(s) = (n _ l)!
s
=
operator
’ 4(t) dt s0 In fact
; (s  O?(t)
dt
(n = 1,2, 3, . ..).
hence
IlZ = 0l 0; . Consequently, p(Z) = 0 by Gel and formula. For any bounded operator A defined on the whole space the circle 111= p(A) contains points of the approximate spectrum. Let A be an isometric operator defined on a closed subspace D(A) c E. Its image Im A is clearly also closed. Since 35Hence it follows that the functional calculus for the operator A operates on the disk )I[ < p(A).
8 4. Theory
of Operators
185
all points L with 111 # 1 are quasiregular. The defect number n, is constant in the disk IA) c 1 and in its exterior 111> 1. Its value on the disk is equal to n, = dimt(E/Im A). We denote its value on the exterior by n,. It follows from the formula A,(A) = A,,(A‘) (111 # 1) that no0 = dimt(E/D(A)) so that n, = OSD(A) = E (clearly n, = OoIm A = E). The pair (n,, n,) is called the deficiency index of the isometric operator. If there exist quasiregular points in the unit disk, then no0 = n,. In order that an isometric operator map the whole space onto itself (in a Hilbert space this means that it is unitary) it is necessary and suflicient that it have deficiency index (0,O). The approximate spectrum of any isometric operator is unitary, i.e. it lies on the unit circle. If the operator is defined on the whole space then its spectrum lies in the disk 111< 1, the circle 111= 1 contains the points of the approximate spectrum but it can also contain quasiregular points; then they are regular and n, = 0. If n, # 0, the spectrum is the disk 111 < 1 and the approximate spectrum is the circle In( = 1. The spectrum of an operator which maps the whole space isometrically onto itself lies on the unit circle. To conclude this section let us consider the structure of the spectrum of a compact operator. This is the next in complexity after the algebraic case. All linear operators on a finitedimensional LT.3 are compact and algebraic; the spectrum of any of these coincides with its set of eigenvalues. Theorem. The spectrum of a compact operator A on an infinitedimensional LTS E consists of zero and a no more than countable set of eigenvalues diflerent from zero. The unique limit point of this set, zf it is infinite, is zero.
Proof. Each point 1 E spec A\(O) is an eigenvalue by the Fredholm alternative. The point A = 0 belongs to the spectrum since otherwise the operator AIA = 1 is compact, which is impossible for an infinitedimensional E. Let V be a balanced neighbourhood of zero whose image is relatively compact. Let us suppose that there exists a set (A,} r of eigenvalues such that I&l 2 E > 0; let e, (n = 1,2, . . .) be corresponding eigenvectors. Now consider L, = Lin{e,}l; clearly dim L, = n. We can choose in L, n 0 a vector x, 4 L,, + V. Since Ax,  &x. E L,, and Ax, E L,l (n > m) we have Ax,  Ax, 4 EV, which is a contradiction. 0 Let us enumerate the eigenvalues which are different from zero (if they exist): . We denote their orders by vl, vl,. . . , and their maximal root subspaces &,L... by WI, W,, . . . . They are all finitedimensional and have invariant complements Vi = Im(A  A1l)yl, . . . . Moreover spec(Alw,) = {A,} and spec(Aj,J = spec A\{&}. Hence it follows that on a Banach space E the resolvent R,(A) has a pole of order v, at the point & and has no other singularities apart from the points 1, and zero, i.e. it is meromorphic on @\ (0).
Chapter 2. Foundations and Methods
186
If a compact operator has no eigenvalues different from zero (i.e. it is quasinilpotent in the Banach situation), it is said to be a volterra operator. The following example explains the origin of this term. Example.
In Lp(O, 1) (1 < p =Gco) Volterra
integral operator
(0 < s < 1) ’ R(s, W(t) dt s0 with continuous kernel R(s, t) is a Volterra operator. In particular the integration operator I is a Volterra operator. Let A be a nonnegative compact operator on a Banach space E with closed cone K. Let us suppose that the cone K is total, i.e. Lin = E. If the spectral radius p = p(A) is positive, then p is an eigenvalue for A and A* and corresponding to it there are an eigenvector x E K and an eigenfunctional f~ K* with f(x) > 0 (M.A. Rutman, 1938). The condition p(A) > 0 is satisfied if there exists a vector u > 0 (u # 0) such that Au 2 cu for some c > 0 (then p(A) > c). (A&(s) =
Let us consider the integral operator 4(s) (O<s
4.5. OneParameter Semigroups. The representations T(t) of the additive semigroup Iw, of nonnegative numbers with T(0) = 1 are designated by the term in the title. The representations of the additive group [ware called oneparameter groups. Each bounded linear operator A on a Banach space E generates a oneparameter group
co A?” T(t) = eAt = 1 I n.
(t E R).
(24)
Moreover ;[T(t)x] i.e. the vectorfunction
= AT(t)x = T(t)Ax
(x E E, t E W),
u(t) = T(t)x is the solution of the Cauchy problem
G(t) = Au(t)
(cOct
u(0) = x
(the solution can easily be shown to be unique). The operator A is infinitesimal for the representation T in the sense that A = T 0). Because the operator A is
9 4. Theory of Operators
187
bounded, the representation T is continuous not only in the strong topology but also in the uniform topology: lim II T(t + z)  T(t)11 = 0
(t E R).
r10
Conversely, if a representation T(t) (t E R,) is uniformly continuous, then T(t) = eAr, where A is some bounded operator. This follows from the general theory which is presented below. Let T(t) be any oneparameter semigroup. Since T(t + s) = T(t)T(s), the function a(t) = In ((T(t)(( is subadditive, a(t + s) < a(t) + a(s), and, by the continuous variant of Fekete theorem, the limit
~ = lim ln IITWII 
T(n) =sm
T(t)e
dt
0
is therefore defined on the halfplane II, = {A: re 1 > g}. Since 1)T(t)(I G MeC’ for some c > e, M > 0, we have
The values of the operator function ?(A) commute pairwise and satisfy the Hilbert identity
F(a) T;(p)= (a  p)T(a)qp).
I
The subspace L = Im F(A) therefore does not depend on 1. The operator f(A) maps E bijectively onto L (injectivity follows from the uniqueness theorem for the Laplace transform, since, if ?(A)x = 0, then f(,u)x = 0 for all p E II,). Let us put A(1) = 11 + [?$)IThis does not depend on 1 and defines a closed operator A (D(A) = L) with R,(A) = F(A). The subspace L is dense since [T(A)]*
= SW T*(t)e
dt
0
and, analogous to the previous situation, Ker@(A)]* = 0. We note that, since R,(A) commutes with all T(s), the operator A also has this property. Let us consider the Cauchy problem ti(t)
=
Au(t)
(t
> O),
u(0) = x E D(A).
IfpEII,andy=Axpx,then x = R,y = 
m (T(s)y)es0
ds,
(25)
Chapter 2. Foundations and Methods
188
and hence m T(t)x = Consequently &(t)x]
i.e.
T(t)x
(T(t + s)y)e
s0 the function
T(t)x
ds = ee”
has a continuous
= ,uT(t)x + T(t)y = T(t)Ax
is a solution of the Cauchy problem
m (T(s)y)eC sf
ds.
derivative for = AT(t)x
t
2 0 and
(t 2 01,
(25). In particular,
putting
t
= 0
we obtain Ax = 1 [T(t)x]I1=o. The Cauchy problem (25) can be considered36 for any operator A. In the situation described above it does not have solutions other than T(t)x, i.e. the corresponding homogeneous problem C(t) = Au(t) (t > 0), u(0) = 0 has only the trivial solution. This is the case (Yu.1. Lyubich, 1960) even under the much weaker restriction of growth of the resolvent on the ray il > 1, > 0:
lim lnll&II coo a++m 1 (this cannot be relaxed further). In fact s R,u(O) = u(t)edt + ehR,u(s) (s > 01, f0 from which it is obvious that R,u(O) is the local Laplace transform for [u(t)]. If u(O) = 0 then u(t) = 0 by the uniqueness theorem for the l.L.t. The subspace I(A) c D(A) of those x for which the Cauchy problem has a solution (independently of uniqueness37) is called the initial subspace. It can fail to be dense (and can actually be the zero subspace) even in the case where D(A) is dense. But if a bound of the form IIRlll = O(lAl is satisfied in some halfplane II, contained in reg A and D(A) is dense, then I(A) is dense (Yu.1. Lyubich, 1964). If x E I(A) and u(t) is a corresponding solution, then clearly its values belong to I(A) for all t (in this sense I(A) is invariant). If the Cauchy problem is determinate, the evolutionary operator U(t) transforming u(O) into u(t) (t > 0) is defined on Z(A). Clearly U(0) = 1 and U(t + s) = U(t) U(s). If U(t) is bounded for each3* t > 0, we have a representation of the semigroup R, in the subspace I(A) (in general it is not closed). However if sup04tSE II U(t)11 c co for some E > 0,
361t is implicit that the solution is continuously differentiable and its values belong to D(A) for all t 2 0 (in a relaxed formulation  for t > 0). 37A Cauchy problem for which the solution is unique is said to be determinate (this property does not depend on x). 3sIn this case the Cauchy problem is said to be pointwise stable on I(A) (simply pointwise stable if I(A) is dense).
189
$4. Theory of Operators
then3’ the extension by continuity of U(t) onto Z(A) preserves continuity of the trajectories and we obtain a representation of the semigroup R, in the closed subspace Z(A). Its infinitesimal operator is the closure of the operator A (S.G. Krejn  P.E. Sobolevskij, 1958). The infinitesimal operator of a oneparameter semigroup is also said to be generating for this semigroup or to be a generator of it since, clearly, the semigroup with a given inhnitesmial operator is unique. The question arises of describing the class of generators of all possible oneparameter semigroups. MiyaderaFellerPhillips Theorem. In order that a densely defined linear operutor A on a Bunuch space be the infinitesimal operator of a oneparameter semigroup, it is necessary and suflicient that reg A contain a halfplane ll, in which the inequality M llR:ll G (re k _ c)”
(n=0,1,2,...;
M a positive constant)
is satisfied. Necessity.
This follows from the bound II T(t)11 < Me” and the formula 1 d , (1)” a3 RI! = (n _ I)! dl ’ = (n  l)! 0 T(t)t s
 dt.
Sufficiency. Under the given conditions the Cauchy problem (25) is determinate and the initial subspace Z(A) is dense. Since s u(t)e
we have by Widder u(t) = lim n+m
dt = R,(u(t)e
s0 inversion formula (0 < ( LRJ lu(0)
+ I

 u(O)), t
< s):
kzo ((n
shlik
.
Ri+ (s) A=ll/t
Hence it follows that Ilu(t < Me” llu(O)ll, i.e. for the evolutionary operator: II U(t)11 < Meet. Since A is closed under the conditions of the theorem, it is the infinitesimal operator of the representation U. 0 An immediate consequence of this result (taking account of the connection between the bounds for the representation and the resolvent) is the HilleYosida Theorem. In order that a densely defined linear operator A on a Banach space be the infinitesimal operator of a oneparameter semigroup of contractions, it is necessary and suflicient that the right halfplane II, be contained in reg A and that the inequality
391n this case the Cauchy problem is said to be stable on I(A) (simply stable if I(A) is dense).
190
Chapter 2. Foundations and Methods
be satisfied in II,.
The operator A in this case is said to be dissipative. Example. The semigroup R, acts by means of shifts on LP(R+): (T(t)&(s) = 4(s + t) (s, t 2 0). Clearly II T(t)11 < 1. The infinitesimal operator of this regular representation
d ds
is equal to . It is defined on the absolutely continuous functions
which, together with the derivative, belong to Lp. This is a dissipative operator. The theory of oneparameter groups is reduced to the previous case by considering representations of the semigroups IR, . In particular, the infinitesimal operators of oneparameter groups of isometries can be characterised (I.M. Gel and, 1939) as those operators whose spectrum lies on the imaginary axis and for which the bound 1
IIWI G Ire
(re 1 # 0)
is satisfied. These operators are said to be conservative. Example. infinitesimal
A regular representation
of the group R! in Lp(R) is isometric. Its d operator  can be described just as in the case of R, but now it is ds
actually conservative. Conservative and dissipative operators have many remarkable properties, in particular the spectral radius of a bounded conservative operator A on a Banach space E is equal to its norm (V.Eh. Katsnel on, 1970). In fact, the function f(eA ) (x E E,fe E*) is an entire function of exponential type
= Idd,(f( eAtx N I<. ~(4
f If(eAW sup
G PCW~I
 Ilxll.
Putting t = 0 we obtain If(Ax)I < p(A)IlfII * (IxIJ and consequently llAl/ < p(A). Incidentally, Bernshtein inequality itself can be interpreted as the coincidence of the spectral radius with the norm of the differentiation operator on the space B, of entire functions of exponential type < 0 which are bounded on the real axis (it is remarkable that the differentiation operator on B, is bounded). A general mechanism underlying inequalities of Bernshtein type and results pertaining to it in spectral theory was brought to light by E.A. Gorin (1980). It was based on the functional calculus associated with integral representations of positive functions. From the point of view of the Cauchy problem there is great interest in the study of asymptotic (as t + co) properties of oneparameter semigroups. For example, boundedness (i.e. supt /IT(t)11 < co) is equivalent to Lyapunov stability
4 4. Theory of Operators
191
for the corresponding Cauchy problem. For bounded oneparameter semigroups on a reflexive Banach space we have the ergodic theorem: the strong limit T(t) dt
exists. It is a bounded projection (for semigroups of contractions it is an orthoprojection) on the subspace of lixed points {x: T(t)x = x (t 2 0)} = Ker A (A is the infinitesimal operator). This theorem owes its origin to the analysis of the foundations of statistical mechanics, which goes back to Koopman and von Neumann40. There is an important class of representations for which the ergodic theorem is true without restriction on the space  almostperiodic representations. A representation T is said to be almostperiodic (a.~.) if all of its orbits are relatively compact41. Example 1. In the space AP(R) of almostperiodic functions on R the representation of the group Iw by means of shifts (the regular representation) is a.p. The definition of an a.p.f. carried over without change to functions on the semiaxis R, . We have correspondingly a regular a.p. representation of the semigroup R, in AP(R+). The general form of an a.p.f. on the semiaxis is 4 = do + h, where boo(t) tends to zero as t + co, and 4r can be extended to the whole axis as an a.p.f. (Frechet, 1941). Example 2. Let A be a compact operator on a Banach space E which does not contain any subspaces isomorphic to the space c of convergent sequences (5,); (in particular, E may be reflexive). Then, if the representation e (t E R) is bounded, it is a.p. In fact, for the function u(t) = e% (x E E) the derivative G(t) = Au(t) is a.p. because of the compactness of A and the function itself is bounded. Consequently u(t) is a.p. (here we are using a generalisation due to M.I. Kadets (1969) of the BohlBohr theorem on the integral of an a.p.f.). By restricting an a.p. representation of the group R to the semigroup R, we obtain, clearly, an a.p. representation but it is such that all the operators T(t) (t E W,) are invertible and their inverses are bounded (in this sense it is a group representation of the semigroup R,). There is a different kind of a.p. representation of the semigroup R, , all the orbits of which tend to zero as t + co. Further, if the space E can be decomposed into the topological direct sum of the spaces 4oThis author established (1932) the ergodic theorem in Hilbert space. The generalisation given here is due to Larch. Usually formulations of ergodic theorems are given for semigroups of powers (I )?. The ontinuous’ case considered in the text is easily reduced to the iscrete’ case (such a reduction occurs quite often). Further, as a rule, we restrict ourselves to only one of these cases. 41 In exactly the same way a.p. is defined for bounded representations of any topological semigroups in any LTS. The property of boundedness is not required a priori in a barrelled LCS since it follows from the relative compactness of the orbits by the BanachSteinhaus theorem. In general we can define a.p.fs on any topological semigroup; they can be not only scalarvalued but also vectorvalued.
192
Chapter
2. Foundations
and Methods
E,, E, in which a.p. representations T,, T, are given (for example, Tl a group representation and To having orbits tending to zero) then the natural representation Tl i To defined on E is a.p. We have the boundary spectrum splittingoff theorem (M.Yu. Lyubich  Yu.1. Lyubich, 1984): let T be an a.p. representation of the semigroup [w, in a Banach space E; then E can be decomposed into the topological direct sum of invariant subspaces E,, E, such that Tl = TIE1 is a group representation and the orbits of T, = TIE0 tend to zero. The subspaces E,, E, are called the boundary and interior subspaces respectively and the projection P associated with them (Im P = E,, Ker P = E,) is called the boundary projection. It arises in the following way. Corresponding to an a.p. representation T there is a Bohr compact set  the strong closure of the family (semigroup) of operators { T(t)},,o. This is an Abelian semigroup which is compact in the strong topology. In any compact semigroup there is a smallest twosided ideal  the Sushkeuich kernel (A.K. Sushkevich, 192843; Numakura, 1952). For an Abelian semigroup (and in certain other cases) the Sushkevich kernel is a compact group. The boundary projection for an a.p. representation is the identity of the Sushkevich kernel of the Bohr compact set. Hence it is clear, incidentally, that [[PII < sup 1)T(t)11 and in particular, if T is a contraction then P is an orthoprojection. In the latter case the representation Tl is an isometry. The theory which has been described carries over to a.p. representations of any Abelian (and certain nonAbelian) topological semigroups, in particular, to semigroups of powers of an a.p. operator 44. A prototype of the general theory of a.p. representations, the boundary spectrum splittingoff theorem for bounded semigroups of operators on a reflexive Banach space, was constructed by Jacobs (1957), after which De Leeuw and Glicksberg (1961) obtained corresponding results in the weak topology for any Banach space. In the weak a.p. case the orbits for T, do not tend to zero in general, even weakly, and they only have zero as an adherent point. Furthermore To can turn out to be a group representation (for example, (T(t)f)(s) = eifsf(s) (s, t 3 0) in L’(R+)). 4.6. Conjugation and Closure. Let A be a linear operator on an LTS E and suppose that its domain of definition D(A) is dense. Then the conjugate operator A* on E* can be defined by means of the relation (A*f)(x) =f(Ax) for those f~ E* for which the right hand side of the identity is continuous on D(A) and, consequently, can be uniquely extended to a continuous linear functional on E. If E is a Hilbert space the definition of the conjugate operator is usually modified on the basis of Riesz theorem so that A* acts on the space E itself: (Ax, Y) = (x9 A*Y)
(x E W),
Y E WA*)).
Therefore (aA)* = ZA*. In this case we call A* the adjoint of A.
43The fundamental work of A.K. Sushkevich is concerned with finite semigroups. 441n particular, any operator of the form A = V + V, where V is compact, 11Y” 11< 1 for some m and sup,,, l[A” 11< co is of this type (Yosida  Kakutani, 1941).
5 4. Theory
of Operators
193
Theorem. Zf a densely defined operator A on an LTS E has continuous inverse A‘: E , D(A) then A* has inverse (A*)‘: E* + D(A*) and (A*)’ = (A‘)*. Thus, for example, (A*)’ is continuous under the topology of uniform convergence on bounded sets and also in the w*topology. Corollary. Zf A is a densely defined operator on a Banach space, then reg A* = reg A and consequently spec A* = spec A. In the Hilbert space context reg A* = reg A and spec A* = spec A (the bar denotes complex conjugation). The defect space E/A, (A E spec,A) is isomorphic to the subspace Ai = N;i z Ker(A*  11). The preceding theorem has a converse in a Banach space: if A is densely defined and closed and A* has a bounded inverse then A also has a bounded inverse (consequently, (A‘)* = (A*)‘). An operator A on a Hilbert space is said to be selfadjoint if A* = A. For example, any orthoprojection is of this type. If A c A* which, in the case of dense D(A), is equivalent to the identity (Ax, Y) = (x, AY)
(~3 Y E WA)), (27) then A is said to be a symmetric operator. Selfadjoint operators play a central role in quantum mechanics, where they are used to give a suitable description of physical quantities (coordinates, impulses, energy, etc.). The principal credit for constructing the spectral theory of selfadjoint operators is due to Hilbert and von Neumann. We say that an operator A on an LTS E admits closure if the closure of its graph is the graph of some operator A, the closure of the operator A. Clearly A c Aand Ais closed. In order that A admit closure, it is necessary and sufficient that a closed extension exist for it; moreover Ais the least closed extension for A. A continuous operator always admits closure; this is just its extension by continuity to D(A). The domain of definition of a closed continuous operator is therefore closed and if it is also dense then it coincides with the whole space. Conversely, any continuous operator defined on a closed subspace is closed. A conjugate operator A* is always closed (even w*closed). In particular, in a Hilbert space E any selfadjoint operator is closed and any densely defined symmetric operator admits closure since A c A*. Moreover it is clear that x is symmetric. If a symmetric operator A is defined on the whole space E, then A is a bounded selfadjoint operator (HellingerToeplitz theorem). In fact, since A c A* and D(A) = E we have A = A*. But A is then closed and consequently it is bounded by the closed graph theorem. Theorem. Zf a densely defined operator A on a Hilbert then A* is densely defined and A** = A.
space admits closure,
A closed densely defined operator on a Hilbert space which commutes with its adjoint is said to be normal. In particular all unitary operators (A*A = AA* = 1) are of this type.
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Chapter 2. Foundations and Methods
4.7. Spectra and Extensions of Symmetric Operators. Let us consider a densely defined closed symmetric operator A on a Hilbert space E. It follows from the identity (27) that IlAx  1x11 2 lim 4 llxll (x E D(A)), (28) from which it is obvious that all 1 with im I # 0 are quasiregular. The defect numbers n, , n associated with the halfplanes im il > 0, im il < 0 are in general unequal, however, if there is a real quasiregular point, then n, = n . The pair (n+, n) is called the deficiency index of the operator A. Theorem 1. In order that A be selfadjoint, it is necessary and sufficient that its deficiency index be (0, 0), i.e. that spec A c R.
Necessity. If A* = A then Nr = 0 whenever im A # 0, since all eigenvalues of a symmetric operator are real. Sufficiency. If n, = n = 0 there exists for any 1 (im I # 0) and any x E D(A*) a y E D(A) such that Ay  ily = A*x  Ix and hence A*(x  y) = L(x  y). Consequently x  y = 0, i.e. x = y E D(A). 0 If A” is a closed symmetric extension of the operator A then A, < n, and ti < n . The operator A”is loser’ to selfadjointness than A since45 A c A”c A”* c A*. Is it possible to choose A o be selfadjoint, i.e. does A admit a selfadjoint extension? This important question was investigated by von Neumann (1929) with the help of the Cayley transform which is defined for an operator A and any A (im 1 # 0) by the formula V, = VA(A) = (A  ill)(A  11)l. The homomorphism V,: AX + An is surjective and isometric46 since if II = a + ifi we have IlAx  Ax112= IlAx  axlIz + /1211xl12 (x E D(A)) and hence 11 Ax  Ix II = IIAx  Ix 11.Moreover the subspace Im( V,  1) = D(A) is dense. Conversely, any isometric operator V for which Im(V  1) is dense is the Cayley transform of the densely defined closed symmetric operator A = (XV  Al) (V  1)l (it follows from the denseness of Im( I/  1) that unity is not an eigenvalue of the operator V and therefore (V  1)l: Im(V  1) + D(V) is defined). By Theorem 1 the Cayley transform of a selfadjoint operator (and only of such an operator) is unitary. The question about selfadjoint extensions therefore reduces to the existence of unitary extensions of the isometric operator F ,which in turn is equivalent to the defect subspaces N,, Nr being isometric. Thus we have 45For any two linear operators A,, A, with dense domains of definition the inclusion A, c A, implies that A: 3 A:. 461mplicit in the construction of the Cayley transform is the idea of spectral mapping: the linearfractional transformation 5 I+ (I  A)(c  ;i,’ maps the real axis onto the unit circle, therefore its operator analogue must transform symmetric operators into isometric operators.
8 4. Theory
of Operators
Theorem 2. In order that A admit a selfadjoint suflicient that its defect numbers be equal.
195
extension, it is necessary and
From what has been said it is also evident that in the case n, = n the set of selfadjoint extensions can be parameterised by the isometries of NA onto N;i, in particular, for differential operators (I.M. Glazman, 1949). Example 1. Let us consider the operator D = f $ on L2(0, 1) which is defined
for all absolutely continuous
functions with derivative in L2. We will denote by
Do its restriction under the boundary conditions d(O) = 4(l) = 0. This is a closed symmetric operator and Dd = D. Each 1 E @ is an eigenvalue for D, the eigen
space NA is onedimensional and it is generated by the function cl(s) = eiL. The deficiency index of the operator D,, is therefore equal to (1, 1). The isometries of NA + Nr take the form V’,(O)e, = BepBer (/I = im 1, 8 a parameter, 181 = 1). The selfadjoint extension De of the operator D, corresponding to the isometry V,(O) is such that (De  ll)(D’
 Il) ,
= 8eMBer.
Putting (De  zl) , = u, we obtain D
 %I = e,,
Dev  Iv = tle8ex,
and hence V=
e,  8eBex nX .
Set 1 = i. Then u(1) = or(O), where o = (1  eQ(e  0)l is a parameter which, together with 0, runs through the unit circle. Thus the extension De of the inimal’ operator D, can be described as the restriction of the aximal’ operator D under the boundary conditions 4(l) = 04(o). Example 2. The operator D on L O, co) defined as in the previous example can be restricted to Do by imposing the condition d(O) = 0. As before, D$ = D but now the deficiency index is (0, 1) since e, E L2(0, co) only if im Iz > 0. Consequently D,, does not have selfadjoint extensions (or even symmetric extensions other than itself). Example 3. The operator D on L2(  00, 00) defined as before is selfadjoint. The operator iD on L2(  co, co) is conservative and only operators of the form iA (A = A*) are conservative on a Hilbert space (see inequality (28)). Therefore if A is a bounded selfadjoint operator we have p(A) = l/All (which can also be
established by elementary means). From this it is easy to establish the nonemptiness of the spectrum of any selfadjoint operator B. In fact if 0 # spec B then A = B’ is a bounded selfadjoint operator and A # 0. Therefore p(A) = IIAlI > 0. Now at least one of the numbers &p(A) is in spec A and then its reciprocal must be in spec B. We note incidentally that p(A) = 0 * A = 0 (i.e. a nonzero self
196
Chapter
2. Foundations
and Methods
adjoint operator cannot be quasinilpotent). Even if the selfadjoint operator A is not presupposed to be bounded but spec A = (0) then A = 0. In fact, in this case R, (A E IX, L # 0) is a bounded seifadjoint operator and spec R, = ( nl}. Consequently 11 Rlll = IL/’ which leads to the inequality IJAxll’  21(,4x, x) > 0 for ail x E D(A) and L E R\(O). But then (Ax, x) = 0 and hence A = 0. The last conclusion can be made on the basis of the polarisation formula tAxp Y) = i kio ik(A(x + iky), x +
Q ),
which is valid for any linear operator A on a Hilbert space E. Incidentally, it follows from this formula that if a Hermitiun quadratic functional (AZ, z) (z E D(A)) is bounded on the unit sphere then the operator A is also bounded and /[All < 2 SU~,,~,,=~I(Az, z)l (the converse is obvious). The functional (Ax, y) (x E D(A), y E E) is sesquilineur, i.e. linear in x and semilinear in y, i.e. it is additive and if y is replaced by ay then it is multiplied by 5. For any sesquilinear functional B(x, y) which is defined for x in some subspace L and for all y E E there exists by Riesz theorem a unique linear operator A such that /?(x, y) = (Ax, y) (x E D(A) = L, y E E). Symmetry of the operator A is equivalent to Hermitian symmetry of the functional, B(y, x) = /3(x, y), which, in turn, is equivalent to the quadratic functional /3(x, x) being real. The operator iD on L 0, 00) is dissipative and in general all operators on a Hilbert space of the form iA (A c A*, n, = 0) and their adjoints4’ are dissipative but these are by no means all the dissipative operators (for example, (I), where I is the integration operator on L 0, 1)). The following criterion for dissipativity of a closed densely defined operator A on a Hilbert space is an easy consequence of the HilleYosida theorem: re(Ax, x) < 0 Remark.
Dissipative
by the requirement
(x E D(A)); operators
re(A*y, y) G 0 (y E WI*)).
on a Hilbert
space are often defined only
re(Ax, x) < 0 (which in addition
they
otate’ through
im(Ax, x) > 0). Under this the halfplane re 1 > 0 is quasiregular but we can have n, > 0. For a dissipative operator in our sense n, = 0. We note in this connection that symmetric operators with zero defect number and only such operators are maximal in the sense that they do not admit symmetric extensions (since their Cayley transforms do not admit isometric extensions). A symmetric operator A is said to be nonnegative (A > 0) if (Ax, x) > 0, positive (A > 0) if(Ax, x) > 0 (x # 0) and positive definite (A >> 0) if (Ax, x) 2 c llxllz (c is a positive constant) (in all cases x E D(A)). For example, any orthoprojection is nonnegative, multiplication by c > 0 is a positive delinite operator. If (e,); is an orthonormal basis and (A,); is a sequence of positive numbers which tends to zero, the corresponding diagonal operator 47Dissipativity
is preserved
under
conjugation
even in the Banach
space case.
Q4. Theory of Operators
197
Ax = f &,<,e, n=O is positive but it is not positive definite. For a nonnegative operator A all points I < 0 are quasiregular and therefore n  n. If A* = A > 0, the spectrum of the operator A is nonnegative. The sieirum of a positive definite selfadjoint operator lies on the corresponding semiaxis I > c. In the real Banach space of bounded selfadjoint operators the nonnegative operators form a closed solid cone (1 is an interior point) as a result of which we can work with inequalities in this space. This approach improves substantially the norm bounds. 4.8. Spectral Theory of Selfadjoint Operators. The simplest48 spectral theorem (after the algebraic case) concerns a compact selfadjoint operator A on a Hilbert space E and asserts that E coincides with the closure of the orthogonal sum of the eigenspaces E, corresponding to all possible eigenvalues4g (Il,}; (zero is included if it is an eigenvalue). This theorem is proved just as in the tinitedimensional case. With a view to generalisation it is convenient to express it in the form of a spectral resolution
where P,, is an orthoprojection
Ax = .f Iz,P,x, n=O onto E, and x = 2 P,x. n=O
A family of orthoprojections (I!,(  cc C I < co) on a Hilbert space E is called an orthogonal resolution of the identity if 1) @n@p= Q% (2 < PL), 2) lim,, +m (EAx = x, lim,, coC?,x = 0 (x E E), 3) the function CE,x (x E E) is continuous on the left with respect to 1 on the whole axis5’. Under these conditions for any interval A = [cr, /I] the operator (E, = (E,  C& is an orthoprojection. Any orthogonal resolution of the identity (El defines a family of measures m, on R which are generated by the nondecreasing functions (CEAx,x) (x E E). Suppose that the scalar function 4 is measurable with respect to all the m,. It is then said to be measurable with respect to EA. We define the integral 48But it does nevertheless include the HilbertSchmidt theorem on integral equations with Hermitiansymmetric kernels. 49To be precise we assume that the set of eigenvalues is infinite. A compact selfadjoint operator with finite spectrum is algebraic. soAn orthogonal resolution of the identity can be given on some interval, for example [0,2x], rather than on the whole axis. In this example the conditions analogous to 2) have the form E, = 0, (E,, = 1.
198
Chapter 2. Foundations and Methods
with respect to the operatorvalued the subspace
measure J da, as the linear operator for which
D(@)={x:xEE,~~
m IW12 4%
4 < ~0 1
(assumed further to be dense) serves as domain of definition (@x, Y) =
m 44 d&x, s 00
Y)
and the identity
(x7 Y E WY)
is satisfied. It can be shown that in addition @* =
m 44 6. sCO Consequently it is necessary and suffkient for selfadjointness of the operator @ that the function 4 be realvalued almost everywhere (with respect to each measure m,). For boundedness of the operator @ it is necessary and sufficient that the function 4 be bounded in the sense that
ll4l =sy(eywIdl)<
00
(in which case 1)@11 = 11411). In general
(x E ww
IlW12 = 1 MU2 4%x, 4 s
The construction which has been described is called the functional calculus for the given orthogonal resolution of the identity a,. The correspondence 4 H @ is linear and & 42 H Qp,Qz if at least one of the functions 41, 42 is bounded. The functional calculus is an isometric isomorphism on the algebra of bounded functions into the algebra of bounded linear operators. Example. Let A4 c R be a Bore1 set and let xM be its characteristic function. Since & = xM and xM = xM, the operator
qf=
is an orthoprojection.
ODx&J s m In particular c&=
The spectral resolution in the form
de, =
sM
6
m del = 1. s U3
of a compact selfadjoint operator A can be expressed
$4. Theory of Operators
A=
199
co 1 da,, s 00
where (IFAX =
c P& A”<1
(x E E).
Theorem.. For any selfadjoint operator A there exists a (unique) orthogonal resolution of the identity (3, ( co < 1~ co) such that the spectral resolution A=
* IIdC& f oo
holds. Moreover 4(A) zr
m W) s a,
d%
for all admissible51 4.
In particular
we have the formula for the resolvent R,=
m daL ~ s mAP
(P E reg A).
It is easy to see from this that the spectrum of an operator A coincides with the set of points of increase of the function 52 (IF,. The eigenvalues are the points of discontinuity of the function (EA.The operator PA = %,+,  GA is an orthoprojection onto the corresponding eigenspace. The dimension of this subspace is called the multiplicity of the eigenvalue 1. The eigenvalues form the socalled discrete spectrum spec,A. The continuous spectrum spec,A is defined as the complement in spec A of the set of eigenvalues of finite multiplicity53. In terms of the resolution of the identity it can be characterised as the union of the set of nonisolated points of increase and the set of eigenvalues of infinite multiplicity. This characterisation combining these two sets is just the result that there exists a normalised quasieigensequence (x,)7 ( llxnll = 1, Ax,  Lx, f 0) with no convergent subsequence. The intersection spec,A n spec,A can turn out to be nonempty. However the whole space E can be decomposed into the orthogonal sum of the closed linear hull E, of the eigenvectors and the closed invariant subspace E, of vectors with purely continuous spectrum, i.e. those x E E for which (@,x, x) is continuous. Then in turn E, = E, @ E,, where the summands are the spaces of singular and absolutely continuous vectors (these concepts are of course defined through the corresponding properties of the function (CE,x, x)). The extraction of the space E,, is particularly important for the theory of scattering (see the end of this section). If ” For rational rp with poles outside of spec A this definition is equivalent to the previous one. ‘*i.e. with the union of the sets of points of increase of all functions (EAx, x) (x E E). s3 SpeccA and spec,A are defined for a normal (in particular a unitary) operator in exactly the same way.
Chapter
200
2. Foundations
and Methods
E,, = E we say that A is an operator with absolutely continuous spectrum and in the general case Ale= is called the absolutely continuous part of the operator A.
Similar terminology can be introduced for the remaining types of spectra. Of the other properties of the resolution of the identity corresponding to a selfadjoint operator A we note that the measure jd6i commutes with A, i.e. (EA commutes with A for any interval A = [a, /I]. In fact B A@, =
J. 6,
@,A = A&&w
s ao One of the approaches to the spectral theorem for selfadjoint through the Cayley transform V = (A  il)(A
operators is
+ il)‘,
a unitary operator for which Im(V  1) is dense. The point is that the spectral theorem also holds for unitary operators for which the proof can be effected by quite simple means. Theorem. For any unitary operator U there exists a (unique) orthogonal resolution of the identity (3, (0 < a < 274 with 2n
U= s0
eiO d@01.
Proof. Let us consider the group of powers (U”)$, . It is easy to verify that the Toeplitz matrix with elements (U”“5, x) (x E E) is positive. By the RieszHerglotz theorem there exists a nondecreasing function a(~; x) (0 < c1< 24 which is a solution of the trigonometric moment problem 2n
eina da(a; x) (n=O, +l, +2 ,... ). (29 f0 Without loss of generality we can assume that a(a; x) is continuous on the left on (0,271) and that ~(0; x) = 0. Under these conditions a(a; x) is unique and (VX,
x)
=
2n
a(2n; x) =
da(a; x) = (x, x).
s0 Consequently a(a; x) < llxll’ for all a. Performing
polarisation
in (29), we obtain
2% (U%
Y) =
eina dz(a; x, y)
(n = 0, *l,
+2 ,... ),
s0
(30)
where z(a; x, y) = t j.
~(a; x + Py)
(x,
Y
E El.
Since there is only one possible function r in (30), it follows that $a; x, y) is a sesquilinear functional of x, y and hence r(a; x, y) = &x, y), where (5, is a
5 4. Theory
of Operators
201
bounded selfadjoint operator. The required properties verified without particular difficulty. 0
of the family (a,} are
In order to pass to the selfadjoint operator A it is necessary to express it using the Cayley transform: A = i(V + l)(V  l)’ = i
Zn eia + 1 do = s oe l=
co f a,
1 d@,,
where @A= E2.Zarcco,l. There are several other proofs of the spectral theorem for a selfadjoint operator which are associated with different variants of the moment problem. For example, we can note that the resolvent R, satisfies the inequality im(R,x, x) > 0 (x # 0) in the halfplane im p > 0 and, as we already knows4, 1 IIRJI G ,im p,
(im P # 0).
We can therefore make use of the integral Nevanlinna class: (R,x, 4 =
and it turns out as before that R, =
where GA is an orthogonal
representation
of functions of the
s
m do@; x) COTy’
m dC& mIq’ s
al s
resolution of the identity. Hence A = pl + R;’ =
Co
I d(E,.
Now we have a further fruitful argument: the operator A is conservative, i.e. it generates a unitary group T(t) (t E [w);since all the functions (T(t)x, x) (x E E) are positive, Stone theorem that T(t) =
m eiAr de, s cc
follows from Bochner theorem and hence by differentiation at t = 0 we obtain a spectral resolution of the operator A. Correspondingly T(t) = eiAr. If A is a densely defined closed symmetric operator with equal defect numbers, its resolutions of the identity are by definition generated by its selfadjoint extensions. For a nonselfadjoint operator A the resolution of the identity is not unique, therefore the question of its invariant (i.e. independent of the resolution) properties arises. 54All this follows from the formula the help of Hilbert identity.
im(R,x,
x) = (im p)
llR,,xjl z which can be easily established
with
202
Chapter 2. Foundations and Methods
Theorem. Zf the defect number n = n, = n of the operator A is finite, then all of its selfadjoint extensions have exactly the same continuous spectrum.
The general property of invariance of the continuous spectrum under mall perturbations which in some sense or other are finite rank or even compact is evident in this result and also in the following. Theorem of Weyl. Zf A, B are selfadjoint spec,(A + B) = spec,A.
operators with B compact, then
In contrast to this the discrete spectrum is not invariant: for any selfadjoint operator A on a separable Hilbert space there exists a selfadjoint compact operator B with arbitrarily small norm such that the system of eigenvectors of the operator A + B is complete (von Neumann, 1935). Remark. Let A, B be closed operators on a Banach space. The operator B is said to be Acompact if D(B) 1 D(A) and BIDCA)is compact under the graph norm of the operator55 A (for this it is necessary and sufficient that the operator BRA(A) be compact56 for some I which is regular for A). For an Acompact operator B the set of points of the spectrum of the sum A + B that lie in any connected component G of the set reg A is at most countable (if it is countable then it condenses to 8G). All these points are normal eigenvalues. Let us consider certain classical situations from the point of view of the abstract spectral theorem. Example
1. The operator 5 1 on L2(  rr, a) with periodic boundary
condi
tions +(  rt) = d(z) is selfadjoint. Its resolution of the identity is generated by the Fourier series expansion: CWM4
= “?A cneins.
The spectrum of the given operator consists of the eigenvalues 1, = in and the corresponding eigenfunctions are eins (n E Z). This is an example of a purely discrete spectrum (there is no continuous spectrum). Example 2. The resolution of the identity for the operator i $ on L2(  co, co)
has the form (@M(S) = &
:m &&@” dp, s where 6 is the FourierPlancherel transform. The spectrum of the given selfadjoint operator fills out the whole axis, however there are no eigenvalues in it since his is llxllA = llxll + llAx]l (x E D(A)). Completeness of the space D(A) under this norm is equivalent to closedngs of the operator A. 56 Hence it is clear that if B is compact then it is Acompact for any A.
5 4. Theory
of Operators
203
the functions eips do not belong to L2. This is an example of a purely continuous spectrum. Example 3. Let us consider the Schrodinger operator y” + q(x)y on L’(O, co) with continuous (for simplicity) real potential under the boundary condition y’(0)  hy(0) = 0 ( co < h < co). First of all we define it only on the functions of finite support that are of class C2 and satisfy the boundary condition; in this role we denote it by Qo. The operator Q,, is symmetric, densely defined but not closed. The domain of definition of the adjoint operator Q$ consists of the functions z E L2 n C1 such that z’ is absolutely continuous, z” + q(x)z, z E L2 and z’(0)  hz(0) = 0. Moreover Q$z = z” + q(x)z. The domain of delinition of the closure Q0 is extracted from D(Q8) by imposing the additional condition lim [y’(x)z(x)  y(x)z’(x)] = 0 (z E NQ,*))x+cc This condition can be satisfied automatically (for example, this is the case if the potential is bounded from below) and then Q. is a selfadjoint operator. In the contrary case the deficiency index of the operator Q. is (1, 1). This is precisely the Weyl alternative, since in the oint case’ the equation Qtz = lz (im L # 0) has only the trivial solution in L2, while in the isk case’it has a onedimensional space of solutions. In any event a selfadjoint extension Q of the operator Q. exists (it is unique in the first case but not in the second). All these extensions are in one to one correspondence with the spectral functions, namely, if p(l) is a spectral function then the resolution of the identity for the corresponding selfadjoint extension Q is generated by the inversion formula associated with p(l): P%m)
(in the notation
=
1. fw~(~~ A dP(PL) f 0D of Chapter 1, Section 2.12). From Parseval
identity
i.e. each measure j d((lF,f,f) is absolutely continuous with respect to the measure Jdp(l), its density being equal to the square of the modulus of the generalised Fourier transform of the function $ In this sense the resolution of the identity 6, for the Schrodinger operator is absolutely continuous with respect to the measure jdp(l). A similar result holds for the threedimensional Schrbdinger operator Au + q(x)u on the whole space (A.Ya. Povzner, 1953) and for more general operators (Mautner, 1953; Yu.M. Berezanskij, 1956). Example 4. Let us consider a Jacobian matrix J with positive offdiagonal elements. Given an orthonormal basis (e$’ of a Hilbert space E the matrix J defines a homomorphism J^: E + s (s is the space of all complex numerical sequences). Restricting 3 to vectors of finite support we obtain a densely defined symmetric operator J,, on E. It is not closed, therefore we have to pass to its
204
Chapter 2. Foundations and Methods
closure IO. The adjoint operator J,* is the restriction of the operator J^to vectors which satisfy the condition .?u E E. In accordance with the Weyl alternative the equation J$u = lo either has only the trivial solution for all L (im 1 # 0) and then the operator .& is selfadjoint (point case) or the space of solutions of this equation is onedimensional for all L (im 1 # 0) in which case the deficiency index of the operator Jo is (1, 1) (disk case). In any event a selfadjoint extension A of the operator Jo exists (it is unique in the first case but not in the second). All these extensions are in one to one correspondence with the extremal solutions a(n) of the power moment problem associated with the given Jacobian matrix. In fact, if CZi is the resolution of the identity for the operator A, then a(2) = (C?,e,, eo). Returning to the Schriidinger operator on the semiaxis, we note that progress in investigating its spectral properties was largely dependent on the discovery and development5’ of the technique of socalled transformation operators (Delsarte, 1938; A.Ya. Povzner, 1948; B.M. Levitan, 1949) which establishes the equivalence of the operator Q and the elementary operator y” with boundary condition y’(0) = 0. In this connection it is important that the transformation operator has the form 1 + K, where K is a Volterra integral operator with continuous kernel, so that the inverse operator has a similar form. V.A. Marchenko (1950) applied transformation operators to a wide range of problems on the spectral analysis of the Schriidinger operator. In particular he obtained in this way the asymptotic expression for the spectral function5*
and established a theorem on the uniqueness of the solution to the inverse problem of reconstructing the operator from its spectral function. A linear procedure for solving this problem, which is also based on the use of transformation operators59, was given by I.M. Gel and and B.M. Levitan (1951). In the physically important inverse problem of scattering theory60 the input data (to begin with the socalled limiting phases) are linked with the asymptotic behaviour61 of the solutions of the equation Qy = ly as x + co. An effective n connection with generalised shift (g.s.). The simplest g.s. (V(t)cp)(s) = (cp(s + t)  cp(s t))/2 arises from the wave equation &@t2 = a2u/8s2; the general theory of g.s.is in many respects parallel to the theory of group representations (B.M. Levitan, 1962). 58The right hand side of this relation is the spectral function of the simplest operator, q(x) E 0. t is a question of the construction of a certan linear integral equation for the kernel K(x, 0, after which the formulae q(x) = 2 dK(x, x)/dx, h = K(0, 0) are applied. All this goes through under certain conditions on ~(1) which are close to being necessary. Necessary and sufficient conditions for p(L) to be the spectral function of a SchrGdinger operator on the semiaxis with continuous potential were obtained in a different way by M.G. Krejn (1951). 60Recently the inverse problem of scattering theory has been at the centre of attention of many investigations thanks to its utilisation for integrating the Kortewegde Ries equation and other nonlinear equations of mathematical physics. 61 The description of which forms the direct problem. We note that both problems (direct and inverse) under consideration are concerned with the stationary theory of scattering, which is formulated in terms associated with the energy spectrum and not with evolution in time.
$4. Theory of Operators
205
method for its solution (V.A. Marchenko, 1955) therefore required the use of transformation operators ttached’ to infinity:
The existence of such operators for potentials which decrease sulliciently rapidly as x + cc was established by B.Ya. Levin (1955). The inverse problem of scattering theory on the whole axis was solved by L.D. Faddeev (1964), also on the basis of the technique of transformation operators. Remark. For differential operators of order n > 2 transformation operators of the form 1 + K, where K is a Volterra integral operator, do not in general exist (V.I. Matsaev, 1960), however in the case of analytic coefficients the transformation operator was constructed by L.A. Sakhnovich (1961). In nonstationary scattering theory the problem is to compare the asymptotic behaviour as t + + cc of a physical system which is evolving in time t with its behaviour as t +  co. Example. The Schriidinger operator Q (just as any other selfadjoint operator) generates a group of unitary operators eiQt. In quantum mechanics there corresponds to this a nonstationary Schriidinger equation for the wave.function +:
where H is a Hamiltonian
h
(energy operator), h = g and h is Plan&s
If a particle with mass m is in a force field with potential H$ =  &A$
constant.
q(x) (x E R3), then
+ q(x)+
(for example, this represents an aparticle in the field of a massive nucleus62 which was the situation in the historical experiments of Rutherford). The effect of scattering is created by the potential q, i.e. it is associated with the deviation of the actual evolution from the free (unperturbed) evolution which is determined by the Hamiltonian
This deviation is described by the wave operators W, = lim eiHteiK3t, t++m
which are clearly isometric. The limits here are naturally treated as strong limits 62This is twoparticle scattering. Nparticle scattering with N > 2 is a much more complicated process. It was investigated for N = 3 by L.D. Faddeev (1963).
206
Chapter 2. Foundations and Methods
and the first question concerns their existence (if not on the whole of L2 then on subspaces which have to be described). Following that we can define the scattering operatoF3 s = (w+) C,
but this construction is satisfactory only if Im W = Im W+ since then S maps isometrically onto D(W+). It is even better if we also have D(W) = D(W+), since then S is a unitary operator on a certain Hilbert space. The elucidation of these cases is the second important question of scattering theory. Clearly the stated questions can be posed in a general form, i.e. for a pair of selfadjoint operators A,, A on a Hilbert space. Such an bstract’ theory is capable of enveloping both the quantum and the classical scattering processes (acoustic, optical etc.). The nontrivial nature of the problem is underlined by the fact that, if A is a selfadjoint operator for which zero is not an eigenvalue, then eiArx (x # 0) cannot have a limit under the norm as t + + co (as also for t +  00). This also prevents the existence of wave operators in the presence of a discrete spectrum for A,. 64 The rigorous theory of scattering is oncentrated on the absolutely continuous spectrum. D(W)
KateRosenblum Theorem. Zf A = A, + B where B is a nuclear operator, the wave operators W, are defined on the subspace E,,(A,) and map it onto E,,(A). Thus the scattering operator S is defined and is unitary on E,,(A,).
Each of the operators W, intertwines the representations eiAot L(Ao)’
eiA
E,,(A)*
Therefore the absolutely continuous parts of the operators A,, A are unitarily equivalent. In applications the conditions of the KatoRosenblum theorem (1957) are not usually satisfied, however Kuroda (1960) discovered more effective conditions for this purpose. They in turn are covered by a theorem of M.S. Birman and M.G. Krejn (1962), in which the resolvents of the operators A,, A are required to be nuclear. It is even sufficient that the difference of the resolvents of q5(A,) and &A) be nuclear, where the function 4 belongs to a certain sufficiently wide class (M.S. Birman, 1962). 4.9. Spectral Operators. Functions of a selfadjoint operator are normal operators. However the spectrum of a normal, for example unitary, operator is not in general real. If
@=
w 44 6, s Q,
(31)
63The Heisenberg Smatrix is unitarily equivalent to this operator by means of the Fourier transform q9H $ (i.e., from the physical point of view, the transition from the coordinate space into the impulse space). 64The case of commuting A, A,, is not of interest in scattering theory.
8 4. Theory of Operators
201
where el. is an orthogonal resolution of the identity, then spec @ is the closure of the set of essential values65 of the function 4 (spectral mapping theorem). The spectral resolution (31) is written more naturally in the form
@= scPd&I,
(32)
where J d@, is the operatorvalued measure in the complex plane C which is the image of the measure J de, under the mapping 4. The support of the measure in (32) coincides with the spectrum of the operator @. The spectral resolution of a normal operator in the form (32) provided the starting point of the theory of spectral operators on a Banach space which was established by Dunford and Bade in the 1950s. In this theory it is postulated that there exists a resolution of the identity G(M) which is a projectionvalued measure on C (E(C) = l), commutes with a given closed densely defined operator A and is such that spec(Al ,,,,eo,J c M (in this case the operator A is said to be spectral). The resolution of the identity for any spectral operator is unique and if A is bounded66 then A=
where R is a quasinilpotent
sc
Aa
+ R,
operator which commutes with A. The term A=
sc
Mqdil)
(33)
is also a spectral operator with the same resolution of the identity, spec A = spec A and moreover the whole spectrum is approximate. We say that operators of the form (33) are of spectral type. Therefore n is called the scalar part of the operator A and R is its quusinilpotent part. The representation of a bounded spectral operator as the sum of an operator of scalar type and a quasinilpotent operator is unique. The class of spectral operators is rather narrow. For example, the shift operator on the twosided lp space (1 < p < co; p # 2) is not spectral (Fixman, 1959). The Id i ds
operator   on Lp(  co, co) or on Lp(  rc, 7~)with periodic boundary conditions also fails to be spectral if p # 2. We shall describe below a much wider class of operators (those with socalled separated spectrum). 4.10. Spectral Subspaces. Let A be a closed densely defined operator on a Banach space E. A closed subspace L c E (L # 0) is said to be spectral for A if
65A value LXis inessential if q(L) # a almost everywhere with respect to each measure m, = j d&x, x). 66For unbounded spectral operators the situation is considerably more complicated than that described below.
208
Chapter 2. Foundations and Methods
1) L is invariant6’ and D(A) n L is dense in L, 2) spec(A1,) c spec A, 3) if a closed invariant subspace M # 0 is such that D(A) n M is dense in M and spec(d(,) c spec(A 1L) then M c L. We shall call a compact set Q c spec A spectral if there exists a spectral subspace L = L(Q) for which spec(Al,) = Q (this subspace is clearly unique). As a trivial example we have: if A is bounded then the compact set spec A is spectral, the corresponding spectral subspace L being E. It can happen that no other spectral compact sets exist. Example. Let us consider the Banach space of scalar functions which are analytic on the disk IL1 C 1 and continuous up to its boundary under the usual norm rnaxu,=r Id(n)]. We have on it the operator n of multiplication by 1. The spectrum of the operator n coincides with the disk D = (1: 111 < l}. Suppose that Q is a nontrivial spectral compact set with corresponding spectral subspace L. We put A = D\Q. If p E A, for any 4 E L the function (A  $‘#(A) is regular for A = p, but then b(p) = 0. Thus if 4 E L we have & = 0. It then follows from the uniqueness theorem that 4 = 0, i.e. L = 0 in spite of the definition of a spectral subspace. Any compact set Q c spec A which is open in spec A is spectral. The corresponding spectral subspace is obtained as the image of the projection
p =  &
s t
R,dl,
where I is a simple closed contour separating Q from its complement Q’in spec A. Moreover 1  P is the projection onto the spectral subspace corresponding to the compacP set Q This construction is contained in the functional calculus of F. Riesz, which we shall describe below under the assumption that A is bounded. Let us put
d(A) =  & j 4(4R, dl, lwhere 4 is a function which is analytic on spec A, i.e. on some neighbourhood G 1 spec A, and I is a closed contour which does not cross itself, lies in G and contains69 spec A in its interior. Formula (34) defines a homomorphism of the algebra of functions which are analytic on spec A into the algebra of bounded operators, such that if #(A) = 1 then 4(A) = A.
67This term requires a more precise definition since A need not be defined everywhere on E. Inuariance of a subspace L means that if x E D(A) n L then Ax E L. The restriction Al, has to be interpreted as the restriction of A to D(A) n L. It is clearly closed along with A, however D(A) n L may fail to be dense in L when D(A) is dense. 68Thus the spectral subspace L = Im P is completely reducing. If Q consists of a single point n and L is finitedimensional, then p is an ordinary eigenvalue and L is the corresponding maximal root subspace. 6qThe neighbourhood mentioned and the contour r can be disconnected. The integral does not depend on the choice of contour.
Q4. Theory of Operators
209
For the calculus just described7’ Dunford (1943) established the spectral mapping theorem, spec 4(A) = 4 (spec A), and the superposition theorem: ($0 Q)(A) = $@(A)) under the condition that Q is analytic on spec A and $ is analytic on hvec 4. If the function qj is analytic on the disk 1II < r, where r > p(A), then, expanding it in a Taylor series and using the expansion (23), we obtain from (34)
i.e. we arrive at the previous definition of the function from the operator. We shall say about an operator A on a Banach space E that its spectrum is separated if each compact set Q c spec A which is the closure of its set of interior points relative to spec A is spectral. For example, any spectral operator is of this type: the spectral subspace L(Q) is the image of the projection
CZQ =s @(dl).
Q
Separatedness of the spectrum of an operator is ensured by a certain bound on the growth of the resolvent close to the spectrum. In fact, suppose that spec A lies on a curve C c @ and that in a neighbourhood of each point p E spec A the inequality
IM
G &W,
C))
(A $ Cl,
(35)
is satisfied, where M,,(6) is a decreasing function of 6 > 0 (6 < do = do@)) such that the Levinson condition do
In In M,(6) da < co s0
(36)
is satisfied. Then A is an operator with separated spectrum (Yu.1. Lyubich V.I. Matsaev, 1960). In the proof of this theorem a special functional calculus is used, with the help of which quasiprojections71 onto the spectral subspaces are constructed. It also allows us to establish (under conditions (35), (36)) the completeness of the system of spectral subspaces corresponding to any covering of the spectrum by finite arcs of the curve C. Example. The Schrodinger operator on L2(  00, co) with periodic complex potential has separated spectrum (V.A. Tkachenko, 1964). For a bounded operator A with real spectrum Levinson condition is equivalent to Ostrowski condition with respect to the weight a(t) = lleiArII (t E R): his can be developed further in the case of a spectral operator. ” These are operators Q, associated with a compact set Q and a neighbourhood U = Q such that @ILCpi= 1 and cD~,~) = 0 for compact sets K lying outside of U. certain heoryof duality’ also holds for spectral subspaces under conditions of type (36) (Bishop, 1959; Yu. I. Lyubich, V.I. Lomonosov, V.I. Matsaev, 1973).
210
Chapter 2. Foundations and Methods
(37,)
The latter signifies the nonquasianalyticity 73 of the Fourier image of the class L,‘(R) of functions which are summable with respect to the weight a. Starting from (37,) it is possible to construct the functional calculus &A) =
m qb(t)eeiA’ dt (4 E mw) f 02 independently of the previous approach; this is the operator Fourier transform, which allows us to obtain functions of compact support of the operator A, in particular quasiprojections onto spectral subspaces. Moreover L(Q) is constructed as the intersection of all the subspaces {x: J(A)x
= x}
(($1, = 1).
The requirement of boundedness of the operator A is not essential, the important thing being rather that it generates a group74 T(t) (t E W). The stated conditions of growth of the resolvent or exponential cannot be weakened: under any more rapid growth rate there arise operators for which the spectrum on any closed invariant subspace coincides with the spectrum on the whole space. For a bounded operator I/ with spectrum on the unit circle the requirement
cm ln IIv I
n=m n 2 +l
(37,)
is the analogue of Ostrowski condition and it is equivalent to Levinson condition; thus separatedness of the spectrum of the operator75 V follows from this. In particular, any isometric operator on a Banach space has separated spectrum but it is certainly not the case that any such operator is spectral. Any closed subset F of the circle (or line in the case of (37,)) is the spectrum of some nonquasianalytic operator ” . If the system of eigenvectors of a nonquasianalytic operator is complete, its spectrum S coincides with the closure of its set ,4 of eigenvalues (N.K. Nikol kij, 1971), since in the contrary case there exists a compact set Q c S such that Q n /i = 125.But then there is a quasiprojection which annihilates all the eigenvectors and is the identity on L(Q). A bounded operator A on a Banach space E is said to be decomposable if for any open covering u;=, Gk 1 spec A there exists a system of spectral subspaces
73An operator A which satisfies condition (37,) is said to be nonquasianalytic. 741n this general case the condition (37,) is stronger than (36). ” Under condition (37,) the operator, as before, is said to be nonquasianalytic. ny nonempty closed subset F c C is the spectrum of some normal operator (selfadjoint if F c R, unitary if F c T (the unit circle)), namely the operator of multiplication by the independent variable on the space L2(F, p). where p is any measure with support F. As regards this we note that a bounded set A c C is the set of all eigenvalues of a bounded operator on a separable Hilbert space if and only if it is of type F, (L.N. Nikol kaya, 1970).
Q4. Theory of Operators
211
(,I,& such that spec(AILk) c Gk (1 < k s n) and c;=i L, = E. The theory of this class of operators has been developed in depth during the last 25 years in the works of Foias and his school. 4.11. Eigenvectors of Conservative and Dissipative Operators. Completeness of the system of eigenvectors (eigenspaces) of a conservative operator A on any Banach space E is equivalent to the almostperiodicity of the corresponding isometric representation” T(t) (t E R). The necessity of almostperiodicity is obvious: if (xk}y is a collection of eigenvectors corresponding to the eigenvalues {i&}y and 11~ En x,II < E, then
T(t)x  k xkeiAkf c E (t E R). k=l II /I Hence it is clear that the orbit of the vector x is relativly compact. The sufficiency of almostperiodicity of the representation T for completeness of the system of eigenvectors of the operator A can be established with the help of the theory of representations of compact groups.‘* In particular, the spectral theorem for compact selfadjoint operators is generalised by this method: if A is a compact conservative operator on a reflexive Banach space E then its system of eigenvectors is complete (Yu.1. Lyubich, 1960). The reflexivity condition can be relaxed: it is sufficient that E should contain no subspace isomorphic to c. Moreover there is a counterexample in c (Yu.1. Lyubich, 1963). Example. Let (A,); be a sequence of distinct positive numbers which converges to zero. Let us consider the diagonal operator
4L) on c. It is compact and conservative: e
= (%t”) C) = (e n’&).
However its eigenvectors S,,,= (6,,,,), (m = 0, 1,2, . . .) only generate the subspace c0 of sequences which converge to zero. Remark. The ergodic theorem for an almostperiodic representation T(t) (t E R) follows easily from the completeness of the system of eigenvectors of the generator A: the existence of the limit 1 S GW.4 dt 5! s s *
for each eigenvector x is clear, i.e. T(t)x = eiCrx for Ax = ipx. ” By the same token this is also valid for any bounded representations: sup IIT(t)ll = M < co. I We note that this approach also leads to the classical theorem of Bohr for almostperiodic functions  the case of a regular representation in AP(R).
212
Chapter 2. Foundations and Methods
According to the ergodic theorem the strong limit s Pn = lim T(t)eP’ dt s%0s ,y exists for each J E R. This is a projection (llPJ < M, therefore if T is isometric, on W, = Ker(A  iA1) and also PAP,, = 0, i.e. PAlw, = 0 (1 # cl). Corresponding to each vector x E E we have a Fourier series in terms of eigenvectors 79 of the operator A:
Pn is an orthoprojection)
(in the case of a regular representation in AP(R) this is the classical Fourier series of an a.p.f.) The uniqueness theorem holds: Vk P2x = 0 * x = 0. The proof reduces to the scalar case if we consider the a.p.f. f(T(t)x) (f~ E*) since the Fourier coefficients of this function are c1 = f(P,x). Using the FejCrBochner method of summation for Fourier series of a.p.fs we can associate with each subgroup I c T a projection Pr (with bound llP,ll < M) on the closure of the sum of the eigenspaces corresponding to eigenvalues 1 E I (N.K. Nikol kij, 1971). The replacement of subgroups by arbitrary subsets is not possible (Rudin, 1962). Thus once again we have evidence of a close connection between harmonic analysis and the spectral theory of operators. There is however another approach to the problem of completeness of the system of eigenvectors of a conservative compact operator, which is based on an analytic technique using a bound for the resolvent (even weaker than (26)). We say that an operator A on a Banach space E is quasiconservative if spec A is purely imaginary and M
IIJUI G m
(re A # 0, M a positive constant).
If this inequality is satisfied for re I > 0 (spec A lies in the halfplane re 1 G 0) we say that the operator is quasidissipative. It is clear that in a Hilbert space each operator of the form iA, where A is similar to a selfadjoint operator, is quasiconservative but the class of quasiconservative operators is not exhausted by these (A.S. Markus, 1966). Moreover even in a Banach space this class has an interesting spectral theory. Its simplest manifestation can be seen in the result that if an operator is conservative then the order of each eigenvalue is equal to one, i.e. there are no root vectors other than eigenvectors. It can be shown further that a quasinilpotent quasiconservative operator must be zero, because in a reflexive Banach space E the system of eigenvectors of a compact quasiconservative operator turns out to be complete and moreover (even without the requirement of compactness) there exists a system of uniformly bounded projections P2( lIPAIl < M, PAP,, = 0 if A # p) onto he set { ik PA # 0) coincides with the set of eigenvalues.
5 4. Theory
of Operators
213
the eigenspaces. Hence it follows that any normalised system {e,} of eigenvectors corresponding to pairwise distinct eigenvalues is distal:
Consequently the power of the set of eigenvalues is no more than dimt E. Generally speaking for operators related to selfadjoint operators we can expect a roperly organised’ system of eigenvectors to have properties close to those of an orthonormal system. MukminovGlazman Theorem. Let iA be a bounded dissipative operator on a Hilbert space and {&,}y a subset of its set of eigenvalues such that
c
m#n
(im A,)(im A.,) IA,;1,12 < O
Then a system {u.}? of corresponding its closed linear hull E,.
(38)
normalised eigenvectors
is a Bari basis of
Proof. Without loss of generality we can assume that the sum of the series (38) is equal to q2 c b. Since im(A(au, + /?u,), au,,, + flu,) 2 0 for all a, B, we have
Ikl~ %)I2 G
4(im A,)(im A,) I4Pl  Xl”
(m # n).
Hence by (38)
In addition
from which it follows that the system (u.}? is olinearly system is a Bari basis in E, by Krejn criterion. lJ
independent. Thus this
The connection between selfadjoint and unitary operators on a Hilbert space, which is realised by means of the Cayley transform, extends to dissipative operators and contraction operators correspondingly. If iA is a dissipative operator its Cayley transform I = (A  ,Il)(A
 ;11)+
(im 1 > 0)
is a contraction and also Im(V,  1) = D(A) is dense. Conversely, any contraction V for which Im(V  1) is dense (since Ker(V  1) = 0) is the Cayley transform of the dissipative operator iA = (TV  Al)(V  1)l. This remark allows us to transform theorems on dissipative operators into analogous theorems on contractions and conversely. Thus, for example, we have the analogue of the MukminovG&man theorem for contractions with condition
214
Chapter 2. Foundations and Methods
 IUZ)
(39)
m#n
which results from replacing Iz,, 1, in (38) by their imagesso with respect to the fractional linear transformation I = i([ + l)(c  1)l. Remark. Condition (39) is only meaningful for eigenvalues which lie in the disks1 111 < 1. This is not surprising if we take account of the theorem of SziikefuluiNagy, Fobs and Lunger on the decomposition of an arbitrary contraction into an orthogonal sum of a unitary operator and a completely nonunitary (i.e. it has no unitary parts) operator.82 A similar remark can be made with regard to dissipative operators. Conditions (38) and (39) can be expressed in terms of Blaschke products: t38) *
jx” (s(
The weaker conditions
m
n
’ 0;
A I, >o; I I
inf n m m mnZnI,  I,
(39) * 8” ( fT,
; / > 0. ln n
44Al >0 I n,i, I
inf n m rn#rl l
arise in the problem investigated by Carleson (1958) of the interpolation cp(&) = ~1,(n = 1,2, 3, . . .) of bounded sequences (a,)? by functions cpof class H” which are analytic and bounded on the halfplane im 1 > 0 or, respectively, on the disk I;11c 1. In fact these conditions are necessary and sufficient for solvability of the stated problem for all (a,)? E I Starting from Carleson criterion, V. Eh. Katsnel on proved that if in the MukminovGlazman theorem condition (38) is weakened to the corresponding condition in (40), then the system {u ? will be a Riesz basis of its closed linear hull (and the situation is similar for contractions). The proof (for a contraction V) is based on the fact that for any finite set S of natural numbers a function cp E H” can be constructed on the disk 111 < 1 such that cp(l,) = 1 (n E S), cp(L,) = 0 (n 4 S) and 11cpII = suplql < C, where C does not depend on the choice of S. If only a finite set of interpolation conditions is retained (n < N), the function cp can be replaced by a rational (pN (in accordance with a theorem of Schur), after which a theorem of von Neumann can be applied to show that 11(pN(V) II i II (pNII. As N , co the sequence of operators ((pN(V)) converges strongly to a projection Ps onto Lin{u,),,s along e. Since lIPsI\ < C for all S, we have that {un>y is an unconditional basis in its closed linear hull. * ith subsequent return to the previous notation. s1 Correspondingly (38) applies to eigenvalues which are not real. **For an isometric operator V this is the socalled Weld decomposition; moreover in this case the completely nonunitary part turns out to be a onesided shift operator, i.e., it acts on a space of type L@ VL@ VZL@... , where L is some suitable subspace (and the unitary part acts on nzI V ).
0 4. Theory of Operators
215
4.12. Spectral Sets and Numerical Ranges. The theorem of von Neumann which has been mentioned is concerned with an aspect of spectral theory which was discovered by him; this is connected with subtle variants of functional calculus. We will discuss this topic briefly. Let A be an operator on a Banach space and S a closed subset of the complex plane C which contains spec A. The set S is said to be spectral for A if the bound II&t)II < sup,lql holds for all rational functions cpwhose poles lie outside of S. Von Neumann theorem asserts that the disk 111< 1 is a spectral set for all contractions on a Hilbert space.83 This is false on a Banach space 84, however, as V. Eh. Katsnel on and V.I. Matsaev (1967) showed, it becomes true for the disk I II < 3 and the radius p = 3 cannot be decreased for the class of all Banach spaces. For a normal operator on a Hilbert space its own spectrum is a spectral set (in consequence of the spectral theorem). Conversely, if any function which is continuous on (spec A) u {co} can be approximated uniformly on this set by rational functions, then the operator A is normal. Another useful substitute for the spectrum of an operator A is its socalled numerical range. In the case of a Hilbert space it is defined as the closure of the set of values of the quadratic functional (Ax, x) for x E D(A), llxll = 1. According to a theorem of Hausdorff the numerical range of any operator A is convex.s5 It contains the approximate spectrum and consequently the convex envelope of this set, but in general the numerical range is more extensive than the convex envelope of spec,A. For a normal operator these two sets coincide. The numerical range of an operator A on a Banach space is defined to be the closure of the set of values j&4x), where x E D(A), I/XII = 1 and f, is a support functional at the point x (the mapping x H f, is said to be supporting, as in the smooth case, but now, in general, it is not uniquely defined). The following example (BonsallDuncan, 1980) shows that the numerical range is not generally convex: for the space C(S) (S compact) with f, = 6,(,,, where s(x) is any point at which maxS Ix(s)! is attained, the numerical range of the operator defined by multiplication by a function n(s) coincides with the set of values of this function. The numerical range depends on the choice of support mapping but, if A is a bounded operator (D(A) = E), all its numerical ranges have the same convex envelope (LumerPhillips, 1961). This set coincides with the intersection of the halfplanes re([e< r(e), where r(0) (0 < 8 < 24 is the minimum of those r for which IleAc(I G efKl (arg [ = 0), i.e. l&II G l/(lcl  r(0)) (arg 5 = 8, I[1 > z(0)). The convex envelope of the spectrum is contained in the numerical range. In all cases the numerical range of an operator A contains its approximate spectrum. * onversely, if the unit disk is a spectral set for an operator A (even on a Banach space) then A must be a contraction. This follows trivially from the definition with q(A) = A. * f the unit disk is a spectral set for all contractions on some Banach space then the space must be Hilbert (Foias, 1957). ssIt is enough to establish this result in 2dimensional Euclidean space. But in this case the numerical range is an ellipse (which can degenerate into an interval) or a point (for a scalar operator).
216
Chapter 2. Foundations and Methods
For any choice of support mapping the numerical range of a dissipative operator lies in the halfplane re 1 < 0. Conversely, if the numerical range of a closed, densely defined operator lies in the halfplane re 1 < 0, the halfplane re 1 > 0 is quasiregular and if in addition n, = 0 the operator is dissipative. 4.13. Complete Compact Operators. A compact operator A on an LTS E is said to be complete if the system C,(A) of root vectors corresponding to the nonzero eigenvaluess6 is complete in Im A. In the case where Ker A = 0 this means that for the operator A‘: Im A + E the system of root vectors is dense in its domain of definition (and it turns out to be dense in E if D(A‘) = Im A is dense). In typical concrete situations, A’ is the initial operator, for example, if A’ = L, a differential operator, then A is the integral operator defined by the Green function of the operator L. Any selfadjoint (and even any normal) compact operator A on a Hilbert space is complete. It is natural to expect that completeness is preserved under small perturbations of these operators. This expectation is justified if the eigenvalues of the unperturbed operator tend to zero sufficiently rapidly. The first general result in this direction is due to M.V. Keldysh (1951). Theorem of Keldysh. Suppose that the operator A on a Hilbert space E has the form A = S(l + R), where S, R are compact operators, S is selfadjoint and the eigenvalues o, # 0 (n = 0, 1,2, . . .) of S, taken according to their multiplicities, satisfy the condition
O b <* for some p > 0.
(41)
Zf Ker A = 0 and Ker S = 0, the operator A is complete.
Condition (41) is equivalent to the requirement that S belongs to the class GP. We shall clarify this from a certain more general point of view. Let T by any compact operator on a Hilbert space E. Then T* T is a selfadjoint nonnegative compact operator. Its positive eigenvalues can be enumerated according to multiplicity in decreasing order*‘: r1 2 rZ > .** . The numbers s,(T) = 4~1, SZ(T) = ,/rz, . . . are called the singular numbers (snumbers) of the operator T (clearly s,(T) = II T II). It turns out that they coincide with the anumbers (D. Eh. Alakhverdiev, 1957). The class GP for a Hilbert space is therefore determined by the condition
For a selfadjoint operator this condition
is equivalent to (41).
Remark 1. It can be shown that if Ker T = 0 and Ker T* = 0, the operators T* T and TT* are similar (even unitarily similar). Therefore T* E GP o T E G,,. 86 We denote the system of root vectors corresponding to all eigenvalues by Z(A). s’ If T is of finite rank r then we put T” = 0 (n > r).
5 4. Theory of Operators
Remark 2. For any compact operator T on a Hilbert expansion
217
space we have a Schmidt
T=c”dT)(., GL, where (u,), (u,) are two orthonormal systems. The first consists of the eigenvectors of the operator A, = m while the second is made up of those of the operator A, = JTT*. This result  is closely  connected  with polar representations: T = U,A, = A,&, where U,: Im T* + Im T, U,.: Im T + Im T* are invertible isometries. Moving on to a concise account of the proof of Keldysh theorem, we recall that $, is a twosided (not closed) ideal in the algebra 5?(E) of all bounded operators on the Hilbert space** E. We denote by P the orthoprojection onto the closed_ linear hull of the system Z,(A) = Z(A) and put Q = 1  P. Then the operator A = QAQ = QA turns out to be a Volterra operator and therefore its Fredholm resoluent F(c) = (1  CA ’ is an entire function of c. Moreover 2 E G,,. We can deduce from this that F(c) has growth of order no higher than p and of minimal type as l[l+ co: In 11 F(C)11 = o( Ill . On the other hand, using the given structure of the operator A, we can show that IIF(LJ/ is bounded on the pair of angles E < (arg (1 < x  E. Boundedness on the pair of angles larg 51 < E, n  E < larg cl < n follows from the PhragmenLindelof principle. Finally, F(c) is bounded on the whole plane. By Liouville theorem F(c) is constant. Consequently 1 = 0, i.e. QA = 0, which establishes the required inclusion Im A c Im P = Lin Z,(A). Corollary. If S, R satisfy B = (1 + R)S is complete.
the conditions
of the theorem, then the operator
In fact Ker (1 + R) = 0, therefore we have from Fredholm theory that 1 + R is an invertible operator; but then B is similar to A: B = (1 + R)A(l + R) 0 Remark. V.I. Matsaev (1961) established that it is sufficient in Keldysh theorem to require that the operator S belongs to the twosided ideal 6, which is defined by the condition: Z n ,( T) < co. He developed a general method for estimating the growth of the resolvent using the distribution function of the snumbers, from which he was able to obtain profound tests for completeness of a system and summability of the corresponding expansion. We note here that if A, B are selfadjoint operators with B E 6, and spec(A + Bi) c R, then A + Bi is an operator with separated spectrum and a complete system of spectral subspaces (corresponding to an arbitrarily line covering of the spectrum), since its resolvent satisfies Levinson condition (V.I. Matsaev, 1961). Example. Let us consider the differential operator L2(a, b) ((a, b) a finite interval) by the expression
A which is defined on
*sThe only nontrivial closed twosided ideal in f?(E) is the ideal of compact operators (theorem of Calkin).
218
Chapter 2. Foundations and Methods
and boundary conditions” nl @j[Y]
=
1 k=O
Under these conditions
{ajkytk'(a)
let the
+
bjky'k'(b))
=
'
(1 <j < 4.
rincipal part’ lo=
ff’ ( > define a selfadjoint operator H and let both operators A, H be injective (the latter does not restrict generality). We put S = HThis is a compact selfadjoint operator. Its eigenvalues have the form 1= io where o runs through the set 52 of roots of the characteristic determinant A(o) = det(@j[ei
]);,kzl
(the Ek are the nth roots of unity). The multiplicities of the eigenvalues do not exceed n  1. Since A(o) is an entire function of exponential type we have
for any 6 > 0. The operator S therefore satisfies the conditions theorem for any” p > l/n. We now have
of Keldysh
But
is a HilbertSchmidt integral operator and is therefore compact. Consequently A’ = S(l + R), where R is a compact operator. Applying Keldysh theorem we establish that the system of root functions of the operator ~4 is complete in L a, b).
Similarly we can obtain completeness theorems for systems of root functions of partial differential operators of elliptic type on bounded domains under boundary conditions, where the principal part of the operator is selfadjoint. Under conditions which are strictly stronger than those of Keldysh theorem it is possible to obtain a theorem on the expansion of each vector x E E in a series * e assume that the boundary conditions are linearly independent and the coefficients pJs) are measurable and essentially bounded. he weaker result that S E Gz follows from the fact that S is an integral operator with HilbertSchmidt kernel.
5 4. Theory of Operators
219
of root vectors of the operator A, which converges after a certain grouping (independent of x) of its terms (AS. Markus, 1962). In a generalisation of Keldysh theorem to operators on a Banach space (A.S. Markus, 1966) selfadjointness has a suitable replacement in the property of being quasiconservative. Theorem. Suppose that an operator on a Banach space has the form A = S(l + R), where S, R are compact operators and in addition S is a quasiconservative operator of class GP for some p > 0. If Ker A = 0 and Ker S = 0, the operator A is complete.
The eigenvalues of any compact operator can be enumerated in order of decreasing modulus, taking account of multiplicity: I,, A,, . . . (if the set of eigenvalues is finite we complete the infinite ail’ with zeros). If the series c. 1, converges unconditionally (and so also absolutely since it is a series of numbers) then its sum is called the spectral trace of the operator A. A natural and important problem is to determine conditions under which the spectral trace coincides with the tensor trace (clearly such conditions must ensure the existence of both traces). The first advance in this direction (not counting the classical theorem for the finite dimensional case) is due to V.B. Lidskij (1959), who showed that for any operator of class 6, on a separable Hilbert space E (in other words, the operator is nuclear since E is a Hilbert space) both the spectral trace and the matrix trace with respect to any orthonormal basis exist and they are equal.g1 By the same token the tensor trace is equal to the spectral trace in the given case. In further investigations (A.S. Markus  V.I. Matsaev (1971), K&rig (1986)) this result was extended to operators of class 6, on any Banach space. One of the important applications of Lidskij theorem is to provide a necessary condition for completeness of the system Z(A) of root vectors of a compact operator A on a separable Hilbert space, whose imaginary componentg2 im A is nuclear: l im Iz, = tr(im A). For the proof it is sufficient to orthogonalise the system L A). The diagonal elements of the matrix of the operator im A with respect to the resulting orthonormal basis are equal to im ;2, (n = 1,2, .. .). For a compact dissipative operator A on a Hilbert space, whose real component is nuclear, the race equation rewritten in the form $i re 1, = tr(re A), 91A fundamental result in this situation is that the matrix trace of a Volterra operator of class B, is equal to zero. If A is a bounded operator (D(A) = E) then re A fr $A
+ A*),
im A zf &A
 A*).
220
Chapter 2. Foundations and Methods
is not only necessary but also sufficient for completeness of the system J&4) (M.S. Livshits, 1954). In fact, for the invariant subspace E, = Lin C(A) the trace of the operator re(AIE1) is given by (42’). If E, # E then E, E Ef # 0 and then for the orthoprojection P on E,, we will have tr(P* re(Al,,)’ P) = 0; hence re(Al,,) = 0 since re A < 0. Consequently ( l/i)A(eO is selfadjoint (E, is invariant for A*). Since A IEOis compact it must have an eigenvector  a contradiction. It is clear from what has just been discussed that the inequality zt re J., 2 We 4 holds for any compact dissipative operator with nuclear real component. The system of root vectors of a nuclear dissipative operator is complete (V.B. Lidskij, 1959) since Inn, = tr A and In& = tr A*. In conclusion we note that completeness of a compact operator is not in general inherited by its restrictionsg3: any Volterra operator on a Banach space is the restriction of a complete compact operator acting on some larger Banach space (N.K. Nikol kij, 1969). 4.14. Triangular Decompositions. The theorem on the existence of a (nontrivial) closed invariant subspace for any compact operator A on a complex LCS E (dim E > l), with the help of Zorn lemma, leads to the existence of a chain of such subspaces which is maximal with respect to inclusion. Any such chain begins with zero and terminates with the space E. In the finitedimensional case it has the form 0 c Lo c L1 c *.* c L, = E, where dim Lk = k (0 < k < n = dim E). Any system of vectors (~1; chosen so that eL E L,\LtI (1 < k < n) is a basis in E with respect to which the matrix of the operator is upper triangular. In the infinitedimensional case the chain Z is not in general discrete (for the order topology) and therefore it cannot generate a basis for a triangular decomposition. The latter has to be connected directly with the chain Z. For each nonzero subspace M E Z we consider the closure M of the union of all those L E Z which are contained in M but are distinct from it. Then M E Z since the chain Z is maximal and clearly M c M. If M # M, the factor space M/M is onedimensional. Those M for which this holds are called points of discontinuity of the chain Z. In the absence of points of discontinuity the chain Z is said to be continuous. If M is a point of discontinuity,
the factor operator A, can be identified with some scalar AM. Ringrose (1962) proved the following theorem (for a Banach space E). Theorem. The set of all nonzero values 1, corresponding to all possible points of discontinuity coincides with spec A\(O). For each I E spec A\(O) the number
931f an operator is complete along with all of its restrictions, then we say that it allows spectral synthesis.The problem of spectral synthesis goes back to a work of Wermer (1952) which is concerned with normal operators. The general situation was studied by A.S. Markus (1970).
8 4. Theory of Operators
221
of points of discontinuity for which 1, = A is equal to the dimension of the corresponding root subspace. Corollary. In order that the operator A be a Volterra operator it is necessary and sufficient that 1, = 0 for all points of discontinuity of the chain 2.
In particular, if the chain 2 is continuous, A must be a Volterra operator. Conversely, if A is a dissipative Volterra operator on a Hilbert space E and Ker A = 0, then all maximal chains of its closed invariant subspaces are continuous (M.S. Brodskij, 1969). A continuous maximal chain 2 for any Volterra operator A on a Hilbert space E results from inessential extension. In this construction a certain Hilbert space G (dimt G depends on A) is adjoined orthogonally to E and we take the operator A”on H = E @ G such that alE = A, A c= 0; .? has the required chain. A Volterra operator A is said to be unicellular if the maximal chain of closed invariant subspaces is unique 94. This term originates from the simplest finitedimensional example  a single Jordan cell. A nontrivial example95 is the integration operator Z on Lp(O, 1) (1 < p < co). In fact, it follows from the Titchmarsh convolution theorem and the completeness of the system of powers that any closed invariant subspace of the operator Z must consist of all functions which vanish almost everywhere on a fixed interval (0, a) (0 < a < 1). In exactly the same way we can establish the unicellularity of all integral operators of fractional order:
V,dW =&s’6VW) dt
(0 < c1< 1).
0
It is interesting that no two of these operators on L O, 1) are equivalent (I.Ts. Gokhberg  M.G. Krejn, 1967), a result which contrasts sharply with the equivalence of all unicellular operators on a finitedimensional space. In a Hilbert space E instead of closed subspaces we can consider the corresponding orthoprojections. Corresponding to the subspace inclusion L, c L, we have the inequality P1 < Pz between the respective projections and therefore the chain of subspaces transforms into a chain of orthoprojections. Suppose we are given an arbitrary strongly closed chain W of orthoprojections and a function F: W + !i?(E). We can define the Brodskij integral A=
F(P) dP sW
as the uniform limit of the integral sums
94This is equivalent to pairwise comparability of the closed invariant subspaces. 95A further example: the weighted shif operator (&)g H (a,<,,)~ (t+ = 0) on 1’ under specific conditions on the weight sequence (a,) (N.K. Nikol kij, 1967).
222
Chapter 2. Foundations and Methods
directed by the partitions PO < PI a* * < P., where all the Pk, Qk belong to the chain W and Pk1 < Qk < Pk (1 < k < n). It is clear that not every function F is integrable in this sense. If A is a Volterra operator, 2 is a continuous maximal chain of closed invariant subspaces and W is the corresponding chain of orthoprojections, then we have the triangular decomposition (MS. Brodskij, 1961) P(im A) dP
A = 2i sW
(the existence of the integral is guaranteed). From this we extract a rich supply of information on relationships concerning properties of the real and imaginary components of a Volterra operator; for example there is the theorem of Sakhnooich that, if the imaginary component of a Volterra operator is a HilbertSchmidt operator, then so also is its real component; we also have the more general theorem of Matsaev: if A is a Volterra operator and im A E E$ then re A E $, (1 < p < co) and moreover N,(re A) < y,N,(im A), where the constant yp depends only on p. In the same direction for p = 1 (i.e. for im A nuclear) I. Ts. Gokhberg and M.G. Krejn (1967) established the deep spectral density theorem for the operator re A:
lim n+(R) 1s =R a,
N,(dP*im
AodP),
RW
where n+(R) is the number of eigenvalues (counted according to multiplicity) the operator re A in the semiaxis CR‘, 00). 4.15. Functional Models. is a commutative diagram
of
A model of a linear operator A acting on an LTS E EE
A
in which B is a linear operator on some LTS G and V is a topological isomorphism (an operator intertwining g6 A with B). Clearly the operators A and B are equivalent (similar) in this situation. Frequently we just call the operator B a model of the operator A. If E and G are Hilbert spaces and Y maps E isometrically onto G, we use the term unitary equivalence. On replacing an operator by its model all its lineartopological properties and its essential nature are preserved (for example, the spectrum, eigenvalues, approximate spectrum). Moreover a 96Here (as in certain other contexts) the term one space into another. As in the theory of ntertwining operator’ in a more general way, but not necessarily continuous invertibility. In
perator’ is used in spite of the fact that V acts from representations it is convenient to define the term namely to require only continuity of the operator V this situation we can speak of a quasimodel.
0 4. Theory of Operators
223
concrete model of an operator may prove simpler to study. For example, if it is a functional model (i.e. G is a function space), some analytic techniques can be applied to it. By way of illustration let us consider a Volterra operator A on a Hilbert space with onedimensional imaginary component9’ and suppose that iA is completely nonselfudjoint (it has no selfadjoint parts). It follows from results of M.S. Livshits (1946) that A is unitarily equivalent to an operator f iZ;, where Z; is the integration operator98 on L’(O, 1) from s up to I (0 < I < co). The unicellularity of the operator Z; (which can be established by analytic means) therefore implies that of the operator A. However the last result can also be obtained directly from the triangular decomposition of the operator A (M.S. Brodskij, 1965) 99. In any event the very fact that the operator Z; is unicellular entitles us to fix it in the role of a standard triangular model which is not subject to further decomposition just as with a Jordan cell in the finitedimensional case. The more general triangular model 1 (0 4 s d 2) WA(s) = +)cp(s) f i d0 dt ss generated by the real function a E L”(0, 1) is no longer a Volterra operator: the spectrum of the operator /i coincides with the set of essential values of the function u (the term a(s)cp(s)is analogous to the diagonal part of a matrix). Any completely nonselfadjoint bounded operator A with onedimensional imaginary component and real spectrum is unitarily equivalent to an operator n defined by some nondecreasing function a and choice of sign in front of the integral. For the proof of this theorem MS. Livshits introduced a powerful unitary invariant, the characteristic function w,(A) = 1  i(R,(A)e,
e)j,
where the channel vector e and the number j = + 1 are defined by the equation im A = &(. , e)e. It turned out that for the given class of operators coincidence of characteristic functions implies unitary equivalence. But for an operator n its characteristic function can be calculated immediately; in fact with j = 1 (to be snecific) (43)
Moreover for the operator A under consideration (with j = 1) the characteristic function maps the upper halfplane im 1 > 0 conformally onto the domain 1w 1> 1 and is regular at infinity. It can therefore be represented in the form (43) for some a(t). But then A turns out to be unitarily equivalent to the corresponding operator /i. ny operator with onedimensional imaginary component is dissipative to within a factor + i. g8 The modification Z I+ Z’is occasioned by the desire to work with an upper (rather than a lower) triangular decomposition. In other respects it is immaterial. 99And Titchmarsh convolution theorem can be deduced from the fact that the operator Z is unicellular (Kalisch, 1962).
224
Chapter 2. Foundations and Methods
Where there is a nonreal spectrum (necessarily no more than countable and consisting of normal eigenvalues), a iscrete’ triangular model is adjoined orthogonally to the previous ontinuous’ model; this is defined on 1’ by the formulae I ?k
=
Aktk
+
i .=g+,
&&i&
+
i
(Pb)jflk
dt
(k=1,2,3
,... ),
s 0
where (A,)? is the sequence of eigenvalues and /lk = im 1,. Correspondingly, Blaschke product
the
* && lIfz=l ;I  & appears in (43). Developing this approach, MS. Livshits (1954) constructed a triangular model for a completely nonselfadjoint bounded operator A with finitedimensional imaginary component. In this case the characteristic function turns out to be a matrix of order r = rk(im A) and the role ofj is taken by a certain Hermitian matrix .Z such that Jz = 1. The characteristic matrixfunction w,(n) (im A > 0) is Jexpanding, i.e. W”(L)JW,(n) > I, and moreover on the real axis outside of the spectrum it is Junitary: W”(n)JW,(L) = I. The corresponding analogue of formula (43) (with products of Blaschke type) was obtained by V.P. Potapov (1950). Passing to the limit as I + cc we can extend the triangular model further to the case where the imaginary component is nuclear. It is impossible to get rid of the condition of being completely nonselfadjoint, since the characteristic function does not change if an arbitrary bounded selfadjoint operator is adjoined orthogonally to A. However this clearly does not matter from the point of view of classification theory. We can say that for operators whose imaginary components are nuclear there is a valuable analogue of Jordan theorem. Incidentally, although the spectral resolution of a selfadjoint operator
reveals its structure sufficiently deeply, it is nevertheless useful to construct in addition a functional model for it. Its structure depends on the multiplicity of the spectrum which we will not define in the general case; we restrict ourselves to the case of a simple spectrum, i.e. where there exists a vector x such that the system (@lx)lo n is complete. lo1 Then, putting c#) = (a,~, x), we can show without difficulty that A is unitarily equivalent to the operator of multiplication by A on the space Li. Now let A be a contraction on a Hilbert space E. A natural measure of the nonunitariness of an operator A is the pair of nonnegative selfadjoint operators To be precise it is assumed further that the nonreal spectrum is infinite. lo1 A vector x with the stated property is said to be generating or cyclic.
5 4. Theory of Operators
1  A*A and 1  AA* (just as the imaginary measure of its nonselfadjointness). Let us put A, = ,/(l  A*A), The characteristic
225
component
of an operator is a
A, = ,/(l  AA*).
operatorfunction
W(A) = (A
+ 1A,(l  AA*)lb,)
I Im
is introduced in this context (Yu.L. Shmul an, 1953). The values of this function are contracting homomorphisms from I%& into Im A,. It is defined and analytic on the domain (reg A*)in particular on the disk 111 < 1, where it is usually considered. If the characteristic functions of completely nonunitary contractions coincide, then these contractions are unitarily equivalent (A.V. Shtraus, 1960). SzokefalviNagy and Foias (1963) found a natural (in terms of characteristic functions) functional model for an arbitrary completely nonunitary contraction A. Of necessity it is rather complicated, since any contraction is the orthogonal sum of a completely nonunitary contraction and a unitary operator and any bounded operator becomes a contraction on normalisation. The construction of the model is significantly simplified if the sequence of powers (A ; converges strongly to zero (although even in this case getting information depends essentially on the measure of nonunitariness, more precisely on the rank of the operator A,). In order to describe the model operator in this situation we consider an arbitrary Hilbert space G and introduce the Hardy class H G) of analytic vectorfunctions
u(A)= jJ t&A”
n=O which satisfy the condition
(IAl < 1, u, E G (n = 0, 1,2, . ..)).
11412= nzo 11%112 < a* This is a Hilbert
space with natural scalar product
The shift operator lo3 (7%)(L) = Au(n) acts on it; it is clearly isometric (but not unitary: Im T = (u: u(O) = O}). The adjoint operator (T*u)(q
is a contraction
= @) ; No) = 2 u,+lA” n=o
(
u(A) = 2 U ” n=O >
(and the sequence of its powers converges strongly to zero). If
ro2This construction arose in another way in joint works of SzokefalviNagy and Foias (1962) and for finitedimensional A,, A, it is equivalent to the definition of M.S. Livshits (1950). A.V. Shtraus (1959) extended the concept of the characteristic function to unbounded operators. lo3 Strictly speaking, the shift is effected on the sequence of Taylor coefficients of the function u(A).
Chapter 2. Foundations and Methods
226
we now take G = Im A,, then T* turns out to be a quasimodel intertwining operator V: E + H2(G) is defined by the formula
for A. The
In fact 11Vxl12 = 2 (AJ , n=O
=
A,A ) = 2 ((1  A*A)A , n=O
A”$
O{ IIA”412  ll~“+w12) = llxllZ1
i.e. V maps E isometrically into H’(G). In addition it is clear that T*V = VA. Therefore the subspace L = Im V c H2(G) is invariant for T* and T*I, is unitarily equivalent to the operator A. If rk A, = 1, the model operator is the adjoint of a shift on the scalar Hardy class HZ. It is possible to describe all closed invariant subspaces for this operator (and hence also those for the initial operator A) (Beurling, 1949). Each such (nonzero) subspace has the form 8H2 = {cp: cp = e$ ($ E HZ)},
where 0 is a fixed function in H2 whose boundary values on the unit circle have modulus one almost everywhere (an interior function). The general approach to models of SzokefalviNagy and Foias is based on the study of the socalled dilatations 2 of the operator. This term is applied to any operator A which acts on some Hilbert space E 2 E and which is associated with A by means of the relations A”=pAI” where P is an orthoprojection Theorem of SziikefalviNagy.
(n = 1,2, 3, . . .),
on E”with Im P = E. For any contraction A on a Hilbert
space there
exists a unitary dilatation. The original proof (1959) of this important theorem was connected with the trigonometric moment problem and the theory developed by M.A. Najmark (1940) of selfadjoint extensions of operators out of their basic space. lo4 There is, however, a more direct proof (SziikefalviNagy and Foias, 1967). First of all an isometric dilatation is constructed on ,& = E 0 E 0.. . in which E is embedded via the first summand. It has the form lo4 Figuring in this theory are nonorthogonal spectral resolutions of symmetric operators Pm Jm
where gA = I ,, 6, being an orthogonal resolution of the identity in a larger Hilbext space k and P an orthoprojection on E onto E.
$4. Theory of Operators
227
x0 0 Xl 0 x2 CD~~~~Ax~@A,x~~x~~x~~....
Now it is possible to extend an isometric dilatation to a unitary one. In fact, thanks to the Wold decomposition, we can suppose that the initial dilatation is a onesided shift and then the corresponding twosided shift will be its unitary extension. The theory of characteristic functions, functional models and dilatations found deep applications in mathematical physics.‘05 In particular, in quantummechanical scattering problems the characteristic matrixfunction coincided with the socalled Smatrix of Heisenberg (MS. Livshits, 1956). This result is of a sufficiently general nature (LaxPhillips, 1964; V.M. Adamyan  D.Z. Arov, 1965). It was used for the investigation of analytic properties of the Smatrix (B.S. Pavlov  L.D. Faddeev, 1972) in scattering theory for automorphic functions, which Lax and Phillips (1976) then developed in detail. The theory of dilatations proved to be an effective tool for investigating the socalled Redge singularity (B.S. Pavlov, 1970). From the physical point of view, nonselfadjointness or nonunitariness of the basic operators which describe the dynamics of physical systems reflects processes of dissipation (of energy or other ubstance’) caused by nonclosedness (openness) of the systems, i.e. by their relations with the external world. Starting from these ideas M.S. Livshits (1963) associated with each open system a mathematical object called an operator node. This consists of a pair of Hilbert spaces H, E (H is the space of internal states of the system, E is the space of channels of communication of the system with the outside world) and a set of three continuous homomorphisms: a dynamical operator A: H + H, a channel operator J: E + E (J* = J, J2 = 1) and a communication operator I: E + H. It is assumed in addition that the relation im A = $TJT* is satisfied. Unitary equivalence is defined naturally in the category of operator nodes. The characteristic operatorfunction w(n)
= 1  iJF’*R,(A)T
is a basic unitary invariant of an operator node. If an open system is decomposed into a chain of subsystems lo6 the characteristic function becomes factorised. Conversely, therefore, if we factorise the characteristic function into factors corresponding to elementary (in the physical sense) systems, we then obtain a model for the initial system in the form of a chain of elementary systems. Many of the concrete situations can be included in this scheme (M.S. Livshits, 1966). Metric. The structure introduced on a Hilbert space E by a operator J such that J2 = 1 (J # + 1) is called an indefinite metric,
4.16. Indefinite
selfadjoint
lo5 From nternal’ applications of this theory we note that of necessary and sufficient conditions for the existence of Riesz and Bari bases composed of root vectors of a contraction on a Hilbert space (N.K. Nikol kij  B.S. Pavlov, 1970). ro6Among the channels of communication are entrance and exit channels. Successive unions of systems are effected by connecting the exits of one system to the entrances of another. This procedure can be formalised in terms of what has been described.
228
Chapter 2. Foundations and Methods
or Jmetric,
since it allows us to consider along with the usual distance the
pseudodistance
d&s Y) = t/CJCx Y), x  Y) (with values in R u ilw) defined by the Jscalar product (u, o) = (Ju, u). The Lorentz metric in fourdimensional space, which is used in the special theory of relativity, is an example of this. Indefinite metrics on finitedimensional spaces had already been studied by Frobenius (at the end of the 19th century). The first work dealing with operators on an infinitedimensional space with an indefinite metric is due to L.S. Pontryagin (1944), further development being associated first of all with investigations of M.G. Krejn and I.S. Iokhvidov. At the present time the theory of linear operators on spaces with an indefinite metric represents by itself an extensive branch of functional analysis with outlets into numerous applications (theory of oscillations, quantum physics etc.). Clearly, we can represent the metric operator as the difference of two mutually orthogonal (nonzero) orthoprojections which form a decomposition of the identity: J=P+P;
P+ + P = 1;
P+P = PP+ = 0.
Correspondingly, any x E E can be expressed as the sum of mutually vectors x, = P+ x, x = Px; moreover Jx=x+
x,
orthogonal
<x, x> = IIx+ II2  lb l12.
The index K = rk P is the easure of indefiniteness of the space. It was assumed in the work of L.S. Pontryagin that ICis finite and many results were obtained initially under this condition. Remark. Suppose that on a complex linear space E there is given a Hermitian symmetric bilinear functional Oforallu~E+, u # 0 (i.e. E, is preHilbert under the Jmetric) and E = E i E,. Starting with the decomposition x = u + u (u E E, u E E,) we introduce a scalar product on E (x1, x2) =  (IQ, u2) + (ul, u2), thereby making E into a preHilbert space. Completeness of the space E is equivalent to completeness of the space lo8 E, since u < co. If we have completeness, E turns out to be a space with indefinite metric: J = P+  P, where PT are the projections associated with the decomposition E = E i E, .
l f E is finitedimensional, K is the negative index of inertia of the form (x, x). lo8 It does not depend on the initial choice of the space E .
$4. Theory of Operators
229
In what follows, E is a Hilbert space with indefinite metric; the notation introduced earlier remains fixed. The general form of a continuous linear functional on E, i.e. f(x) = (x, y), is easily reformulated in Jterminology: if z = Jy, then f(x) = (x, z). A subspace L c E is said to be positive (nonnegative) if (x, x) > 0 for all x E L, x # 0 ((x, x) > 0 for all x E L). The following terms are introduced in a similar way: negative (nonpositive) subspace. A subspace L is said to be neutral if (x, x) = 0 for all x E L. From the geometrical point of view (x, x) = 0 is the equation of a second order cone log; the positive vectors lie nside’ the cone, the negative ones utside’ it. The subspace E, = Im P+ is nonnegative and maximal (with respect to inclusion) among the nonnegative subspaces. Phillips (1959) and Yu. P. Ginzburg (1961) obtained the following important characterisation of maximal nonnegative subspaces. Theorem.
In order that L be a maximal nonnegative subspace it is necessary that there exist a contracting linear operator ’ K: E, + E(E = Im P) such that L = Im(1 + K).
and suficient
Let A be a linear and (for simplicity) bounded operator on E. The operator A’ = JA*J is called the Jadjoint of A on the grounds that the identity (Ax, y) = (x, A ) is satisfied by it (and only by it). The formal properties of the adjoint are preserved in the Jadjoint: A” = A; (aA + /ID)’ = EA’ + FB (A@’ = B 1’ = 1. If A’ = A we say that A is Jselfadjoint. If A is invertible and A’ = A’ then A is said to be Junitary. The study of Jselfadjoint operators reduces to that of Junitary operators with the help of the Cayley transform. For the sake of brevity therefore, unless otherwise stated, the assertions made in the sketch below relate only to a Junitary operator A. First of all we note that it follows from the equality A’ = JlA*J that the spectrum of the operator A is symmetric with respect to the unit circle, i.e. if Iz E spec A then 1* E (1)l E spec A. The spectrum cannot be unitary even in the finitedimensional case. In general, a meaningful spectral theory for Junitary operators requires that the index K is finite or at least the operator P AP+ is compact. In this case A = U + I/ where U is unitary and I/ is compact. Consequently, the nonunitary spectrum of the operator A is no more than countable (if it is infinite, it condenses on the unit circle) and consists of normal eigenvalues L,, A:, A,, A,*, . . . (I 1,1 < 1 (k = 1,2, . . . )). The corresponding maximal root subspaces IV,, IV:, W,, W2*, . . . are neutral; each W, (Wk*) is lo9 Here the word one’ is used in the sense of elementary analytic (if preferred  algebraic) geometry. rr” Because of its geometric character we call this an angular operator. This is equivalent to the identity (Ax, Ay) = (x, y) in combination with the requirement of surjectivity; by itself the identity, which is known as Jisometry, is equivalent to the equality A*JA = J. The inequality A*JA d J (i.e (Ax, Ay) C <x, y)) defines the Jcontractions. We can define Jdilatations similarly.
230
Chapter 2. Foundations and Methods
Jorthogonal to all R$ y* except W,*(W,); the subspaces W,, W,* (k = 1,2, . . .) are in duality with respect to the Jscalar product (consequently, dim Wk* = dim W,). Suppose that the index ICis finite. Then the dimension of the spectral subspace L, corresponding to the part of the spectrum which lies in the disk 111< 1 does not exceed IC and is equal to the dimension of the spectral subspace L, corresponding to the symmetric part of the spectrum. The direct sum L, = L, i L, is called the hyperbolic subspace of the operator A. The spectral subspace L, corresponding to the unitary spectrum is Jorthogonal to L, and L, + L, = E. By the same token (for finite IC) we can restrict attention to the case of unitary spectrum in the spectral analysis of Junitary operators. In this connection substantial use is made of the Theorem of Pontryagin. If the index K is finite , any Junitary’ l3 operator A has invariant maximal nonpositive and nonnegative subspaces.
The dimension of an invariant maximal nonpositive subspace M is equal to K. It can be chosen in such a way that the spectrum of the operator Al, lies outside the disk 111 < 1. Let p(L) be the characteristic polynomial of the operator Al, (so that p(Al,) = 0). Then114 Im p(A) c M(l). But M(l) is a nonnegative subspace. Consequently (p(A)y, p(A)y) 2 0 (y E E). Putting q(1) = p(n)p(n , we obtain (q(A)y, y) > 0 and also q is a Hermitiansymmetric (real on the unit circle) polynomial in L and 1l. This procedure for achieving definiteness (I.S. Iokhvidov, 1950) served as the key to the spectral theorem which is finally established for all’ ’5 specified Junitary operators with unitary spectrum (Langer116, 1967). Although the procedure depends on the choice of subspace M, the final spectral resolution is unique. The main complication in the proof of the spectral theorem is associated with the roots of the polynomial q which lie on the circle 111= 1. Near to these ritical points’ the resolvent has power order of growth higher than the first. M.G. Krejn and Langer (1964) applied the indefinite metric to investigate the quadratic equation X2+BX+C=0
(44
in the algebra f!(H) of operators on a Hilbert space H. Here the coefficient B is assumed to be selfadjoint while C is compact, selfadjoint and nonnegative. The operators And also under the condition that the operator P AP+ is compact (M.G. Krejn, 1950). It is interesting to note that M.G. Krejn proof is based on the fixed point principle. 3A Jselfadjoint operator was considered in the work of L.S. Pontryagin. The transition to a Junitary operator was effected by LS. Iokhvidov (1949) using the Cayley transform. I14 For any subspace L the subspace ~5 = {x: (x, y) = 0, (y E I,)} can be called its Jorthogonal quasicomplement. In general L n L ) # 0. ’I5 Without restriction of the type to compactness or finiteness of the index. 116Jselfadjoint operators were considered in the work of Langer (the finite index case is due to M.G. Krejn and Langer (1963)).
231
5 4. Theory of Operators
on the space E = H OH were considered. Since A is Jselfadjoint and P AP+ = ,/C is compact, there exists a maximal nonnegative subspace which is invariant for A. If K is the corresponding ngular’ operator, then X = KJC turns out to be a root of the equation (44). Also X*X < C since lllyll < 1. We note that equation (44) is naturally associated with the equation of linear oscillations Tii + Rzi + Vu = 0
(45)
Here ZJ= u(t) is a vectorfunction with values in a Hilbert space H and T, R, V are selfadjoint (generally unbounded) operators with T > 0, R 2 0 and V >> 0 (the forms generated by T and Vare the kinetic and potential energies respectively, R is a dumping operator). Under the substitution u = flu, equation (45) becomes Cfi+Bti+u=O
(46)
where C = V“2TV”2, B = V“2RV’12. The operators C and B are usually bounded in applications and after closure they satisfy the conditions formulated above. Exponential solutions u = uOeQf arise as eigenvectors of the quadratic pencil of operators S(p) = Cp2 + BP + 1, S(p)uO = 0. If X is an operator root of equation (44), 1 is any of its eigenvalues (2 # 0) and u0 is a corresponding eigenvector, then u,, is an eigenvector of the pencil S for the eigenvalue p = 1l. A root X gives rise to another root Y = X*  B, which is also a source of eigenvalues and eigenvectors of the pencil S. If we now put Z = X  B, the pencil factorises: S(P) = (1  Pm
 Pw.
The pencil S is said to be strongly damped if m
4 > 2J{ (x9 4 (Cx, xl>
(x # 0).
It is easy to see that in this case all the eigenvalues of the pencil are nonnegative. Taking account of multiplicity, we shall denote by (0;) those of them which are generated by the operator X and by ( 0:) those generated by the operator Y. It turns out that the pencil has no other eigenvaluesl” (and there are no other roots of equation (44)). Each of the corresponding normalised systems of eigenvectors {u:}, (0:) is a Riesz basis in H. In this sense the system of all eigenvectors of the pencil is doubly complete. The phenomenon of multiple completeness of the system of root vectors of a polynomial operator pencil had already been discovered by M.V. Keldysh (1951). Since then the theory of polynomial pencils has developed into an
And even no other points of the spectrum, i.e. the operator S(p) is continuously invertible for p different from (CD;), (  u$ .
232
Chapter
2. Foundations
and Methods
independent chapter of functional analysis and has found various applications in mathematical physics. Remark. The investigation of quadratic equations in the algebra 52(H) is, on the whole, a problem of transcendental difficulty. This is clear from the fact that for a continuously invertible operator A the existence of a root of the equation X2  AX = 0 different from 0 and A is equivalent to the existence of a nontrivial closed invariant subspace. 4.17. Banach Algebras. This important topic is closely connected with the spectral theory of operators. In fact, let ‘3 be a complex commutative Banach algebra (in the sequel it will simply be called an lgebra’). Let us consider the isometric embedding T: ‘9I + !2(9I) which associates with each a E VI the operator T,x = ax (x E ‘3l).Invertibility of an element a is equivalent to invertibility of the operator T, in Q(a). Consequently, spec a = spec To and so information about the spectra of bounded operators carries over to ‘3. In particular, the spectrum of any element a E ‘$I is nonempty and compact, Gel and formula holds for the spectral radius p(a) etc. Example. Let A be a bounded operator on a Banach space E and let [A] be the smallest closed subalgebra of 2(E) which contains A, i.e. the uniform closure of the algebra of all polynomials in A. We denote the resolvent set and the spectrum of the element A in the algebra [A] by reg,A and spec,A respectively. Clearly, reg,A c reg A and hence spec A c spec,A; moreover p,,(A) = p(A) as a result of Gel and formula. It turns out that reg,A coincides with the unbounded connected component G, of the set reg A; correspondingly specJ = spec A u G, where G is the union of the bounded connected components of the set reg A. In fact, if 112)> p(A) we see by expanding in a Laurent series at infinity that R, E [A]. But then it follows from the Taylor expansion of the resolvent at the point 1 that R, E [A] for p in the maximal neighbourhood of the point I which does not intersect spec A. Connecting the point 1 to any point [ E G, by means of a polygonal line, we deduce after a finite number of steps that R, E [A]. Thus G, c reg,A. Now let H be a bounded connected component of the set reg A and let p E H n reg,A. Then for any E > 0 there is a polynomial such that II R,  p(A) II < E. Any point J, E 3H is contained in the approximate spectrum of the operator A. Let (x,,)? be a quasieigensequence: Ax,  Ix, + 0, 11x,11= 1. Applying the operator R,  p(A) to x,, we obtain in the limit: I(n  p)’ p(n)1 < E. Thus the function (A  pL)’ can be uniformly approximated by polynomials on aH, which contradicts the existence of a singularity at the point p E H. It is now clear that the equality spec A = spec,A holds if and only if spec A does not disconnect the plane. It is not difficult to determine that algebra defined by an operator A for which the spectrum of the element A coincides with that of the operator A. This is the smallest closed subalgebra [[A]] of f?(E) which contains A and the family of operators (&IA, n, where n is a set obtained by choosing one point from each bounded connected component of the set reg A (the algebra [[A]] does not depend on the choice of set A).
5 4. Theory of Operators
233
Returning to an arbitrary algebra ‘$I, we note that for each closed ideal J c 2I the factor algebra ‘2X/Jis a Banach algebra under its usual norm. But all maximal ideals are closed since the closure of an ideal J # !!I is an ideal .i # ‘$I, as a consequence of the fact that the set of invertible elements is open. The factor algebra E = %/M for a maximal ideal M is therefore a Banach field over @. If E # @ then for any < E E\@ and for all 1 E @ the nonzero elements r  I are invertible in,E. But then spec 5 = 0. With this we have established the Gel’&&Mazur theorem that a complex commutative Banach algebra is spectral. The characters x of an algebra ‘+Iare continuous because of the closedness of their kernels, these being maximal ideals. Since x(a) E spec a, we have Ix(a)1 < p(a) < Ilull. Moreover x(e) = 1 and so llxll = 1. Now let us consider the set ‘%R= 9R(‘%)of maximal ideals of the algebra 2I. It can be regarded as a subset of the unit sphere of the conjugate space ??I*, in which case it is closed since it is determined by the system of equations &,a,) = &,)&) (al, a, E 2I), x(e) = 1, which are defined by continuous functions of x. Consequently, %I is compact. Example 1. If ‘?I = C(S) (S compact) then !Ul@I) = S. Example 2. If 2I is an algebra with a single topological generator a, i.e. ‘8 = [a], the uniform closure of the algebra of all polynomials in a, then !IR(%) = spec a. We note that the problem of giving an explicit description of the maximal ideal space of a concrete algebra can be extremely difficult. An example of this is the famous problem which was solved by Carleson (1962); this is concerned with the algebra of functions which are analytic and bounded on the unit disk A = {A: 1 E @, IL1 < l}. The result is that A is dense in 9X. The canonical functional representation (Gel and transformation) aH Z (ii(x) = x(u)) is a morphism118 of 9I + C( I(2I)) which is clearly contracting. For it to be an isometry (i.e. for the identity p(u) = [lull to hold), it is necessary and sufficient that the identity llu2 II = IjaIl should hold in %. The Gel and image of the algebra Iin C(9R) is a subalgebra (generally not closed) which separates the points of the space 9.X It can fail to be dense in C(!U2). In the given context and also for many applications it is important to have an intrinsic characterisation of dense subalgebras of C(S) (S compact). The effective solution of this problem includes the Weierstrass approximation theorem (the subalgebra of polynomials is dense in C[O, 11). It was obtained by Stone (1937). StoneWeierstrass Theorem. Suppose that a subalgebra 9I c C(S) is symmetric (i.e. along with each function cp it contains its complex conjugate @) and separates the points of S. Then !JI is dense in C(S). For the proof (and also independently of this goal) it is useful (following de Branges, 1959) to deal first of all with the real case. sIn the category of complex commutative Banach algebras over C (i.e. a continuous algebra homomorphism) YJI(‘%)is a contravariant functor into the category of compact spaces.
234
Chapter 2. Foundations and Methods
Real Variant of the StoneWeierstrass Theorem. Let C,(S) be the Banach algebra of real continuous functions on a compact set S and let %,, be a subalgebra. Zf the functions in 912,separate the points of S, then ‘%Ois dense in C,(S). Proof. Suppose that the subalgebra I&is not dense. Let us consider the set N # 0 of real measure on S which annihilate ll0and satisfy the condition 11v 11< 1. This is. a convex, centrally symmetric, compact subset of the space of measures on S under the w*topology. By the KrejnMil an theorem N has an extreme point cr. Clearly, /loll = 1 and by construction Js cp do = 0 (cp E ll,,),in particular Jsda = 0. Hence it follows that the support of the measure (r contains at least two distinct points sl, s2. But there is a function $ in the subalgebra !I& which separates these points: I,&~) # $(sz). Without loss of generality we can assume that 0 < $(s) < 1 (s E S). Since cp$ E !lQ, for all cp E !l&, we have Is cp$ do = 0, i.e. the measure d? = + da annihilates 9$, and so the measure dp = (1  +) da also has this property. Putting ? = r/llrll, 3 = p/IIpII we have Q = pz* + qfi (p, q > 0, p + q = 1). Consequently f = aa (a = constant), i.e. $lsuppa = constant. 0
For the proof of the complex StoneWeierstrass theorem it is now sufficient to consider the set ‘%c= (0: 0 = re cp (cpE ll)}. If cpE % we also have (p E % according to the condition and so re cpE Consequently, !RO c and by the same token !RO is a subalgebra of the real algebra C,(S) and separates the points of the compact set S. According to the real StoneWeierstrass theorem, ‘%eis dense in C,(S). But then % is dense in C(S). 0 A complex commutative Banach algebra canonical image is symmetric. Corollary.
If the algebra
is symmetric,
is said to be symmetric
if its
then its canonical image is dense in
wJw)). Remark. The Banach space C,(S) is also a vector lattice with respect to the usual ordering and moreover the two structures are compatible: if [lx/l < 1 and IyI < 1x1 (I.1 is the lattice modulus), then llyll < 1. We can say that C,(S) is a Banach lattice. The following variant of the StoneWeierstrass theorem holds: any sublattice of the vector lattice C,(S) which contains 1 and separates the points of the compact set S is dense in C,(S). Closed subalgebras of the algebras C(S) (S compact) are called uniform algebras. An example of such an algebra on the unit circle T is the algebra A(T) offunctions which are the boundary values of analytic functions on the unit disk A. It is called the disk algebra and it is maximal among the nontrivial closed subalgebras of C(T) (Wermer, 1953). The maximal ideal space of the algebra A(T) is naturally identified with the closed disk A = A u T. In general, if a uniform algebra 2I on S separates points, then S is homeomorphically embedded in the maximal ideal space %Jl(%). For any algebra I and any element a E 2I we can construct, just as in the theory of operators, a functional calculus by putting
5 4. Theory of Operators
235
cpw =&srp(l)(a r
le)’ dA,
where cp is an analytic function on the spectrum, i.e. on some neighbourhood G r> spec a, and the contour r contains spec a in its interior. Moreover cp@Nw = cp@W)) (M E WW), f rom which we obtain immediately the spectral mapping theorem:
spec q(a) = cp(spec a). Example. Let W be the Wiener algebra, let f E Wand let cpbe a function which is analytic on the set of values of the function J Then the composition cp of belongs to W (theorem of Levy; the case q(c) = [’ is a theorem of Wiener). An algebra ‘2I is said to be regular if for any compact set Q c 9X(%) and any maximal ideal MO # Q there exists an element a E % such that 51, = 0 and Z(M,) = 1. It now follows from this that for any two disjoint compact subsets Q1, Qz of 9JI there exists an element x E 2I such that ZIQ1 = 0 and )?Ipl = 1. If 2I is symmetric, x can be chosen so that 0 < Z(M) < 1 for all M. The theory of regular algebras was constructed by G.E. Shilov (1941). One of the fundamental results of this theory is the Local Theorem. Suppose that the algebra VI is regular and that the function cp belongs to 9l locally on ‘93,i.e. for each point M, E !JJIthere exists a neighbourhood U and an element a, E 2l such that ii,(M) = q(M) (M E U). Then cp E ‘&
The point is that, because of the regularity of the algebra, for any finite open covering of the compact set !JJI we can find a partition of unity as a sum of functions in a, each of which is concentrated on the corresponding element of the covering. Example. If on a neighbourhood of each point a periodic function cp can be expanded as an absolutely convergent trigonometric series (depending on the choice of point), then it has an expansion of this type on the whole period (Wiener, 1933). An important consequence of the local theorem is a further generalisation of the WienerLevy theorem: suppose that the algebra 9I is regular and that the function rp on !DI is such that for each point M,, E ‘2Xthere exists a neighbourhood U, an element a,, E 2X and a function . which is analytic on spec a0 and such that q(M) = cpo(iio(M)); then cp E ai. In certain cases this theorem allows us to apply multivalent analytic functions’ lg to elements of the algebra. Example. If a function satisfies the conditions
a E L R) is such that its Fourier
1  A(1) # 0 (n E R),
transform
A(1)
ind(1  A(I)) = 0,
1I9 An interesting variation on the topic of multivalent functions of an algebra element is the solution of algebraic equations with coefficients in the algebra. Nontrivial homomorphisms of the fundamental group nl(%R) into braid groups are global barriers to solvability (E.A. Gorin  V.Ya Lin, 1969).
236
Chapter
2. Foundations
and Methods
then the function ln( 1  A(1)) is the Fourier transform of some function in L’(R). The role of this result in the theory of equations with a difference kernel is wellknown (see Chapter 1, Section 4.5). By way of explanation we state that for the Banach algebra L’(R) (with identity adjoined) the Gel and transform and the Fourier transform coincide (1 = 1). The maximal ideal space of this algebra (or of the Wiener algebra which is isometrically, isomorphic to it) is naturally identified with the onepoint compactihcation Iw = IR u (co}. In any algebra Cuan element a is said to be logarithmic if it can be represented in the form a = eX, where x E . All such elements are clearly invertible but in general not all invertible elements are logarithmic. Example 1. In the algebra C[O, l] all functions that are never zero are logarithmic, i.e. the functions which are invertible in the algebra. Example 2. In the algebra C(T) the logarithmic nonvanishing functions cp for which
functions are precisely those
ind cp = & A,(arg cp) = 0. We note that any invertible
II/ E C(T) can be expressed in the form
where )2 = ind $ and cp is a logarithmic function. The set exp VI of logarithmic elements of any algebra !!I is a subgroup of the group inv I of invertible elements. It turns out that the corresponding factor group depends only on the homotopy class of the maximal ideal space of the algebra. ArensRoyden Theorem. The factor group inv (U/exp I is isomorphic to the group H W, Z) of onedimensional tech cohomologies of the maximal ideal space of the algebra 2I.
In the proof of the ArensRoyden theorem (as also in many other questions in the theory of Banach algebras) we make use of analytic functions of n algebra elements: a,, . . . , a,. In the corresponding functional calculus12’ the role of the spectrum is played by the combined spectrum spec a of the system a = (ai, . . . , a,), which is defined as the complement of the combined resolvent set. The latter consists of regular points (A,, . . . , A,) E C for each of which there exists a system b = (b,, . . . . b.) of algebra elements such that
k$l (ak A&
= e.
It is easy to see that spec a coincides with the image of the mapping lzoIt
was developed
in the works
of G.E. Shilov
(1953,
1960) and of Arens
and Calderbn
ii: W + C” (1955).
4 4. Theory
of Operators
231
which is defined by the formula a”(M) = (5,(M), . . . , c,(M)). Since this mapping is continuous, spec a is compact. If it is polynomially convex, then, by the wellknown OkaWeil theorem, each function which is analytic on spec a admits arbitrarily close uniform approximation by polynomials. This allows us to extend by continuity the intrinsic construction of the element p(a) = rp(a,, a,, . . . , a,) from the algebra of polynomials in n variables to the algebra of functions which are analytic on the spectrum. A deeper analysis is required in the absence of polynomial convexity for the combined spectrum. Returning to regular algebras, we note their significance for spectral theory. Example.
Let a = (a,),, z be a real sequence such that a, >, 1, a,+, G anam,
In a, C 7 < *. n.zn +1
Then the Banach algebra 1; (with convolution) of scalar sequences that are summable for the weight a is regular (G.E. Shilov, 1947). Its Gel and image, the Wiener algebra W, with weight a, consists of those functions on the circle with Fourier coefficients in 1, The corresponding maximal ideal space is identified with the unit circle T. If V is a nonquasianalytic linear operator on a Banach space (spec V c ), we can put a, = 11V” 11.The functional calculus q(V)=
z
II=al
a,V”
q(l) =
f
al
a&
111= 1
> arises on the algebra W,. The spectrum of the operator turns out to be separated thanks to the fact that, for any compact set Q c T and any neighbourhood U 3 Q, there exists a function rp E W, such that lo = 1, cp(T\ U) = 0,O < cp < 1 (clearly q is not only nonanalytic but it also does not belong to any quasianalytic class). We thus have convincing evidence that the methods of the theory of analytic functions are of paramount importance for the theory of Banach algebras (as indeed for spectral theory in general). Conversely, the theory of Banach algebras is a useful instrument for studying various classes of analytic functions, particularly in the setting of approximation theory. The concept of a boundary of the algebra plays an important role here. This term is applied to any subset X of the maximal ideal space W such that for each function u(M) (a E 2I) its maximum modulus is attained at some point A4, E X. The trivial boundary is X = W and, for example, in the case of the algebra C(S) (S compact) it is the only one. The boundaries for the algebra A(T) are precisely those subsets of the disk A which contain the circle T. Since in any algebra an extension of a boundary preserves the boundary property, it is natural to look for a smallest boundary. G.E. Shilov (1940) proved that any algebra has a smallest closed boundary (subsequently called the Shilou boundary). Without the requirement of closedness a smallest boundary may not exist, however such a boundary does exist if the maximal ideal space is metrizable and in this case it consists of the peak points, i.e. those MO E 9R (
238
Chapter 2. Foundations and Methods
for each of which there exists a, E ‘8 such that ii, = 1 and I&(M)] < 1 (M # M,). The results were obtained by Bishop (1959), who also gave the following application of them. Theorem. A sufficient condition for the set R,(S) of rational functions with poles outside of the compact set S c C to be dense in C(S) is that (with respect to plane Lebesgue measure) almost every point of S be a peak point for some function in R(S) = ,,(S).
A.A. Gonchar (1963) showed that the following condition is suflicient for a point s0 of a compact metric space to be a peak point of a uniform algebra 9I c C(S): there exist constants c < 1, N > 1 such that for any neighbourhood U of s0 there is a function cpE % for which cp(s,) = 1, Iv(s)] < c (s E U) and llqll < N. It follows from this that the set of peak points of the algebra ?I is of type G. Let % be a complex Banach algebra (in general noncommutative). A mapping a H a* of the algebra into itself is called an involution if 1) (a + b)* = a* + b*, 2) (au)* = Za*, 3) (ab)* = b*a*, 4) a** = a, 5) Ila*II = Ilull. The element a* is called the adjoint of a. If a* = a, then a is said to be selfadjoint. If a* commutes with a, we say that a is normal. In particular, all unitary elements (a*a = aa* = e) are normal. Example. The identity e is a selfadjoint unitary element. If the element a is invertible, then a* is also invertible and (a*)’ = (a *. Hence it follows that reg a* = i%gZ and spec a* = spec. 1 Any x E rU can be uniquely represented in the form x = a + bi, where a, b are selfadjoint :a = re x = (x + x*)/2, b = im x = (x  x*)/(2i). The selfadjoint elements of 2I form a closed real subspace re 2I. For any a E ‘$I the element a*a is selfadjoint. The closure of the nonnegative linear hull of the set {x: x = a*a (a E ‘$I)} is a wedge %Xx,whose elements, as usual, are said to be nonnegative (x > 0). Using ‘$I+ we define in the canonical way the collection 9Iz*, of nonnegative linear functionals. Any f E !!I*, satisfies the inequality If(a)1 < f(e) /la/l and is consequently continuous ( 11f II = f(e)). Moreover, it is real on re ‘%or equivalently: f (a*) = f(a) (a E 2l). Such functionals are said to be symmetric. Any functional g E (re ‘$I)* can be uniquely extended to a symmetric functional g E %I*; also II 811= II 911. In the category of Banach algebras with involution the continuous homomorphisms h: VIZ,+ a2 which are compatible with the involutions, i.e. (ha)* = ha* (or, equivalently, h preserves selfadjointness of elements), are called morphisms ((*)homomorphisms). A subalgebra of an algebra with involution is said to be symmetric if the involution does not leave it. The class of symmetric ideals (one or twosided) is defined similarly. Under factorisation by a symmetric twosided (onesided) ideal the involution carries over naturally to the factor algebra (factor space). Here the bar denotes complex conjugation.
5 4. Theory of Operators
A Banach algebra ‘?I with involution
239
is called a C*algebra if
lb*4 = 11412 for all a E 2I. We mention
Example 1. The algebra C(S) of continuous
involution
(47)
two examples of C*algebras. functions on a compact set S (the
is complex conjugation).
Example 2. The algebra f?(E) of bounded linear operators on a Hilbert E (the involution is the operator adjoint). In fact, if A E g(E) we have
IIA*AII = ,vl x
(A*&
4 = ,yl x
space
lbw2 = llAl12.
Any closed symmetric subalgebra of a C*algebra clearly belongs to the same category. Any closed twosided ideal of a C*algebra is symmetric and the corresponding factor algebra is a C*algebra. Gel andNajmark Theorem. 1) Let be a commutative C*algebra and !lJI its maximal ideal space. Then the Gel$znd representation is an isometric (a)isomorphism. 2) Any Palgebra is isometrically (*)isomorphic to some operator algebra, i.e. a closed symmetric subalgebra of the algebra of operators !i!(E) for some Hilbert space E.
For the proof we note that for any C*algebra the spectra of unitary elements lie on the unit circle, since if a ’ = a* it follows from (47) that lluV1 II = Ilull = 1, hence p(a) < 1 and ~(a < 1; the spectra of selfadjoint elements are rea1 2, since if a* = a we have that eia is unitary. Consequently, if 3is commutative and x is a character, then x is real on re 2I and hence it is symmetric. By the StoneWeierstrass theorem is dense in C(%R). It remains to show that p(a) = Ilull for all a E W. If a = a* this is obtained by iterations of the identity lla211 = llal12, while in the general case we have llal12 = llu*al/ = &*a) < da*)&) = P2M. cl The noncommutative situation is significantly more complicated not enter into it. Let us consider two applications of the preceding theorem.
and we will
Example 1. Let S be a completely regular topological space. We will consider the commutative C*algebra U(S) on it. The canonical mapping of S + .Xis a homeomorphism onto its image since CB(S) separates points from closed subsets. We can therefore regard S as a topological subspace of %R.But then S is dense in W since, if cpE C(roZ) and & = 0, we have cp = 0 because of the isometric property of the Gel and representation (which in this context becomes the I** We note in passing that the spectra of nonnegative elements are nonnegative.
240
Chapter 2. Foundations and Methods
extension by continuity). Consequently ‘9JIcan be regarded as a compactification of S. If coincides with the StoneTech compactification. All possible compactifications are in onetoone correspondence with the closed symmetric subalgebras ‘3 c CB(S) which separate points. However the topology of the compactilication 9Jl(‘%)induces on S a topology which is coarser than its initial topology and may not coincide with it. For example, the Bohr compactification of the space R, which is defined by the algebra AP(R) of almost periodic functions, has this property. Example 2. Let A be a bounded normal operator on a Hilbert space. Then A and A* generate a commutative C*algebra (this is the smallest closed subalgebra of i?(E) which contains A and A*). It can be shown that the spectrum of the operator A coincides with its spectrum as an element of this algebra ‘$I and can be identified with its maximal ideal space 93. The Gel and representation of , C(YJI) is an isometric (*)isomorphism. The inverse mapping can be regarded as a functional calculus. It can be extended (by continuity in the topology of pointwise convergence) to the algebra of bounded Baire functions cp(l) (A E spec A). The orthopojections forming an orthogonal resolution of the identity for the operator A correspond to the characteristic functions of the Baire subsets of the spectrum. Thus in this way we obtain the spectral theorem for a bounded normal operator. Banach algebras with involution play a significant role in the theory of representations and harmonic analysis. If G is a locally compact group and p is a right Haar measure, the convolution operation
(cp *bwd =scpb+knJMY) (9 EG) G
turns L G, ,u) into a Banach algebra (without identity if G is not discrete; we then have to adjoin an identity formally. This group algebra is commutative if and only if the group G is commutative and in this case its characters can be identified with the unitary characters of the group 3 by means of the Fourier transform
4(x) =sddx(d 44s) (x EG*) G
(if G is not discrete, the group algebra has another character  the coefftcient of the identity). The maximal ideal space of the group algebra coincides with G* if G* is compact (0 G is discrete), while in the general case it is the onepoint compactification of G*. The Gel and representation for the group algebra L G, ,u) coincides with the Fourier transform, i.e. for G = [wwe have the classical Fourier transform and for G = T it associates with each function its sequence of Fourier coefficients. We have correspondingly T* = Z and R* = R. The group
lz3These form the dual group G* (pointwise multiplication). compactopen topology, under which G* is locally compact.
The usual topology on G* is the
Q4. Theory
of Operators
241
algebra of a locally compact abelian group is semisimple, which is equivalent to the uniqueness theorem for the Fourier transform. An involution can be introduced on a group algebra by means of the formula cp*(g) = cp(g‘). Although this is not a C*algebra, the presence of an involution on it nevertheless turns out to be extremely useful, for example, in such a central question as the generalisation of Plancherel theorem 4 (A. Weil, 1940, M.G. Krejn, 1941). In this way D.A. Rajkov (1941) obtained a new proof of the Pontryagin duality principle, according to which for any locally compact abelian group the canonical mapping of G + G** is a topological isomorphism 5. The principle of duality contains, in particular, the assertion that the unitary characters of a locally compact abelian group separate its points (in this sense there are sufficiently many of them). This result extends to the noncommutative situation if we replace characters by irreducible unitary representations (GellfandRajkov theorem). Moreover, any unitary representation of a locally compact group can be expressed as the direct integral of irreducible unitary representations (M.A. Najmark, 1956). This theorem bears the same relationship to the previous one as Choquet theorem does to the KrejnMil an theorem and the technique of extreme points is actually used in the corresponding proofs. For a compact group the expression reduces to a discrete orthogonal sum (with a suitable modification this is also true for representations in a Banach space). For the regular representation of a compact group the decomposition into an orthogonal sum of irreducibles is a basic result from the PeterWeyl theory. We note that ail irreducible representations of a compact group (even in a locally convex space) are linitedimensional and can therefore be taken to be unitary. Compact groups (even nonabelian) have their own particular principle of duality (Tannaka, 1939; M.G. Krejn, 1949); the object which is dual to the group is the socalled blockalgebra. The study of any Banach algebra 2I with involution reduces to the case of a C*algebra in the sense that there exists a C*algebra @I, unique up to (*)isomorphism, which is associated with YI by means of a morphism p: + @ in such a way that, for any morphism h: 2I + 2 (B a C*algebra), we have the commutative diagram
i2
suitably
ormal&l’
and correspondingly also a generalisation
Haar
measure
p* on G* appears
in the inuersion
formula
in the generalised Plancherel theorem. In this context it is appropriate of Bochner theorem on the representation of positive functions:
to note
CPM = x(g)WA where u is a finite measure (a > 0). The original proof (L.S. Pontryagin, 1934) was based on structure theory. The requirement contained in it of a countable base was subsequently removed (van Kampen, 1935).
242
Chapter 2. Foundations and Methods h 9Xi?
where ,Ais some morphism. The C*algebra !8 is said to be universally enveloping for 9l. This is an extremely useful construction. For example, it works effectively in the proof of the Gel andRajkov theorem (where 9I is a group algebra). In the 195Os, Yu. M. Berezanskij and S.G. Krejn extended the scope of harmonic analysis, defining the generalised convolution on L S, m) for any locally compact space S with measure m:
where for fixed t C(s, r; t) is a Bore1 measure with respect to s and r separately and for fixed subsets in place of s and r it is a continuous function of t with compact support; it is required in addition that
s s
C(A, B; t) dm(t) < m(A)m(B).
We then have IIv+M < lldlll~l1 an d so a commutative structure
Banach algebra with
meusure126 C(A, B; t) arises on L S, m). It is assumed further that an
involutory homomorphism s H s* acts on S, preserving the measure m and acting on the structure measure so that an involution is defined on the Banach algebra under consideration by the formula q*(t) = @(t*). From these standpoints, for example, ordinary (group) and generalised (hypergroup) shifts can be given a unified treatment.
fj 5. Function
Spaces
5.1. Introductory Examples. The development of functional analysis is connected not only with the working out of its bstract’ intrinsic structures but also with the expansion of the stock of concrete function spaces. This has both an applied and a conceptual meaning. The revolutionary introduction into mathematical usage of the language and techniques of generalised functions can serve as a striking example (S.L. Sobolev, 1936; L. Schwartz, 195051). The general theory of interpolation of linear operators, which came into being right at the
lz6This is the continuous analogue of the structure constants of finitedimensional algebra. It is assumed that relationships hold which guarantee associativity and commutativity. An identity may be lacking but it can be adjointed formally in the usual way.
0 5. Function Spaces
243
beginning of the 1960s (Gagliardo, Lions, Calderon, S.G. Krejn), developed on the one hand from concrete material relating to spaces of type Lp, while on the other hand it opened up a broad field of activity in new function spaces (for example, in symmetric spaces of measurable functions). Interpolation methods are intrinsically linked with embedding theorems of S.L. Sobolev, which are concerned with spaces of differentiable functions of several variables. In connection with multivariate approximation theory important advances were made in the theory of embeddings during the 1950s thanks to work of S.M. Nikol kij and O.V. Besov, in which new classes of differentiable functions were introduced, leading, in particular, to valuable converses of embedding theorems. At the present time we can say that function spaces form an independent topic, in the investigation of which abstract and concrete analytical methods are intertwined. The successful solution of many problems in the theory of partial differential equations depends on the right choice of function space (or family of spaces) appropriate for the given problem. Example 1. The Cauchy problem
au& t) =
at
a2u(x, t)
ax2
(t
for the heat equation
2 0, co < x < co),
4% 0) = 44x).
is initially set in the class of all sufficiently smooth functions but in this situation the solution is not unique. In fact in the nonquasianalytic class C(n ) (0 < E < 1) on the interval [0, l] there exists a function t+b(t)which together with all its derivatives vanishes at the end points. Let us put $(t) = 0 for t > 1. Then cm i) ‘(t) 4% t) = x0 (2n)IxZ”
(t 2 0, co < x < co)
is a solution of the heat equation and U[~=~ = 0. The reason for this phenomenon lies in the possibility of rapid growth of u(x, t) as [xl+ co. Taklind (1937) showed that if h(x) is a nondecreasing odd function then the bound lu(x, t)l < A4exh@) defines a uniqueness class if and only if * dx = s 1 h(x)
al.
(48)
This result is closely connected with a criterion for quasianalyticity. In particular, condition (48) is satisfied by the function h(x) = x, which defines the wellknown uniqueness class lu(x, t)I < MeX2 identified by A.N. Tikhonov (1935). Example 2. In the Dirichlet problem for Laplace
Au = 0,
equation
4r = cp
in a domain G c R” with (sufficiently wellbehaved) boundary I a key role is played by the following quadratic functional (known as the Dirichlet integral):
244
Chapter
2. Foundations
D[u] =
sG
(dx is Lebesgue measure). The Hilbert
and Methods
jVuj2 dx
space of functions W:(G)
lbll = JtDbl
+
with norm
lbll&(G))
(introduced by S.L. Sobolev) is intrinsically connected with this problem. This norm requires the existence of partial derivatives if only almost everywhere, however this is not enough. An adequate description of the space is obtained by the introduction of generalised derivatives (see below). Since Au = 0 is an EulerLagrange equation for the functional D[u], it is appropriate to change the formulation of the Dirichlet problem by requiring that D[u] = min, ulr = cp, u E W:(G). The question of second order smoothness is thereby separated from the questions of existence and uniqueness (after which it can be solved by methods specially adapted for it). However the variational formulation also contains a subtle aspect which concerns the boundary condition: an admissible function cp must be the limit of functions in W:(G) in a certain sense (requiring more precise definition) and it is desirable to have an intrinsic characterisation of the class of these functions. The answer to this question requires the introduction of the new function space W:12(IJ of functions which are smooth of order 3 in a certain integral sense. It turns out that the restriction homomorphism u H ulr acting from W,‘(G) into Wi’2(r) is continuous and surjective’ (Aronszajn, 1955; V.M. Babich  L.N. Slobodetskij, 1956). 5.2. General&d Functions. The simplest generalised function (although not a function in the usual sense) is the bfunction which is in fact a continuous linear functional on the asic’ space of continuous functions C(W) (for the time being we restrict ourselves for simplicity to the onedimensional case):
WI
= f(O),
and formally m G(x)f(x)
dx = f(0).
s a, By formally integrating by parts we can extract from this equality an heuristic definition of the derivative 6’: m sl(x)f(x)
dx = 
m G(x)f’(x)
dx = f’(O).
s 00 s 03 Thus we can define a linear functional s f] = f’(O), however C’(W) will be the natural basic space for this (and 6’turns out to be continuous). For the simultaneous definition of all derivatives 8(“)[f] = ( l) ( (O)(n > 1) we have to take ’ The boundary sense.
function
cp = I& is defined
almost
everywhere
and is the limit
of u in a certain
integral
8 5. Function Spaces
245
Cm(R) as the basic space. On the other hand, however, the bfunction is an elementary measure, namely the unit measure concentrated at zero.2 If we wish to regard all measures as generalised functions, then, because a measure can grow arbitrarily at infinity, we have to take for the basic space C,(R), the space of continuous functions with compact support. C~(R), the space of infinitely differentiable functions with compact support, is the universal receptacle of all the formulated requirements. It is clear from what has been said that a generalised function has to be defined not as an individual object but as an element of some class. Let G be an open set in R” and E some linear topological space of functions f: G + @which we will call the basic functions. The continuous linear functionals on E, i.e. the elements of the conjugate space E* (with its w*topology), are called generalised functions over E. The first and most important example is E = C:(G) = 9(G) (to say that f: G + c has compact support means that there is a compact set Q/ c G outside of which f(x) = 0; the smallest such compact set, supp f, is the support of the function f). We introduce the locally convex (nonmetrizable) DieudonndSchwartz topology on 9(G) by means of the family of seminorms ak,++k,
p,(f)
=
k ,,...,
2
k,=O
suP xsG
Pkl,...,k
n a.+.
f . . ax
’
where p = (P~,,,.,,~,) is an arbitrary nmultisequence of continuous functions on G which is locally finite in the sense that for each compact set Q c G the set of multiindices3 k = (k,, . . . , k,) for which pkla # 0 is finite. With 3(G) as the choice of basic space the generalised functions are called distributions. Thus 1 is a distribution if it is a linear functional on .9(G) which is subordinate to a finite family of seminorms pPcl), . . . , pPcl,. An effective criterion for continuity of a functional 1 E 9(G)’ is that for each compact set Q c G there exists an integer m = m(Q) (we shall assume it to be as small as possible) and C = C(Q) > 0 such that (49)
for all f E 9(G) with supp f c Q. If sup0 m(Q) = I < cc, we say that I is a distribution of finite order r. A distribution of order zero is a complex measure. We can also regard a locally summable function 0 as a distribution by identifying it with the measure J 8 dx. Such distributions are said to be regular and form a t is convenient to interpret the &function physically as a unit charge concentrated at zero. Then 6’ is a dipole concentrated at zero with unit momentum. However in quantum mechanics the concept of the Sfunction is essentially as a functional. he use of multiindices in work with functions of n variables is very convenient since it leads at,++k. f aff It is also convenient to put Ikl = to abbreviations of the form: x& = x:1 . . . x2; ax)r =
ax:l... a+'
k, + ... + k,.
246
Chapter 2. Foundations and Methods
dense subspace L:,,(G) c g(G)*. In general it is far from being the case that we can speak of the values of a distribution 1 at points x E G (although the conventional notation A(x) is widely used because of its formal convenience4). If an open set H c G is given and A[f] = 0 for all f E g(H), we set5 &, = 0. The complement in G of the union of all such H is called the support of the distribution I and is denoted by supp A. The distributions with compact support form a space which is the conjugate of Cm(G). The topology on C”(G) is defined by the collection of seminorms appearing in (49), i.e. it is the topology of uniform convergence along with all derivatives on each compact set. Clearly, Cm(G) is a locally convex Frechet space. Remark 1. Without going into the general definitions we note that 5@(G)is the inductive limit of the family C*(X), where X runs through all closed balls
contained in G. The topology on g(G) is stronger than that induced by Cm(G). Remark 2. If 1, ,u E $3(G)*, H c G is an open set and (A  p)IH = 0, then we say that the distributions A, p coincide on H. If 8 E L,QH) and 21, = 8, then the values A(x) are defined at all Lebesgue points of the function 8. Selecting the cone of (pointwise) nonnegative functions in g(G) we obtain the cone of nonnegative distributions in g(G)*. It turns out that it coincides with the cone of measures. Multiplication by a function cpE Cm(G) is a continuous linear operator6 on g(G). Its conjugate operator on g(G)* is also called an operator of multiplication by ~1 WI U1 = ACdl. F or example, (~6 = (p(O)& This operation does not allow extension to multiplication in g(G), which would convert g(G)* into a commutative algebra.’ Thus g(G)* is only a module over the algebra P(G). The differentiations Dj = a/ax, (1 < i < n) are also continuous linear operators on g(G) and consequently the conjugate operators D,+ act on $2(G)*, however, as already explained by the example of the bfunction, it follows that we should put an/ax, =  DFA for any distribution A..
4For example, in this form it is convenient to make the change of variables x = Fy (F: G + G is a diffeomorphism): qx)f(F ) WY)f(Y) dY = s0 sG Consequently, it is necessary to take as definition
dx det F Flx)’
sThe natural embedding of g(H) into g(G) (extension by zero) is a continuous homomorphism whose conjugate is the homomorphism which restricts a functional to s(H). 6Continuous operators of multiplication on function spaces by a function (possibly not belonging to the given space) are called multiplicators. his circumstance can be regarded as a formal cause of the appearance of divergence in quantum field theory and the removal of divergence (i.e. normalisation of the theory) as the regularisation of multiplication of generahsed functions (N.N. Bogolyubov  O.S. Parasyuk, 1955).
247
0 5. Function Spaces
All linear differential .9(G)*:
operators with coefficients in Cm(G) are now defined on
They are all continuous under the w*topology, which is taken as the standard topology on Q(G)*. As an example to illustrate the flexibility and convenience of the language of distributions, let us mention the general definition of one of the principal concepts of the theory of partial differential equations. For any linear differential operator L with coefficients in P(G) a distribution r,, which depends on the parameter y E G and satisfies the equations is called a fundamental solution. For example, the Laplace operator in R” has as a fundamental solution when n > 2 the function9 1 r,(x) = ux9 Y) = @ _ 2)a”,x _ y,n2v where O, is the area of the unit sphere of R” with the canonical Euclidean norm
IxI=J,$x;. A fundamental solution r, exists for any operator L with constant coefficients (Ehrenpreis, 1954; Malgrange, 1955) and clearly it has the formlo T,(x) = l(x  y) (x, y E UP). An immediate application of this fundamental solution is the construction of solutions to the equation” L [A] = p by means of the formula Iz = r * p. Here we are using convolution of distributions. For its introduction it is necessary to consider first the convolution of the distribution p with an arbitrary function f E S+!“), i.e. the function
(cl *f)(x) = Am
 .)I
This function belongs to Cm(W) but if the distribution
p has compact support
* 6, is the unit measure concentrated at the point y. In R” we can write 6,(x) = 6(x  y). This notation carries over to any domain G. 9If n = 2 we have l(x, y) = k lnlxl. If n = 3 we can interpret l(x, y) physically as the potential of a point charge. lo In a more complete formulation of the theorem it is asserted that I is a distribution of finite order. We note that in theorems of this type not only the formulations are important but also the constructions which allow us to establish further useful properties of the solutions. The generalised Fourier transform is usually applied in the construction of fundamental solutions of equations with constant coeffkients (see below). ii In the case where L is the Laplace operator this is Poisson equation describing the potential of a charge with ensity’ p.
248
Chapter 2. Foundations and Methods
then p *f E ~(I?‘) and, by definition, we can put (I * ,u) [f] = I * (p * f). Thus the equation L[A] = p is successfully solved (in the class of all distributions) for any distribution p with compact support. In particular, p can be a summable function with compact support but the solution 1 nevertheless turns out to be only a distribution in general (generalised solution). However, if the operator L is elliptic12 (and only in this case), all generalised solutions will be real analytic functions whenever the right hand side of the equation is a real analytic function (I.G. Petrovskij, 1937; Hormander, 1955). An analogous result relating to infinite differentiability holds for the socalled hypoelliptic operators13 (and only in this case) (Hormander, 1955). Generalised solutions are entirely natural from the physical point of view since they satisfy the equation in an integral sense: the equality L[A] = p means that 1[Laf] = ~[f] for all f E g(G), i.e. in the usual notation (L,* = L). W(Loff)(x) dx = PL(Mx) dx sG sG This form of the equation can have even greater physical meaning than the traditional one. The onjugate operator principle’ has already been used repeatedly to transfer the standard operations of analysis to generalised functions: if a continuous linear operator A is given on the basic space, then A* is its transfer to the space of generalised functions. In a more general situation we can be given a continuous homomorphism h: E + E, of the basic space into another function space. Since h * p E E* (,u E E:), it follows that h * p is a generalised function (on E). We can regard this as the transfer of the homomorphism h to the generalised functions over the basic space El. The most important example of this type is the generalised Fourier transform.
Suppose that the basic space E consists of functions on R” and E c L’(W). Then the ordinary Fourier transform14 f(c) =
f(x)ei(c*x’ s o4”
dx
is defined on E. It maps E bijectively to a certain function space ,? (contained in the space of continuous functions which tend to zero at infinity). Let us provide E with the topology carried over from E. As a result the Fourier transform can be transferred to the generalised functions on the basic space E. For example, the Fourier transform of any distribution on R” is a generalised function over the basic space of entire (after analytic continuation with respect to x) functions of lZAn operator L is said to be elliptic if its principal
part
at is defined by a homogeneous ax’
‘&,,, a,
polynomial P(r,, . . . , rN) (symbol) with no real roots < # 0. i3 All elliptic operators are hypoelliptic. Infinite differentiability of solutions of elliptic equations with infinitely differentiable right hand sides was established by L. Schwartz (1951). r4(& x) is the canonical scalar product on R”.
0 5. Function
Spaces
249
exponential type that decrease more rapidly than any power of 1x1.In particular we can now speak of the Fourier transform of any locally summable function 0(x) without imposing any restrictions as 1x1+ co. It is clear that the generalised Fourier transform has the usual perational properties: differentiation Dj passes over to multiplication by irj, therefore any linear differential operator with constant coefficients passes over to the operator of multiplication by the corresponding polynomial; convolution becomes a product and so on; hence differential, convolution and some other equations are as usual transformed into algebraic equations. For example, the equation L[n] = p takes the form P(t)n = ,& and the problem reduces to determining the possibility of dividing a generalised function by a polynomial (division problem). The general approach to the Fourier transform was developed gradually (Bochner, 1932; L. Schwartz, 1951) and took its final form, which has been described above, in the work of I.M. Gel and and G.E. Shilov (1953), where they obtained with the help of the Fourier transform uniqueness classes for the Cauchy problem for systems of differential equations of the form
2 = ktl Ljk(t)CUkl
t1 Gj G m)9
(50)
the Ljk(t) being linear differential operators in R” with coefficients which depend continuously on t (but are constant with respect to x). If p is the least order of the operators Ljk, then the uniqueness class is defined by the bound max luj(x, t)I < Mealx14, j where15 q = p/(p  1) and 0, M are positive constants. In this connection the consideration of generalised functions over suitable spaces of entire functions turned out to be essential, since it allowed the possibility of establishing uniqueness by using the PhragmenLindelof principle. The application of this sort of analytic technique requires an intrinsic characterisation of the space16 E* for those spaces E being used as basic spaces. We stress the lack of sensitivity of the method described to the type of system (for parabolic (according to Petrovskij) equations the uniqueness class was established by O.A. Ladyzhenskaya (1950) and for parabolic systems by SD. Ehjdel an (1953); in this situation G.H. Zolotarev (1958) obtained a criterion of Taklind type). The general result is well illustrated by the following example: for the equation
a~ a% z=a= ISIn fact we can take q = ~ PO
PII1
where
PO c P is the socalled
(51)
reduced order. There is an effective
formula for its determination, from which we can see that p,, is a rational number tor c m; in particular, p,, = p if m = 1 (V.M. Borok, 1957). leThis is a certain (not very great) cost on account of the restrictions at infinity.
with
denomina
250
Chapter 2. Foundations and Methods
with complex a the uniqueness class in the Cauchy problem bound Iu(x, t)l < MeuX2.
is defined by the (52)
For a > 0 this is the heat equation, for a < 0 it represents everse’heat flow and for a = i we have the simplest case of a nonstationary Schriidinger equation. We note that for a < 0 the Cauchy problem for equation (51) is incorrect in the sense that we do not have continuous dependence of the solution (in the uniform metric on an arbitrarily small time interval) on the initial function.” The technique of generalised Fourier transform allows us to identify correctness classes in the Cauchy problem for the system (50) (G.E. Shilov, 1955). Certainly they are, in general, no longer the uniqueness classes, although sometimes the two coincide. For example, the Cauchy problem for the heat equation in the class (52) is correct. Now let the basic space E be an extension of the space 9 = g(R with a weakened topology and also let 9 be dense in E for the weak topology. Then all continuous linear functionals on E remain such when restricted to 9 and are uniquely defined by their restrictions. Thus generalised functions on E can be regarded in the given case as distributions. The most remarkable example of this type is the Schwartz space S of functions of class Co3(R which decrease together with all their derivatives more rapidly than any power 1x1’ (v = 0, 1,2, . . .). This is a locally convex Frechet space defined by the family of seminorms
All linear differential operators with polynomial coefficients act continuously on S. The Fourier transform is a topological automorphism of the space S. Generalised functions on S are called tempered distributions. All linear differential operators with polynomial coefficients act continuously on this class (i.e. on S*). The Fourier transform on S* is a topological automorphism. Any linear differential equation with constant coefficients has a tempered fundamental solution (Hbrmander, 1958; Lojasiewicz, 1959). This is a nontrivial refinement of the EhrenpreisMalgrange theorem which was formulated above; it is connected with lower bounds for polynomials in several variables, which allow the division problem to be solved successfully in the given situation. If the basic space E is a contraction of 9 with a stronger topology and also E is dense in ~3, then the generalised functions over E will not now be distributions in general (but all distributions will be generalised functions over E). Natural examples of this type of basic space are the nonquasianalytic classes of infinitely differentiable functions (generalised functions over such spaces are given the name ultradistributions). However quasianalytic and especially analytic classes are of no lesser interest in the scheme being discussed. We call continuous linear he physical significance of this fact is that the process of heat conduction is not reversible in time.
$5. Function
Spaces
251
functionals on such spaces analytic function& As a specific case let us consider the space A(@“) of entire functions of n complex variables under the topology of uniform convergence on compact sets, In order that a linear functional u on this space be continuous, it is necessary and sufficient that there exist a compact set Q c @” and a constant C such that
IdYll G C yEa; IA41
(f E 3).
We can say that the functional a is concentrated ml8 Q but this term is also used in the case where for any neighbourhood I/ 3 K there exists a constant C, such that
bCfll G G Example.
SUP
If(
Let (at) be any multiindexed
(f E 4@ ).
(53)
scalar sequence satisfying
the
condition lim Ikl
;l/lael= 0.
Then”
is an analytic functional concentrated at 0 and (54) gives the general form of such functionals. Let us denote by ,4*(Q) the set of analytic functionals which are concentrated on the compact set Q c @ Restricting ourselves now to real compact sets2’ (i.e. such that Q c 08”c Q=”where the last embedding is the canonical one), let us consider the linear space NV
= ,lJ A*({ x: x E R llxll < r}).
Martineau (1961) proved 21 that any a E A lR has a support, i.e. a smallest compact set supp c1on which c1is concentrated. This opened up a relatively simple path to the theory of socalled hyperfunctions on [w”which was created by Sato (1960). Hyperfunctions with compact supports are just the same as functionals from A R . If G is a bounded open set in Iw the space of hyperfunctions on G is by definition the factor space A*(G)/A*(aG). The passage to hyperfunctions on the whole of R” is effected by means of a covering by bounded neighbourhoods. Thus, roughly speaking, a hyperfunction is a ocally analytic’ functional. On the whole,
Thus all continuous linear functionals on A(a) have compact support. I9 Here we use the abbreviation k! = k,! . . . k,! 2o But for such compact sets it is still necessary in (53) to consider complex neighbourhoods. 21 In the proof Martineau uses cohomology theory. An lementary’ proof was given by HGrmander (1983).
252
Chapter 2. Foundations and Methods
this construction is such that hyperfunctions are not generalised functions in the sense used earlier. However in many respects we can work with them just as with generalised (and so also as with classical) functions; in particular we can consider differential equations in hyperfunctions. It is interesting that hyperfunctions were intended first of all to play the role of limiting values of analytic functions in situations where these do not exist in any usual sense. Because of this the setting of hyperfunctions allows us to obtain, for example, adequate generalisations of Bogolyubou theorem n the edge of the wedge’ (Martineau, 1968; Morimoto, 1979; Zharinov, 1980). We note that the original version of this theorem was obtained by N.N. Bogolyubov (1956) in connection with the construction of dispersal relations in quantum field theory. The theorem asserts the local analytic continuability of one into the other for a pair of functions f+, f which are analytic on the domains G + iK (G is a domain and K is a cone in R under the condition that their limiting values on G coincide. The extended function turns out to be analytic on some complex neighbourhood V 2 G (we even have V 1 I, where I is the Cconvex hull of the domain G (VS. Vladimirov, 1960)). 5.3. Families of Function Spaces. Although the list of families presented below contains many of the wellknown function spaces it is certainly by no means complete and illustrates rather the variety of possible constructions. We note that the nature of a function space is determined to a considerable extent by the structure which is given on the domain of definition X of those functions from which the space is composed. For example, if X is a space with a measure, it is natural to consider measurable functions on it; on a topological space X we consider continuous functions (if we also look at discontinuous functions, they should at least be Borel” functions); a metric on X allows us to introduce the modulus of continuity and to define, for example, Lipschitz classes; on a smooth (analytic) manifold we can consider smooth (analytic) functions and so on. I. Let M(u) (U > 0) be a convex function such that M(u) > 0 (U # 0) and
lim u (u)
= 0,
U+0
Then we say that this is an Nfunction.
lim u (u) “+a3
= co.
Its Legendre transform
M*(u) = sup (uu  M(u))
(u 2 0)
II>0
is also an Nfunctionz3 and it is said to be dual to M (it is easy to see that iif** = M). Let X be a space with a ofinite measure p. We consider the set LM(X) of measurable functions u(x)@ E X) which satisfy the condition M*(u) =
sX
M*(l44l)
dp < 00.
.e. such that the inverse images of Bore1 sets are of the same type. 23The nonnegativity of M*(u) is the classical Young inequality.
5 5. Function Spaces
253
We denote by L;$(X) the set of functions u(x) such that uu E L’(X, p) for all u E J+(X). This linear space has a norm Ilull =
SUP M*(u)<1
IS X
uu d,u I
(it can be shown that the supremum is finite) under which it turns out to be a Banach space; it is called the Orlicz space with norming function M. If M(u) = %( p > l), then M*(u) = 4 p + 4 = 1 and the Orlicz T l > space L;I, coincides with Lp (to within a multiple of the norm). Now let us assume that M(u) is of slow growth in the sense that Example.
M&4 fp M(u)
The general form of a continuous
< O
linear functional
on L$ is then
where v E L,,. By the same token L,,. turns out to be topologically isomorphic to the conjugate space of L$. II. We associate with each measurable function u(x)(x E X) its indicatrix24
m,(t) = I+: le4l > t>
0 2 0).
This is a nonincreasing function which is continuous on the right. Let $(r) (r > 0) be a concave, nondecreasing function such that e(O) = 0, Il/(co) = co. The condition
II4 = oa+h(t)) s
dt < 00
defines the Banach space A,, which is called a Lorentz space. The conjugate of space M,, which consists of the measurable functions u(x) (x E X) for which A, is the Marcinkiewicz
1
ll4l = sup ~ II/(&a
I4 dp < ~0
sA (the general form of a continuous linear functional on ,4 ti is standard). In the space of measurable functions which are described above it is possible to construct a rich theory of nonlinear integral equations, by choosing the space in accordance with the nature of the nonlinearity (power, exponential, etc.). This 24 In the probabilistic situation this is a variant of the distribution function of a random variable. Just as we speak of identically distributed random variables in probability theory, here we can speak of equimeasurable functions (i.e. functions whose indicatrices coincide  this is also an important concept).
254
Chapter 2. Foundations and Methods
programme was realised in the 1950s and 1960s by M.A. Krasnosel kij in collaboration with several coauthors. III. Let X be a metric space with metric d and let U(X) (x E X) be a uniformly continuous function. The quantity
is called its modulus of continuity. Clearly, w,(6) is a nondecreasing, nonnegative function with o(0,) = 0. Each nondecreasing positive function Q(6) (6 > 0, 52(0+) = 0) defines a space of Lipschitz type 25 8, consisting of the continuous functions u(x) (x E X) such that Ilull = lu(xJ
w@) <
+ sup _ a>0 Q(6)
00
(x0 E X is a fixed point). This family of Banach spaces serves first and foremost the needs of approximation theory. IV. Among the spaces of smooth functions let us mention first of all the Soboleu spaces which are defined in their general form by the following procedure. Let G be an open set in R let 1 < p < cc and let m be an integer 2 1. The space WPm(G) consists of those functions whose generalised partial derivatives up to order m inclusive are functions of clasP U(G). Correspondingly we introduce the norm
The discrete series of spaces W, G) (m = 1,2,3, . . .) is included in the continuous family W, G) (s > 0) of Slobodetskij spaces. Ifs = m + r (0 < I < l), it is required of a function f E W, G) that f E W, G) and
Ilfll, = Ilfllm + J
m
(Ss G
G
> l/P
Ix’
_
Xyn+rP
The first differences of the partial derivatives g
dx’ dx”
< 00.
appearing here can be replaced
by differences of higher orders ” (this is advisable). The Nikol kijBesoo spaces, which we now mention, arise in this way in combination with certain other useful modifications. The norms by means of which these spaces are introduced can be replaced by equivalent norms defined in terms of best polynomial approximations in the LPmetrics. This fact is the multivariate analogue of the JacksonBernshtein theorem. In the case G = R” the Fourier transform is an effective mechanism for constructing spaces of smooth functions; in terms of it smoothness is regulated by 25For Q(6) = 6” (0 < cz< 1) we have the class Lip a. 26i.e. regular distributions. 27Here it is necessary to assume that G is a convex domain.
9 5. Function Spaces
255
the rate of decrease at infinity. For a more refined account of smoothness it is appropriate to form a decomposition of the Fourier transform with the help of a partition of unity into a sum of functions of compact support ((P&Y in the class S which are concentrated on a sequence of spherical layers r&i < 1x1 < rk (P0= 0, r, + co):
Finally, we can require that the sequence of norms (e.g. in Lp) of the inverse Fourier transforms of the terms in (55) decrease at a predetermined rate in accordance with the growth of the radii r,. An alternative to the last step is to first calculate a certain norm (taking the radii into account) for the sequence (qpkf)? and then to require that this function behaves itself sufficiently well at infinity (e.g. belongs to ~5~).These approaches, which were outlined in the mid1960s (S.M. Nikol kij, P.I. Lizorkin, Peetre), were systematically developed during the 1970s by Peetre, Triebel and Stijckert and led to the construction of very extensive families of spaces of smooth (of power order) functions which include the Nikol kijBesov spaces, the Lizorkin spaces associated with derivatives of fractional order and so on. A corresponding theory for the general modulus of continuity has been constructed during the last ten years (G.A. Kalyabin, M.G. Gol man). Function spaces on manifolds are constructed naturally from those on charts (the same can be said about spaces of differential forms and, in general, about sections of vector bundles on manifolds). As has already been said above, in the theory of boundary value problems it is important to know to what space the boundary values of functions from the given space belong. Moreover it is important to have a collection of embedding theorems for individual spaces into others and also to know when the embedding mappings are compact. At the present time extensive information has been accumulated on this account. 5.4. Operators on Function Spaces. We have already touched upon this topic repeatedly when speaking of integral and differential operators. A formal notation of the type f’(x) = Jm
6’6  YMY) dy a, allows us to regard linear differential operators as integral equations with generalised, highly singular kernels. The kernels of ordinary integral operators are regular so that these operators (or their powers) are compact on spaces of type’* LJ’. An intermediate position between the two cases mentioned is occupied by singular integral operators whose kernels are functions with discontinuities of
281n particular, kernels with bound K(x, y) = 0(1x  yl“)( m c n) on a bounded domain G c R” are of this type.
Chapter 2. Foundations and Methods
256
such a type that, while compactness is lost,2g we still have boundedness on spaces of type Lp. Example.
The Hilbert operator
<x<00) Wd(x) =& s1$f$ (co
(56)
is defined and bounded on Lp( R) (1 < p < co) (theorem of M. Riesz). The integral (56) is obtained as a principal value. If cpE L’(R), it exists for almost all x (Zygmund, 1959), however the Hilbert operator acts neither on L’ nor on L”: if (~(5) = 1 (151 < l), rp(O = 0 (ItI > 11, then WV)(~) = ln z . It is useful for I I the investigation of the operator H to pass to the Fourier transforms 4, eq. Then it transforms into the operator of multiplication by $sgn 1. However the Fourier transform itself acts on Lp only for p = 2, so that for the time being only boundedness of the Hilbert operator on L2 is clear. From certain considerations of a special character we can obtain in addition its boundedness for p = 4, 6, . . . , after which the passage to arbitrary p 2 2 is effected with the help of the following. M. RieszThorin Interpolation Theorem. Let 1 < pi < ri < co (i = 0, 1) with pi finite, put pa = [(l  a)p;’ + ap; ’ (0 < a G 1) and let r, be defined similarly. Zf an operator A is defined on the set of piecewiseconstant functions (p(t) of compact support (co < 5 < co) and maps them into functions in Lro n L in such a way that IIAqlI,, < Mi IIrp IIpi (i = 0, l), then (0 < a < l),
IIA~II,~ G WIMlp,
where h4. = M~ ;
(11. IISis the norm on L .
This theorem together with its immediate generalisations (Marcinkiewicz, 1939) became the starting point of the general theory of interpolation of linear operators which was referred to above. We will describe briefly the subject of this theory which at the present time is already far advanced and finds diverse applications. Let E,, E, be a pair of Banach spaces both of which are continuously embedded in the same LTS (a Banach pair). We can then construct the Banach spaces E, n E,, E, + E, by putting
lbll % I= max(IlxllE,, II%J llxll E,+E,
=
inf x=x0+x,
(II%
IlEo
+
llxl
liEI).
A Banach space E which is contained in E, + E, and contains E, n E, is said to be intermediate for the Banach pair (E,, E,) if the embeddings E, n E, + f fact it is already appropriate in these cases to regard the kernel as a generalised function.
4 5. Function Spaces
251
E * E, + E 1 are continuous. In particular, E, and E 1 are themselves of this type. We also note that, if E, c E, and the embedding is continuous, then E, n E, x E,andE,+E,xE,. An intermediate space for a Banach pair (E,, E,) is said to be an interpolation space if it is invariant for all linear operators A on E, + E, for which both E, and E, are invariant and the operators AlEo, AlE1 are bounded for the corresponding norms3’ (the linear space of such operators is denoted by Z(E,, E,)). The Interpolation Inequality. Zf E is an interpolation (E,, E,), then Al, is bounded and
MMI G ME max(ll&,II,
space fir
a Banach pair
IL%, II),
where ME is a constant independent of A (interpolation
(57)
constant)3’.
Proof. The operator Al, is closed, since it is closed (even bounded) under the weaker norm induced from E, + E,. It is therefore bounded. Further the space Z(E,, E,) is a Banach space with respect to both the norm which appears on the right hand side in (57) and the weaker norm [IA IIEo+s,. These norms are therefore equivalent. Finally, the restriction homomorphism of Z(E,, E,) + 2(E) is closed (which is easily verified with the norm IIAIIE,+E, on Z(E,, E,)). It is therefore bounded, i.e. (57) holds. 0 Remark. In the more general situation to which, in particular, the RieszThorin theorem relates we have to consider two Banach pairs (E,, E,), (F,, F,), spaces E, F which are intermediate for these pairs, and an interpolation in the sense that, for all homomorphisms A: E, + E, , F. + F1 such that AE, c F,, AE, c F1 and AlEo, AlEI are bounded, we have AE c F. There is no problem in carrying over the interpolation inequality to this situation. As an example let us consider the Banach pair (L1 (W), Em(~)). The role of the embedding LTS is taken here by the space @ of measurable functions with the topology of convergence in measure on each finite interval. The space Q. = L’ + L” is topologically isomorphic to the space of functions rp E Q, for which the indicatrix m. f co. A Banach space E which is continuously embedded in Q. is said to be symmetric if 1) whenever q E E, + E Q. and 11&5)1< Iv(<)1 almost everywhere, it follows that $ E E and jl+llB < llqllE, 2) whenever cpE E, $ E Q. and cp, II/ are equimeasurable, it follows that I/ E E and ll$llE = IIqIIs. For example, all Lp (1 < p < co) are of this type, as are Orlicz, Lorentz and Marcinkiewicz spaces. E.M. Semenov (1964) proved that all symmetric spaces are intermediate for (L’, L”) and any interpolation space for a pair of symmetric spaces is topologically isomorphic to a symmetric space (it is symmetric if the interpolation constant is equal to 1). Calderon (1966) obtained a complete description of interpolation spaces for (L’, L”). In any event all separable symnder these conditions A is bounded both on E,, + E, and on E, n E 1 for the noms introduced on these spaces. 31 It can at once be assumed to be the least such constant. If we then have ME = 1, the interpolation space is said to be exact.
258
Chapter 2. Foundations and Methods
metric spaces are interpolation spaces for (L’, L”). The particular symmetric spaces mentioned above are interpolation spaces for (L’, Lm). Nevertheless there does exist a symmetric space which is not an interpolation space for the pair (L’, L”) (G.I. RUSSU, 1969). In situations where it does not seem to be possible to describe all interpolation spaces for some Banach pair, a sensible approach is to find an effective construction for producing a sufficiently extensive family of interpolation spaces. The interpolation methods3’ proposed for this purpose are functorial constructions and it is precisely in this that the guarantee of their success is contained. This point of view, which goes back to work of Aronszajn and Gagliardo (1965), has recently taken on a fundamental character (Yu.A. Brudnyj, N. Kruglyak). One further significant aspect of the theory of interpolation spaces is connected with oneparameter families of Banach spaces E, (0 < CI< 1) having the following properties: 1) for /I > LYthe space E, is continuously and densely embedded in E,; 2) for y > B > ci llXllEB
<
C(a,
p,
y)
IIxII~~~)‘(y=)IIxII~=)‘(y=)
(x
6 Ey),
where C(a, fi, y) is a positive constant. Such a family is said to be a scale (normal if IIxIIs, < llxllEb (B > a) and C(a, /I, y) = 1). Many scales (Ea),,4.Gl constructed under specific conditions turn out to consist of interpolation spaces for the pair (E,, E,) (Lions, 1958; S.G. Krejn, 1960; S.G. Krejn  Yu.1. Petunin, 1966)33. Because of this occurrence, scales prove to be a suitable tool for investigating boundary value problems for elliptic differential operators (Magenes, 1963). Returning to the consideration of specific operators on function spaces, we note first of all the multivariable analogue of the Hilbert operator:
Here q,, = y/lyl, the function 52(x, 0) is continuous
(for simplicity)
and
0(x, 0) d0 = 0.
s lel=l The integral (58) is obtained as a principal value. The singular integral operator Hc is defined and bounded on Lp( I?) (1 < p < co) if 52 belongs to Lq with respect to 13  +  = 1 and its LPnorm is bounded with respect to x (the case n = 2, c : > p = 2 is due to S.G. Mikhlin, 1953; the general case is due to Calderon and Zygmund, 1956). 32 eal’ and omplex’ methods differ here. The latter are based on results from complex analysis such as Hadamard threelines theorem. We note that with these methods it is possible to show, for example, that Nikol kijBesov spaces are interpolation spaces for a pair of spaces from this same family or for a pair of Sobolev spaces etc. 33 In particular, the scales which can be constructed by the complex interpolation method.
6 5. Function Spaces
259
The Fourier transform with respect to y of the kernel of the operator Ho is called the symbol of this operator:
eY) 4x, 5) =ssz(xy ei(C.y)
b4”
dy.
IYI”
If 52 and consequently also c do not depend on x, then
In this case the composition theorem holds: the product of the symbols corresponds to the product of the operators (CalderonZygmund, 1952). This result has a natural connection with the convolution theorem but it is not so obvious since the integrals concerned are principal values.34 Since the correspondence between operators and symbols is clearly also linear, we have here a homomorphism of an algebra of operators into an algebra of functions, i.e. a certain variant of operator calculus. This is an effective route to the solution of multivariate singular integral equations (this was in fact projected in the late 1920s by Tricomi). In the general case it is useful to consider along with HB the operator
We naturally expect that the operator Go, whese investigation can turn out to be simpler than that of Ho, will at the same time imitate HO sufficiently well. The operator Gc is pseudodifirential. To explain this term let us consider an operator of the form U%)(x) = m
1 s
R”P(X, bN5)ei(~*x) d5,
where p, an arbitrary given function of power growth rate3’ with respect to <, is the symbol of the operator P. If p(x, 5) is a polynomial in r (with coefficients depending on x), then P is a difirential operator (with the same coefficients). If we eliminate the intermediate Fourier transform, we can represent the operator P as a double integral: u+m)
1 = pq
R” Rnp(x, &p(y)ei(e*xY) ss
d( dy.
(59)
34The concept of the symbol was introduced differently at an early stage in the development of the theory of singular integral equations and correspondingly the composition theorem was proved in another context (S.G. Mikhlin, 1936; Giraud, 1936; S.G. Mikhlin, 1955). Then the equivalence of the two definitions of the symbol was established (S.G. Mikhlin, 1956). Uniformly with respect to x on each compact set. The exponent of the power of growth (not necessarily integral) is called the order of the operator. Usually p is a function of class C” with respect to all its variables with suitable bounds on the derivatives.
260
Chapter
2. Foundations
and Methods
The final step leading to the concept of a pseudodifferential operator (PDO) in full generality consists of replacing p(x, <) in formula (59) by p(x, y, 5) (it now follows that we should call an operator of the form (59) a PDO in the restricted or proper sense). This class of operators is sufficient for studying many questions in the theory of elliptic differential equations (including the solution of the index problem (AtiyahSinger, 1968), in which PDOs played an important role). We can regard the construction of a PDO as a quuntization which associates with each scalar function (symbol) an operator on a suitable function space. Thus, for example, problems such as quasiclassical asymptotics (passage to the limit from quantum mechanics to the classical situation by letting Plan&s constant /i tend to zero) are naturally studied in the setting of PDOs and their further generalisations (V.P. Maslov, 1965; V.P. Maslov, M.V. Fedoryuk, 1976; V.L. Rojtburd, 1976). The generalisations mentioned are Fourier integral operators (FZOs), whose theory was formulated during the mid 1960s and early 1970s thanks to works of V.P. Maslov, Hijrmander and other authors. PDOs were already in use at the beginning of this period, however many problems (including quasiclassical asymptotics and the Cauchy problem for hyperbolic equations) required the introduction of FIOs. The operators are constructed on Lagrange manifolds by gluing together local FIOs which have a similar form to PDOs but with exponential term exp(iS(x, y, t)), where in general S is some nonlinear function. In conclusion we note that in the development of functional analysis functions of infinitely many variables played and continue to play a significant role. Such, for example, are the functionals which arise in variational calculus, states with an unrestricted number of particles in quantum field theory, and so on. Rigorous analysis has been successfully constructed on spaces of functions of infinitely many variables. A systematic account of it is contained in the monographs by Yu.M. Berezanskij (1978) and by Yu.L. Daletskij and S.V. Fomin (1983). It is most important here to have a satisfactory theory of measure and integration (in the field of applications we point out the Wiener measure in the theory of random processes, the Feynman integral in quantum mechanics and electrodynamics, secondary quantization). A fundamental difficulty is caused by the lack of a translation invariant measure on an infinitedimensional LTS since it is not locally compact (the last condition is close to being necessary in the theorem of Haar). However for many purposes the socalled quasiinvariance is sufficient. In particular, aussian measures’, which include the Wiener measure, have this property. It is clear from what has been said that on an infinitedimensional space we have to construct not measures n general’ but special classes of ood measures. A certain axiomatic scheme for measure theory in an inlinitedimensional space was proposed by A.M. Vershik (1968) and he and V.N. Sudakov (1969) conducted an extensive investigation of probability measures. Generally speaking the connections between functional analysis and probability theory are deep and varied. A meaningful account of this aspect lies beyond the scope of the present volume.
Bibliography
261
Commentary on the Bibliography The principal events in the foundation of functional analysis have been elucidated in the articles of Bourbaki and in the book by Dieudonne (1983). This discipline attained a definite maturity in 1932, which is testified to by the simultaneous appearance in this year of three fundamental works: heorie des operations lintaires” by Banach, athematische Grundlagen der Quantenmechanik” by von Neumann and inear transformations in Hilbert space and their applications to analysis” by Stone. However many important topics (linear topological spaces, ordered linear spaces, Banach algebras etc.) were developed later on and surveys and monographs devoted to them appeared correspondingly later. At the present time there are many hundreds of titles of individual books dealing just with functional analysis and its applications. We hope that the selection presented below is representative of the subject (but it is by no means representative in terms of priority). As far as articles are concerned the list includes only certain surveys from spekhi Matematicheskikh Nauk” (in order to supplement the monographs) and works from previous volumes in the present series which are connected with our subject. Several books are also mentioned from the pravochnaya Matematicheskaya Biblioteka”, including the publication unctional Analysis” (1972) which was produced by 17 authors under the general editorship of S.G. Krejn. References in the text to articles (author, year) are not associated with the Bibliography. In the majority of cases the original source will be found without difficulty from the given information.
Bibliography* I. Basic
Texts
Kantorovich, L.V., Akilov, G.P. [1977]: Functional Analysis. Nauka: Moscow. English transl.: Pergamon Press: Oxford, 1982,Zbl.484.46003 Kolmogorov, A.N., Fomin, S.V. [1972]: Elements of the Theory of Functions and Functional Analysis, 3rd ed. Nauka: Moscow. French transl.: Mir.: Moscow, 1974, German transl.: VEB Deutscher Verlag der Wissenschaften: Berlin, 1975, Zbl.57,336; Zbl.90,87; Zbl.167,421; Zbl.299. 46001 Lyustemik, L.A., Sobolev, V.I. [1965]: Elements of Functional Analysis. Nauka: Moscow, Zbl.141,116; Zbl.44,325. English transl.: J. Wiley & Sons: New York, 1974 Riesz, F., SzokefalviNagy, B. [1972]: Lecons d nalyse Fonctionelle. Akad. Kiado: Budapest, Zbl.64,354; Zbl.46,331 Rudin, W. [1973]: Functional Analysis. McGrawHill: New York, Zbl.253.46001
II. Additional
Akhiezer, transl.: Akhiezer, Transl.
Educational
Literature
N.I. [1965]: Lectures on Approximation Theory. Nauka: Moscow, Zbl.133,16. German AkademieVerlag: Berlin, 1967 N.I. [1984]: Lectures on Integral Transforms. Vishcha Shkola: Khar ov. English transl.: Math. Monogr. 70: Providence, 1988,2b1.546.44001
*For the convenience of the reader, references to reviews in Zentralblatt fur Mathematik (Zbl.), compiled using the MATH database, have, as far as possible, been included in this bibliography.
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VI. Reference
Rooks
from
the Series Spravacknayr
Matematichesknya
Bibiioteka
Babich, V.M. et al. [1964]: Linear Equations of Mathematical Physics. (Mikhlin, S.G. (ed.)) Nauka: Moscow, Zbl.122,330 Birman, MSh. et al. [1972]: Functional Analysis. (2nd ed.) (Krejn, S.G. (ed.)) Nauka: Moscow, Zb1.261.47001 Brychkov, Yu.A., Prudnikov, A.P. [1977]: Integral Transforms of General&d Functions. Nauka: Moscow, Zbl.464.46039 Ditkin, V.A., Prudnikov, A.P. [1971]: Integral Transforms and Operational Calculus. Nauka: Moscow, Zb1.99.89. English transl.: Pergamon Press: Oxford, 1965 Guter, R.S.,Kudryavtsev, L.D., Levitan, B.M. [1963]: Elements of the Theory of Functions. (Ul anov, P.L. (ed.)) Phys.Math. Section of Nauka: Moscow, Zb1.115,47. English transl.: Pergamon Press: Oxford, 1966 Zabrejko, P.P. et al. [1968]: Integral Equations. Nauka: Moscow
Author Index Adamyan, V.M. 227 Akhiezer, NJ. 29,61 Alakhverdiev, D.Eh. 216 Alaoglu, L. 145 Aleksandrov, P.S. 175 Amir, D. 143 Appel, P.E. 65 Arens, R.F. 148,236 Aronszajn, N. 175,258 Arov, D.Z. 227 Arzela, C. 154 Ascoli, G. 113, 154 Atiyah, M.F. 95,260 Atkinson, F.V. 162, 167 Babenko, K.I. 135 Babich, V.M. 244 Bade, W.G. 207 Baire, R. 122 Banach, S. 14, 112,122, 123, 145,146, 151153, 157160,261 Bari, N.K. 159,169 Berezanskij, Yu.M. 203,242,260 Bernoulli, D. 19 Bernoulli, Johann 10 Bemshtein, S.N. 26, 29, 30,43, 55, 57, 63 Besov, O.V. 243 Bessel, F.W. 24,28, 52 Beurling, A. 226 Birkhoff, G.A. 121 Birman, MS. 206 Bishop, E. 130,209,238 Blaschke, W. 68,224 Bochkarev, S.V. 134 Bochner, S. 35,52,53, 141,241,249 Bogolyubov, N.N. 179,246,252 Bohl, P. 191 Bohr, H. 51,52,191 Bonsall, F.F. 215 Borel, E. 53,69, 151 Borok, V.M. 5,249 Bourbaki, N. 261 Brauer, A. 174 Brodskij, MS. 221223 Brudnyj, Yu.A. 5,258 Bunyakovskij, V.Ya. 13 Burnside, W. 177,180 Calderon, A.P. 236,243,257259 Calkin, J.V. 217
Caratheodory, K. 108 Carleman, T. 30, 70 Carleson, L. 23,214,233 Cauchy, A. 12, 13, 144 Chebyshev, P.L. 53,54,58,59 Choquet, G. 143 Clairaut, AC. 20 Clarkson, J.A. 130, 158 Cotes, R. 64 Courant, R. 47 D lembert, J. 20 Daletskij, Yu.L. 260 Day, M.M. 131, 158 De Leeuw, K. 192 Delsarte, J. 44, 204 Denjoy, A. 26 Dieudonne, J. 152,261 Dirac, P.A.M. 11 Dirichlet, P.G.L. 17,22,42 Dolzhenko, E.P. 66 Du BoisReymond, P. 23 Duncan, J. 215 Dunford, N. 143,207,209 Dvoretzki, A. 136, 172 Dynin, A.S. 172 Eberlein, W.F. 151 Etimov, N.V. 147 Ehjdel an, SD. 249 Ehrenpreis, L. 247 Eidelheit, M. 113 Enflo, P. 134, 158, 166 Esclangon, E. 93 Euclid 4 Euler, L. 10, 19,20 Faber, G. 63,66 Faddeev, L.D. 205,227 Fedoryuk, M.V. 260 Fejer, L. 23,53, 117 Fekete, M. 184 Feller, W. 189 Fermat, P. 10 Fischer, E. 14,47 Fixman, V. 207 Foias, C. 211,214,215,225,226 Fomin, S.V. 260 Fourier, J. 19,20, 50 Frechet, M. 191
274
Author Index
Fredholm, I. 74, 165, 166 Frobeoius, G. 175180,228 Fubioi, G. 15 Gagliardo, E. 243,258 Gakhov, F.D. 82 Gamkrelidze, R.V. 5 Gel aod, I.M. 27,48,107,140,190,204,232, 233,239241,249 Gel ood, A.O. 62 Giozburg, Yu.P. 229 Giraud, G. 259 Glazmao, LM. 213 Glicksberg, I. 192 Giidel, K. 97 Gokhberg, I.Ts. 138,162, 167,221 Gol mao, M.L. 255 Goochar, A.A. 238 Gorio, E.A. 5,190,235 Grotheodieck, A. 152,170173 Haar, A. 56,179 Hadamard, J. 10,69, 144,258 Hahn, H. 112 Hamburger, H.L. 30 Hartmao, P. 51 Hartogs, F. 66 HausdortT, F. 34154,215 Heaviside, 0. 40 Hellioger, E. 31, 193 Helly, E. 35, 145 Herglotz, G. 35 Hermite, C. 62 Hilbert, D. 14, 27, 74, 103, 105, 187, 193, 201 Hille, E. 189 Holub, J.R. 166 Hopf, E. 79 Hopf, H. 175 Hormaoder, L. 248,250,260 Huff, R.E. 143 Hunt, R.A. 23 Iokhvidov, IS.
228,230
Jackson, D. 55,57 Jacobs, K. 192 James, R.C. 130,147,158 Jordan, G. 22,97,98,224 Jordan, P. 130 Kadets, M.I. 5, 122, 135, 158, 191 Kadets, V.M. 5, 136 Kakutaoi, S. 121, 177, 192 Kalisch, G.K. 223
Kalyabio, G.A. 255 Kaotorovich, L.V. 117,163 Karlio, S. 135 Kato, T. 206 Katsnel oo, V.Eh. 5,190,214,215 Keldysh, M.V. 65,217,231 Khejtits, A.I. 133 Khiochio, A.Ya. 35 Klee, V.L. 131 Koch, H. 171 Kolmogorov, A.N. 12,23,56,93, 100,107, 122,126 Kooig, H. 219 Koopmao, B.O. 191 Korkio, A.N. 58 Kostyucheoko, A.G. 48 Kotel ikov, V.A. 69 Krasoosel kij, M.A. 137,182,254 Krejn, M.G. 28,61,79, 115, 129, 137139, 146,153,162,169,182,204,206,221,228, 230,241 Krejo, S.G. 189,242,243,258,261 Kruglyak, N. 258 Krull, W. 101 Krylov, N.M. 179 Kuroda, S.T. 206 Kuz io, R.O. 64 Ladyzheoskaya, O.A. 249 Lagrange, J.L. lo,17 Lang, S. 105 Laoger, H. 214,230 Lavreot v, M.A. 65 Lax, P.D. 227 Lebesgue, H. 12,15,23,55,151 Leiboiz, G.W. 6 Leray, J. 165 Levio, B.Ya. 29,56,205 Levinsoo, N. 135 Levitao, B.M. 48,204 Levy, P. 80,235 Lidskij, V.B. 219,220 Lio, V.Ya. 235 Liodemann, F. 62 Liodeostrauss, J. 143, 157 Lions, J.L. 243,258 Liouville, J. 46,47, 184 Litvioov, G.M. 170, 171 Livshits, M.S. 220,223225,227 Lizorkio, P.I. 255 Lobachevskij, N.I. 20 Lojasiewin, S. 250 Lomooosov, V.I. 176,2 10 Larch, E.R. 191
Author Index Lumer, G. 215 Luzin, N.N. 23,26 Lyubarskij, Yu.1. 5 Lyubich, M.Yu. 5, 192 Lyubich, Yu.1. 43,44,92, 188, 192,209,211 Mackey, G.W. 148,149 Magenes, E. 258 Mairhuber, J.C. 56 Malgrange, B. 247 Mal sev, AI. 97 Marchenko, V.A. 204,205 Marcinkiewicz, J. 256 Markov, A.A. 5760 Markov, A.A. (junior) 177 Markus, A.S. 5, 138, 170,212,219,220 Markushevich, A.I. 135 Martineau, A. 251,252 Maslov, V.P. 48,260 Matsaev, V.I. 170,205,209,210,215,217, 219,222 Mautner, F.I. 203 Mazur, S. 107,113,122,131,146,233 Mellin, H. 39 Mercer, T. 78 Mergelyan, S.N. 29,65,66 Mikhlin, S.G. 168,258,259 Mil an, D.P. 130,137139,162, 182 Milyutin, A.A. 122 Minkowski, H. 14, 138 Mityagin, B.S. 172 Miyadera, I. 189 Molchanov, A.M. 51 Molin, F.Eh. 180 Montel, P. 151 Morimoto, M. 252 Morris, P. 143 Mukminov, B.R. 213 Miintz, C.H. 133 Najmark, M.A. 226,239,241 Nevanlinna, R. 31 Newton, I. 6 Nikodjrm, 0. 142 Nikol kaya, L.N. 210 Nikol kij, N.K. 5,210,212,220,221,227 Nikol kij, S.M. 243,255 Noether, E. 89 Noether, M. 82 Numakura, K. 192 Ostrowski, A.M.
70,210
Paley, R.E.A.C.
38, 163
Parasyuk, OS. 246 Parseval, M. 25, 28, 37,46,48,49, 53, 135 Pastur, L.A. 5 Pavlov, B.S. 227 Peetre. J. 255 Pelczyirski, A. 135 Perron, 0. 175 Peter, F. 180,241 Petrovskij, I.G. 248,249 Pettis, B.J. 140,143 Petunin, Yu.1. 258 Phelps, R.R. 130,143 Phillips, R.S. 189,215,227,229 Picard, E. 75,164 Pie&h, A. 169, 172 Plancherel, M. 37,50,241 Pontryagin, L.S. 228,230,241 Potapov, V.G. 224 Povzner, A.Ya. 203,204 Ptlk, V. 152 Pythagoras 13,25 Radon, J. 142 Rajkov, D.A. 149,241 Read, C.J. 175 Riemann, B. 12,23,39, 136 Riesz, F. 14, 16, 35, 116, 117, 129, 131, 132, 165,208 Riesz, M. 31,256 Ringrose, J.R. 220 Roberts, J.W. 139 Rogers, C.A. 136,172 Rojtburd, V.L. 260 Rosenblum, M. 206 Rosenthal, A. 66 Royden, H. 236 Rudin, W. 212 Runge, K. 65 Russu, G.I. 258 Rutman, M.A. 146,162,175,186 Sakhnovich, L.A. 204,222 Schauder, J. 167,174 Schmidt, E. 74 Schoenberg, I.Y. 57 Schrodinger, E. 46,47,203 Schur, I. 177,180,214 Schwartz, H.A. 13 Schwartz, L. 45,242,248,249 Semenov, E.M. 257 Shilov, G.E. 235237,249,250 Shmul an, V.L. 130, 139, 151153,158 Shmul an, Yu.L. 225 Shtraus, A.V. 225
215
276 Sieklucki, K. 56 Singer, I.M. 95,260 Slobodetskij, L.N. 244 Smith, K.T. 175 Sobolev, S.L. 242244 Sobolevskij, P.E. 189 Sokhotskij, Yu.V. 83 Stechkin, SB. 147 Steinhaus, H. 123 Steinitx, E. 136 Steklov, V.A. 47 Stieltjes, K. 12, 28 Stiickert, B. 255 Stone, M.H. 104,201,233,234,261 sturm, c. 46,47 Sudakov, V.N. 260 Sushkevich, A.K. 192 Sylvester, J.J. 99 SzokefalviNagy, B. 214,225,226 Taklind, S. 243 Tannaka, T. 241 Thorin, G.O. 256 Tikhonov, A.N. 145,174,243 Titchmarsh, E.C. 45,48,223 Tkachenko, V.A. 5,44,209 Toeplitz, 0. 96, 123, 159, 193 Tonelli, L. 56 Tricomi, F.G. 259 Triebel, H. 255 Tzafriri, L. 157
Author Index Van Kampen, E.R. 241 Vershik, A.M. 5,260 Vitushkin, A.G. 65 Vladimirov, V.S. 252 Volterra, V. 10 VonNeumann, J. 124,130,178,179,191, 193,194,202,214,215,261 Walsh, J.L. 65 Weierstrass, K. 24,28,65,68, 151,233, 234 Weil, A. 241 Wermer, J. 220,234 Weyl, H. 31,48,50, 51, 180,202 Widder, D.V. 40,189 Wiener, N. 27, 38, 39, 79, 80, 163, 235, 236 Williamson, J.H. 165 Wintner, A. 51 Wold, H. 214 Yang, C.T. 57 Yentsch, R. 175 Yosida, K. 189, 192 Young, L.C. 252 Zharinov, V.V. 252 Zobin, N.M. 172 Zolotarev, E.I. 58 Zolotarev, G.N. 249 Zygmund, A. 256,259
Subject Index Action 10, 178 Aftine manifold 88 Algebra, Banach 27,232   commutative 27  Boolean 04  function 101  group 240  local 101  of sets 12  operator 239   reducible 176  semisimple 102  spectral 102  symmetric 234  uniform 234  universally enveloping 242  Wiener 27   with weight 237  with involution 238 Algebraic interior 108 Almostperiod 52 Annihilator of a subspace 89 Ball, qunit 111 Banach limit 178 Barrel 122 Bases, quadratically close 169 Basis, Bari 169  Hamel 85  Riesz 159  Schauder 134  unconditional 135 C*algebra 239 Capacity, analytic 65   continuous 65 Central symmetry 108 Centralizer of a subalgebra 176 Centre of an algebra 176 0fmass 143 Character of a representation 180  of a semigroup 180  of an algebra 104 Charge 131 Chebyshev alternant 53 Class, Hardy 38,225  Holder 26  Lipschitz 26  Nevanlinna 32  quasianalytic 69
Closure of a linear operator 193 Codefect of a homomorphism 90 Codimension of a subspace 88 Coefficients, Christoffel 60  Fourier 21,52   generalised 27 Coimage of a homomorphism 90 Cokemel of a homomorphism 90 Combination, convex 107   absolutely 108  nonnegative 109 Commuting operators 181 Compact set, Bohr 192 spectral 208 Complement of a subspace, algebraic     topological 157 Condition, boundary 11,19,71  distributed 71  initial 18 
Levinson
85
209
 Lipschitz 9, 163  of regularity of a functional 16 Cone 109  minihedral 118 total 186 Constant, interpolation 257  Lebesgue 55 Convergence, absolute 101,135  mean square 13  unconditional 135 Convolution 22,240  of sequences 27 Coordinates of a vector 86, 134 Corank of a homomorphism 90 Crossproperty of a seminorm 173 Decomposition, triangular 220,222  Wold 214 Defect of a homomorphism 90 Deficiency index of a symmetric operator of an isometric operator 185 Definiteness, procedure for 230 Sfunction 11 Derivative, Frechet 163  generalised 244  of fractional order 42  RadonNikodym 142 Determinant, Fredholm 77  of a linear operator 170 Deviation 53, 147
194
278 Differential of a function 6  of a mapping 163 Dilatation 226 Dimension, algebraic 86  topological 136 Direct sum of linear operators Disk algebra 234 Dissipation 18 Distribution 245  of compact support 246  of finite order 245  probability 12  regular 245  tempered 250 Duality of linear spaces 147
Subject Index
92
Edge of a wedge 109 Equation, Abel integral 42  characteristic 45  EulerLagrange 16  Fredholm integral 74 heat 19  Kepler 9  Lagrange 17  Laplace 243  Poisson 247  singular integral 82  Volterra integral 74 wave 18 Equimeasurability of functions 253 Equivalence of metrics 157  of norms 158  of operators 181  of representations 180   unitary 180 Extension of an operator 181 Flow 178 Formula, CauchyHadamard 144  D lembert 19  EulerFourier 20  Gel and 184  interpolation   Hermite 62   Lagrange 62   Newton 62  inversion, Fourier 37   FourierPlancherel 37   RiemannMellin 39   Weyl 49   Widder 40  KochGrothendieck 171  Lagrange 72  LagrangeSylvester 99
 of finite increments 163  quadrature, Cotes 64   Gaussian 59  Taylor 7 Function, absolutely continuous 15  absolutely monotonic 43  almostperiodic 51  analytic 7  basic 245  Cauchy 41  characteristic 223  differentiable 6, 163  eigen 51   generalised 5 1 flat 7  general&d 245  Green 71  integrable 12  interior 226  Lagrange 16  measurable 12  normalised 13  of compact support 38,86  periodic in mean 44  positive 15, 36  Riemann t; 42  smooth 6  summable 12   locally 15  Weierstrass 54 Functional, additive 11  analytic 251  bounded 127 locally 118119  convex 110  coordinate 88 gauge 111  Hermitian quadratic 77  homogeneous 11   positively 110  linear 11  Minkowski 111  monotone 15  multiplicative 104  nonnegative 238  positive 15  sesquilinear 196  subadditive 109  sublinear 110 support 113,115  symmetric 238   Hermitian 196 Functional calculus 99, 104, 170, 198,208, 234
Subject Index Gap between subspaces 137 Generalised sum of a series 100 Group, amenable 177 178 dual 240  oneparameter 186  unimodular 179 Halfopen interval 108 Halfspace 113 Hamiltonian 46,205 Harmonic oscillator 17 Homomorphism, absolutely summit xg 172  bounded 153 closed 155  compact 165  conjugate algebraic 90   topological 155  finite rank 93  HilbertSchmidt 168  Noetherian 93  nuclear 169  * 238 Hull, convex 108   absolutely 108, 138   closed 138  linear 88 Hyperfunction 251 Hyperplane 88  of support 115  separating 113 almost 114   strictly 128 Ideal, maximal 101   standard 106  simple 102  symmetric 238 Index of a homomorphism 93  of an indefinite metric 228  of nilpotence 97 Inequality, interpolation 257  triangle 13, 109 Intimum 118 Integral, Bochner 142  Brodskij 221  Dirichlet 243  Fourier 36  Gel andPettis 140  Lebesgue 12  LebesgueStieltjes 15  of Cauchy type 82  of fractional order 42 Invariant mean 177
279
Jordan cell 97 98  structure 97 Kernel, Cauchy 168  difference 78  Dirichlet 22  Fejtr 23  Hermitian symmetric 77  HilbertSchmidt 77  of a homomorphism 89  of a seminorm 111  of an integral operator 71, 76  Sushkevich 192 Lproblem of moments 60 Imeasure 14 Lattice, Banach 234  vector 118   Dedekind complete 119   Dedekind acomplete 119 Logarithmic property of the index Matrix, Hankel 30  Heisenberg 206  identity 86  Jacobian 34  Nevanlinna 33  Toeplitz 35 Measure, Bore1 14  Choquet 143  finite 12  general&d 13 1  Haar 179  invariant 178  Lebesgue 15   Stieltjes 14  of compact support 132  operatorvalued 198 outer 14  probability 12  regular 116  structure 242  vector 142 absolutely continuous 142 Method, Fourier 19  of least squares 147  of summation 100 Cedro 23 Fejer 23  of tangents 8 Metric, indefinite 227  invariant 120 Model of a linear operator 222  functional 223
93
280
Subject Index
Modulus of continuity 26,254 Moment problem, abstract 116  ChebyshevMarkov 58  exponential 43  NevanlinnaPick 68  power 29  trigonometric 35 Monoid 178 Multiplicators 246 Multiplicity of an eigenvalue 199 Nil radical 102 Nilpotent 102 Nodes of a quadrature formula 59 Norm, graph 202  HilbertSchmidt 168  nuclear 169  of a function 13  of a homomorphism 127,153  of a measure 132  of a vector 14, 111  of a vector measure 142 Normal eigenvalue 182 Normal mode 18 Number, approximation 168  defect 182  singular 216 Numerical range of a linear operator Rtopology 153 Operator, adjoint I92  algebraic 97  communication 227  compact relative to another operator  complete 216  completely nonselfadjoint 223  completely nonunitary 214  conservative 190  contracting 157   uniformly 157  damping 23 1  decomposable 210  differential 70,259   elliptic 248  differentiation 11,99  dissipative 190  dynamical 227  expanding 224  Hilbert 256  identity 11  infinitesimal 186  integral 71   Fourier 260  isometric 226
 Laplace 18  linear 11,91  metric 228  nonquasianalytic 209  normal 193  normally resolvable 162, 166  pseudoditferential 259  quasiconservative 212  quasidissipative 212  quasinilpotent 184  scattering 206  selfadjoint 73,193  shift 99, 225   weighted 221  symmetric 46,193   maximal 196   nonnegative 196  unicellular 92,221  Volterra 186 wave 205 Order of a pseudodifferential operator  of an eigenvalue 97
215
202
259
Parallelogram identity 130 Part of an operator 181  quasinilpotent 207  scalar 207 Pencil of operators, quadratic 231  strongly damped 23 1 Point, algebraic interior 108  extreme I38  interpolating 61  Lebesgue 23  quasiregular 182  regular 98, 181  stationary 10  strongly exposed 143  Weyl 32,50 Polar 145 Polar representation of an operator 217 Polynomial, Bemshtein 30  characteristic 41  Chebyshev   of first kind 54   of second kind 58 Faber 66  Hermite 28  Laguerre 28  Legendre 28 minimal 97  of best approximation 53 Principle, contraction mapping 9  fixed point 175  of duality 241
Subject Index  of uniform boundedness 123,154  variational 10 Problem, Cauchy 164,186   pointwise stable 188   stable 189  Dirichlet 243  of scattering theory direct 204   inverse 204  RiemannHilbert 82   additive 82  SturmLiouville 45 Process, interpolation   of Lagrange type 63   of Newton type 63  Newton 8,163  quadrature Cotes 64   Gaussian 64 Product, Blaschke 68  of measures 15  scalar 13  tensor 173 Projection 91  boundary 192 root 98 Property, approximation 165  of reciprocity of polars 148  RadonNikodym 142 Pseudodistance 228 Pseudonorm 120 Quasimodel of a linear operator Quasinilpotent 103 Quasinorm 121 Quasipolynomial 40 Quasiprojection 209 Radical of an algebra 102 Radius, spectral 184  Weyl 31,SO Random events 12  variables 12 Rank of a homomorphism 90  of a set of vectors 88 Ray 109 Regularisor 94 Representation 179  almostperiodic 191  bounded 179  canonical functional 103  contracting 179  isometric 179,233  regular 180
222
 uniformly continuous 179  unitary 179 Resolution  of the identity 197  spectral 98, 197, 199 Resolvent 103  of a linear operator 98, 181   Fredholm 217  of an element of an algebra 103 Restriction of an operator 181 Scale of Banach spaces 258 normal 258 Schmidt expansion 217 !kmiflow 178 Semigroup, amenable 177178  oneparameter 186 Seminorm 111  factor 111 Separation of variables 19 Sequence, locally finite 245  quasieigen 182  subadditive 184 weight 25 Series, Dirichlet 42  Fourier 20,21,52,134,136   generalised 27,45  interpolation   Lagrange 68   Newton 62  Neumann 162 Taylor 7 Set, absorbent 110  algebraically open 108   solid 108  balanced 108  bounded 126  Chebyshev 147  circled 108  conal 109  convex 107   absolutely 108  measurable 12  precompact 154  relatively compact 154 dense 52  resolvent 98, 103, 181  spectral 215 Shift, generalised 204  semigroup 179 Shilov boundary 237 ualgebra 12 Solution, fundamental 247
282
 general&d 248 Space, Banach 14  barrelled 122  configuration 16  conjugate algebraic 88   topological 129  coordinate 134  countably Hilbert 173  defect 182 factor 88  Frechet 122  fully complete 152  function 11  Hilbert 14  intermediate 256  interpolation 257  linear topological 119   complete 122  locally bounded 126   convex 124  Lorentz 253  Mackey 149  Marcinkiewicz 253  maximal ideal 103  Monte1 150  Nikol kijBesov 254  normed 14 nuclear 171  of Lipschitz type 254  ordered 117  Orlicz 253  phase 178  preHilbert 14  probability 12  Ptak 152  reflexive algebraically 90   topologically 130, 150  Riesz 118  Schwartz 250  second conjugate 150  Slobodetskij 254  smooth 130  Sobolev 254  strictly convex 131  superreflexive 158  symmetric 257  uniformly convex 130   smooth 130 Spectrum 51,97, 103,181  absolutely continuous 200  approximate 182 Bohr 52  combined 236
Subject Index  continuous 51, 199  discrete 51, 199  residual 182  simple 224  singular 199200 Subordination of metrics 158  of norms 158 Sum of spaces, external direct 89    orthogonal 87 Sum of subspaces, direct 87 topologicaf 157    orthogonal 157 Support of a distribution 246  of a function 86  of a measure 143  of a vector 85  of an analytic functional 251 Supremum 118 Symbol of a differential operator 248,259  of a pseudodifferential operator 259 System, biorthogonal 61  Chebyshev 56  complete 27, 132133  distal 136 i dynamical 178  Markov 57   normalised 59  minimal 135  wlinearly independent 134  orthogonal 13  orthonormal 13,134 Theorem, boundary spectrum splittingoff 192  closed graph 155  convolution 22  ergodic 191  expansion 40  implicit function 9  inverse homomorphism 156  local 235  open mapping 156  separating hyperplane 113,114,128  spectral 98,197,200,230  spectral mapping 99,104,207,209,235  stability of the index 95  superposition 209 Theory, Fredholm 76  HilbertSchmidt 77  Noether 84  scattering, nonstationary 205   stationary 204 Topological isomorphism 121 Topology, consistent with duality 148
Subject Index  DieudonnCSchwartz 245  locally convex 124  Mackey 149 norm 150  of convergence in measure 120  of uniform convergence 125   on bounded sets 150   on compact sets 125  strong 145  tensor product 173  uniform 150, 154 weak 145  weakened 145 Total family of linear functionals 124   of projections 92   of seminorms 124 Trace 170 matrix 170  spectral 219  tensor 169 Transform, Cayley 194  Fourier 36   generalised 49, 248   operator 210   Plancherel 37   Stieltjes 37  Gel and 233  Laplace 39 finite 44 local 43
283
  Stieltjes 42  Legendre 252 Travelling wave 18 Ultradistribution 250 Ultrapower 106 Unit sphere 127 Variation of a functional 16 Vector, channel 223  cyclic 224 root 97 Vectorfunction, absolutely integrable  analytic 144  Bochner integrable 142 entire 144  Gel andPettis integrable 140 simple 141142 w*topology 144 Wedge 109  generating 109 Weight 28 Weights of a quadrature formula Weyl disk 32, 50  point 32, 50  radius 31,50 Young
relation
132
59
141