Absolute Continuity and the Uniqueness of the Constructive Functional Calculus

Math. Log. Quart. 40 (1994) 519 - 527

Mathematical Logic Quarterly

@ Johann Ambrosius Barth 1994

Absolute Continuity and the Uniqueness of the Constructive Functional Calculus')

Douglas Bridges' and Hajime Ishihara2

'Department of Mathematics and Statistics, University of Waikato,

'School of Information Science, Japan Advanced Institute for Science and Technology, Hokoriku, Tatsunokuchi, Ishikawa, 923-12, Japan2)

Private Bag 3105, Hamilton, New Zealand

Abstract. The constructive functional calculus for a sequence of commuting selfadjoint operators on a separable Hilbert space is shown to be independent of the orthonormal basis used in its construction. The proof requires a constructive criterion for the absolute continuity of two positive measures in terms of test functions.

Mathematics Subject Classiflcation: 03F60, 46330, 47330.

Keywords: Constructive functional calculus, Absolute continuity of positive measures, Functional calculus measure.

1 Introduction

In his constructive theory of operators on a separable Hilbert space, ERRETT BISHOP proved in [ l , Chapter 7, 8.221 the following spectral theorem for selfadjoint operators:

Let T = (Tn)F=' be a sequence of commuting selfadjoint operators, each with bound b > 0 , on a separable Hilbert space H , and let ( e n ) be an orthonormal basis of H . n,"==l[-b,b], and a bound- preserving homomorphism f ++ f ( T ) of L , ( p ) onto an algebra of commuting selfadjoint operators on HI such that

(i) i f f = Eyl ,.,., i N = O c(i1, . . . , i ~ ) ail . . . x $ , where c ( i1 , . . . , i ~ ) E C and ai is the

Then there exist a positive measure p on X

i th projection of X on [-b, b], then f ( T ) = Eyl ,_,,, i N = O c( i1 , . . . , Z N ) c1 . . .TkN, and

( i i ) ~ ( f ) = E,"=' 2 - " ( f ( T ) e n , e n ) for each f E Lw(p).

Moreover, if (f,,) i s a bounded sequence of elements of L, (p ) that converges in measure t o an element f of L, (p ) , then the sequence (fn(T)) converges strongly to f(T).

')The research leading to this paper was carried out while ISHIHARA was a Research Visitor at the University of Waikato. The authors wish to thank the Japan Advanced Institute for Science and Technology, Hokoriku, for its generous support of that visit.

')e-rnail [email protected]

520 Douglas Bridges and Hajime Ishihara

The mapping f I-+ f ( T ) in the spectral theorem is called the funct ional calculus f o r TI and p is called the functional calculus measure f o r T corresponding to the orthonormal basis ( e n ) .

Surprisingly, BISHOP did not discuss whether the operator f ( T ) depends on the orthonormal basis (en). Following the constructive path laid down by him3), we prove in this paper that it does not so depend; in order to do so, we provide a constructive characterisation of absolute continuity for positive measures on a locally compact (metric) space.

We shall assume that the reader has access to [l] and [3]; the first chapter in each of these books gives an introduction to the spirit and methods of constructive analysis. Our work draws largely on the measure theory developed in [l , Chapter 61, some features of which we summarise for the convenience of the reader.

A metric space (XI e ) is said to be compact if it is totally bounded and complete, and it is said to be locally compact if each bounded set in X is contained in a compact set. Note that, following BISHOP, we require totally bounded sets to be nonvoid, so that a compact or locally compact subset S of a metric space (XI e) is located, in the sense that the distance

e (2 , S) = inf(e(t , y) : y E S}

from z to S exists for each c E X. The metr i c complement of a located set S is the set -S of all points t in X such that e(z, S) > 0. A test funct ion on a locally compact metric space S is a uniformly continuous mapping f : X - R.such that f(t) = 0 for all t in the metric complement of some compact set (called a compact support of f ) ; the set of test functions on X is denoted by C(X).

A posit ive measure on X is a nonzero linear mapping p of C(X) into R such that p(f) 3 0 whenever f(t) 2 0 for all z E X. As in classical analysis, a positive measure p gives rise to a set L 1 ( p ) of integrable functions. A subset F of X is said to be full if it is the domain of an integrable function.

(A' , A') of subsets of X such that p(z,y) > 0 for all t E A' and y E A'; the characterist ic funct ion of A is the map X A : A' U A o - (0,1} such that X A ( A ' ) = (1) and XA(AO) = (0). Operations on complemented sets A and B are defined in terms of their characteristic functions; for example, A - B has characteristic function X A . (1 - xe); and A < B means that X A 5 XB on a full set (in which case we say that A i s a subset of B ) . An integrable set is a complemented set A whose characteristic function is integrable; A then has measure p ( A ) = X A dp. A compact integrable set is an integrable set K = (K', KO) such that K' is compact and KO is the metric complement of Ii' in X; we usually identify K with K ' , and -K with KO. If K is a compact set and A is a complemented set such that K c A', then A' C -K and so K < A (see [l, Chapter 6, (6.6)]).

A complemented set in X is an ordered pair A

I f f is an integrable function, then the complemented set

lf 2 11 = ((2 : f(2) L 2 1 , t. : f(z) < 1 ) )

3)For reasons that are discussed fully elsewhere (see [3, Chapters 1 and 61 and [2, Section 2]), all the work in this paper can also be regarded as a contribution to both intuitionistic and recursive analysis.

Uniqueness of the Constructive Functional Calculus 52 1

is integrable for all but countably many t > 0. (In contrast to the classical situation, this theorem is difficult to prove constructively; see [l , Chapter 6, 541.) Note that [ f 2 t 1 is a complemented set, as all functions f considered in integration theory are strongly eztensional: that is, if I f ( x ) - f(y)I > 0, then e ( x , y) > 0.

We say that a compact subset K of X is strongly integrable if there exists a constant c such that the following condition holds: for each E > 0 there exists 6 > 0 such that i f f E C ( X ) , 0 5 f 5 1 , f ( x ) = 1 for all x E K', and f(z) = 0 whenever e ( x , K) 2 6 , then I J f d p - c I < E . In this case, K is integrable and p ( K ) = c (see [l, Chapter 6, (6.2)]). The following is the fundamental result about strongly integrable sets.

T h e o r e m 1. If p is a positive measure on a locally compact space X I and A an integrable set with positive measure, then for each e > 0 there exists a strongly integrable compact set Ii' c A' such that p ( A - K ) < E .

Note that although the statement of this theorem in [l, Chapter 6, (6.7)] does not mention the condition that K c A', the proof of the theorem shows that K can be constructed as a subset of A'. We shall use Theorem 1 several times in the work below.

2 A criterion for absolute continuity

Let p , p' be positive measures on a locally compact space X. We say that p' is absolute continuous relative to p if, for each set S that is both p- and p'-integrable4), and for each E > 0, there exists 6 > 0 such that if A is a p- and p'-integrable subset of S with p ( A ) < 6, then p'(A) < E (see [4, Chapter IV, (18.3) and (18.4)]). Our aim in this section is to prove

T h e o r e m 2. Let p , p' be positive measures on a locally compact space X. Then p' is absolutely continuous relative to p i f and only if the following condition holds:

(AC) For each nonnegative f in C ( X ) , and for each E > 0 , there ezists 6 > 0 such that zf g E C ( X ) , 0 5 g 5 f , and J g d p < 6 , then S g dp' < E .

In order to prove this theorem, we need a number of preliminary lemmas. L e m m a 1. Let X be a locally compact space, and let p , p' be positive measures

on X such that condition (AC) of Theorem 2 holds. Then each strongly p-integrable compact set K is strongly $-integrable.

P r o o f . Let K be a strongly p-integrable compact set. Construct a (necessarily decreasing) sequence (gn) of test functions such that for each n, 0 5 gn 5 1, gn(z) = 1 if e ( x , Ii') 5 2-"-', and gn(x) = 0 if e(z , K) 2 2-". Choose a sequence (rn) of positive numbers decreasing to 0 such that for each n,

Kn { x E x : gn(x) 2 (1 + ~ n ) - ' )

is compact and strongly integrable with respect to both p and p' (see [l, Chapter 6, (6.3)]). Then

gn+l I XK, I (1 + rn)gn and p' are positive measures on the same locally compact space X, we shall give such ')If

expressions as p-infegrable, p'-full, . ..their obvious meanings.


on the p’-full set Kn U -Kn, and Kn+l c Kn. If m > n, then

0 I XK, - XK, I gn - gm+l+ rngn on a p’-full set, so that

0 I p’(Kn) - p’(Km)

(1) I J ( g n - gm+l) dp‘ + rn J gn dp’

I J ( g n - gm+l) dp’ + rn J 91 dp’.

Since K is strongly p-integrable, J ( g n - g m + l ) dp -, 0 as m, n -, 00; taking f = g1

in (AC), we now see that J ( g n - gm+l) dp‘ + 0 as m, n + 00. Since also rn 0, it follows from (1) that (p‘(Kn));=l is a Cauchy sequence in W, and so converges to a limit 1 E R. Hence, by [I , Chapter 6, (3.9)],

is p‘-integrable, with p’(Km) = 1. But X K = X K , on the p’-full set Kk U K L , so K is p’-integrable and p ‘ ( K ) = 1.

To prove that K is strongly p’-integrable, let E > 0 and choose N such that 0 5 ~ ’ ( K N ) - p ’ ( K ) < E . Let g be a test function such that 0 I g 5 1, g ( z ) = 1 if I E K, and g ( z ) = 0 if ~ ( z , K ) 2 2 - N - 2 . Then X K I g I gN+1 5 X K ~ on a p‘-full set, so

0 I Jsdp’ - p’(K) I p w v ) - p’(K) < E . 0

L e m m a 2. Let p , p‘ be positive measures on a locally compact space X , let the complemented set A be integrable relative t o both p and p‘, and suppose that either p ( A ) > 0 or $ ( A ) > 0. Then for each E > 0 there ettsts a compact set K c A’ such that K is strongly integrable relative t o both p and p’, p ( A - K ) < E , and

P r o o f . Let v be the positive measure p + p’ on X. By (1, Chapter 7, (3.31)] a complemented set S is v-integrable if and only if it is both p- and $-integrable, in which case v ( S ) = p(S) + p’(S). Since condition (AC) clearly holds with v replac- ing p , we see from Lemma 1 that each strongly v-integrable compact set is strongly p’-integrable; similarly, each strongly v-integrable compact set is strongly p-inte-

0

L e m m a 3. Let p , p‘ be positive measures on a locally compact space X , and let g be a nonnegative test function on X . Then there etists a sequence ( x n ) of functions such that

p ’ ( A - K ) < E .

grable. The desired conclusion now follows from Theorem 1.

(i) for each n, 0 I xn 5 g , (ii) for each n, xn is both p- and p’-simple, (iii) J X n d j i + J g d p andJXndp‘- ,Jgdp’ a s n - o o .

P r o o f . Construct a compact support K for g that is strongly integrable relative to both p and p’; this is possible in view of [l, Chapter 6, (6.3)]. A simple adaption of the proof of [I, Chapter 6, (7.4)] enables us to construct a sequence ( x n ) of functions satisfying (i) and (ii), and such that for each n, 0 5 Xn 5 g 5 Xn+n-’XK on a set that

Uniqueness of the Constructive Functional Calculus 523

is both p- and p’-full. Then 0 5 s g dp - S X n dp I n - ’ p ( K ) , so S X n d p --* S g dp; 0

We say that a simple function x 3 Cr=l Ci X A , is in canonical form if X A , ‘ X A , = 0

We now arrive at the P r o o f o f T h e o r e m 2. Suppose that p’ is absolutely continuous relative to p.

Given a nonnegative element f of C ( X ) , construct a strongly p-integrable compact support K o f f and write

similarly, S Xn dp’ + s g dp’.

whenever i # j .

CY 3 (1 + 2p‘(K)) - ’ .

Let E > 0, and choose 6 > 0 such that if A is a complemented subset of K that is both p- and p’-integrable, and if p(A) < 6, then p’(A) < Q E . Consider a simple function x C ~ X A , in canonical form, such that 0 5 x _< XK and the complemented sets A’, . . . , An are both p- and p’-integrable; note that 0 I ci 5 1 for each i. Write { 1 , . . . , n} as a union of two sets P, Q such that ci > CYE if i E P , and ci < 2 a ~ if a E Q. Suppose that s x dp < a 6 ~ . Then

QEp(VicpAi) I CiEpci p(Ai) 5 SX dp < whence p(V icp Ai ) < 6 and therefore p’ (v icp Ai) < CYE. It follows that

S ~ d ~ ‘ = C i c p C ‘ 8 p‘(Ai) + CiEg ci p’(Ai)

- < p’(Vicp Ai ) + 2@&~’(ViEg Ai) < CYE + 2CYEpyK)

= a ~ ( i + 2 p y q = E .

Now consider any g E C ( X ) such that 0 5 g 5 f and S g d p < a 6 ~ . Without loss of generality, assume that 11 f I I 5 1. By Lemma 3 and [l, Chapter 6, (7.8)], there exists a sequence (xn) of functions, each of which is both p- and p‘-simple and in canonical form, such that 0 5 xn I g on a p- (and therefore p‘-) full set, s xn d p + s g dp , and Sxn dp’ --* S g dp’. For each n, since 0 5 Xn I f I XK and Sxn dp < Q ~ E , we have s xn dp‘ < E . Letting n + 00, we obtain S g dp’ I E .

Conversely, assume (AC) and consider a complemented set S that is both p- and $-integrable. Assuming, to begin with, that S is compact, construct a test function f such that 0 5 f I 1, and such that f(z) = 1 whenever e(z,S) I 1. Given E > 0, choose 6 > 0 as in condition (AC), and let A be a complemented subset of S that is both p- and p’-integrable, such that p(A) < 6. Either p’(A) < E or, as we may assume, $ ( A ) > 0. Choosing t such that 0 < t < 6-p (A) , use Lemma 2 to construct a compact set K such that K c A’ , I< is strongly integrable relative to both p and p’, p (A - I<) < 1 , and p’(A - K ) < t . Choose a test function g such that 0 5 g I 1, g(z) = 1 for all z E K , g(z) = 0 whenever e ( z , K ) 2 1/2, and s g d p - p(K) < t . Then 0 5 g 5 f and S g dp < 6, so S g dp‘ < E ; whence

p’(A) = $ ( K ) + p’(A - K ) I Sg dp’ + t < E + t .

Since t is arbitrary, we conclude that $ ( A ) 5 E . This completes the proof in the case where S is compact.


In the general case, either p’(S) < E or, as we may assume, p’(S) > 0. Then there exists a strongly $-integrable compact subset KO of S such that p’(S-KO) < ~ / 2 . By [l, Chapter 6, (6.3)], there exists a compact set K that is strongly integrable relative to both p and p’, and that includes KO. By the first part of the proof, there exists 6 > 0 such that if B is a complemented subset of I< that is both p- and p’-integrable, and if p ( B ) < 5 , then p ’ ( B ) < ~ / 2 . Now consider a complemented subset A of S that is both p- and p’-integrable, such that p(A) < 6. Since p ( A A K) < 6, we have

p’(A) = p’(A A K O ) + p’(A - KO) 5 p’ (A A I<) + p’(S - KO) 5 E / 2 + E / 2 = E .

This completes the proof in the general case. 0

The reader may wonder why we did not define absolute continuity in the usual classical fashion, using null sets. We chose our definition, which is classically equivalent to the one using null sets, because the notion produced by the latter is too weak to be of any constructive value; for example, the stronger notion is needed in the constructive version of the Radon-Nikodim theorem (see [I, Chapter 7, $31). Moreover, the equivalence, for finite measures p and p’, of absolute continuity in terms of null sets and absolute continuity as we have defined it is essentially nonconstructive. To see this, let a be a real number such that - (a 5 0), and define positive measures p , p’ on X 5 { - l , O , 1) by setting

P(f) = f(-1) + af( l ) , P’(f) = f(-l) + f(1) for each f E C ( X ) . The only set of measure 0 relative to p is {0), which also has measure 0 relative to p’. Suppose p’ is absolutely continuous, in our sense, relative to p. There exists 6 > 0 such that for each complemented subset A of X, if p(A) < 6, then $(A) < 1. It follows immediately that p ( ( 1 ) ) 2 6, so that a > 0. Thus the equivalence of the two notions of absolute continuity under discussion would entail a constructive proof of the statement

v2 E w (T(2 5 0) 3 2 > O),

which is equivalent to Markov’s principle:

If (a,) is a binary sequence such that +n (a,, = 0 ) , then there exists n such that an = 1.

Since Markov’s principle represents a form of unbounded search, most constructive mathematicians do not accept it, and regard as essentially nonconstructive any proposition from which it can be derived.

It may be suggested that the Radon-Nikodim theorem could simplify the proof of the first half of Theorem 2. However, the Radon-Nikodim theorem in its full classical form is essentially nonconstructive, as the following Brouwerian example shows. Let a be any nonnegative number, and define positive measures p , p‘ on W as follows:

P(f) = f(0) + f(a), P’(f) = f(0). Then p’ is absolutely continuous, in our sense, relative to p. Suppose there exists a p-measurable function fo on R such that for each p’-integrable function f, ffo is


p-integrable, and J f f o dp = f f dp'. Note that if a # 0, then we can construct f E C(R) such that f(0) = 1 and f ( a ) = 0, so that fo(0) = 1. In general, by considering J- 1 dp', we see that 1 -fo(O) = fo (a ) . Either fo(0) > 1/2 or fo(0) < 1. In the first case, f o ( a ) # fo(0) and so a # 0; in the second case, T(a # 0) and therefore a = 0. Thus a constructive proof of the usual Radon-Nikodym theorem would led to one of the proposition

V t E R (t = 0 or t # 0), which is equivalent to the essentially nonconstructive limit principle of omniscience:

If (a,) is a binary sequence, then either a, = 0 for all n o r else there exists n such that a, = 1.

Note that the limited principle of omniscience is false in the recursive interpretation, even with classical logic (see [3, pp. 52-53]).

3 The uniqueness of the functional calculus

Throughout this section we shall adopt the notation of the spectral theorem at the beginning of the paper. For convenience, we sketch here the construction of p and f ( T ) . For a function f of the form

C ~ l , . . . , i N N = ~ c ( ~ l , . . . , i N ) n f ' "'n$, which we shall call a general polynomial function on X, we define f(T) by

f(T) ~ ~ l , , , , , i N = o C ( i ~ j . . . , iN) . . .TiN and we set

(2) ~ ( f ) = C,"==,2-"(f(T)en,en). It is then simple to show that p extends to a positive measure on the product space X 3 n,"=l[-b,b]. For a general f in L, (p ) we construct a uniformly bounded sequence ( p , ) of general polynomial functions converging to f in the L1 (p)-norm. Then the sequence (p,(T)) converges strongly to a selfadjoint operator f ( T ) which is independent of the choice of the sequence ( p , ) and satisfies (2).

Our aim now is to prove that the functional calculus is independent of the orthonormal basis (en) and the corresponding measure p. Note that B ( H ) is the space of all bounded linear operators on H , and &(H) is the unit ball of B ( H ) :

&(H) = {T E B ( H ) : llTtll 5 1 1 ~ 1 1 for all t E H}. L e m m a 4. Let (en) be an orthonormal basis of H , llell 5 1, and E > 0. Then

there exist N and 6 > 0 such that ifA E &(H) and IIAe,II < 6 (1 5 n 5 N), then

&/2N. If IlAell < E -

P r o o f . Choose N such that (IC,"=N+l(elen)enll < &/2, and let 6 A E &(H) and llAe,II < 6 for 15 n 5 N, then


Two positive measures on the same set are said to be equivalent if each is absolutely continuous relative to the other.

P r o p o s i t i o n 1. Let ( en ) , ( e l ) be orthonormal bases of H ; T = (Tn) a sequence of commuting selfadjoint operators on H , each with bound b > 0; and p, p' the functional calculus measures for T corresponding to ( en ) , ( e l ) , respectively. Then p and p' are equivalent positive measures on X E n;''[-b, b].

P r o o f . I t will suffice to prove that condition (AC) is satisfied: for in that case, p' is absolutely continuous relative to p , by Theorem 2; so, interchanging the rdes of p and p', we see that p is absolutely continuous relative to p'. Accordingly, let f be a nonnegative element of C ( X ) ; without loss of generality, we may assume that l l f l l 5 1. Given E > 0, choose no such that C;=notl 2-" c ~ / 2 . By Lemma 4, there exist N and 6 > 0 such that if A E &(H) and lIAen)I < 6 (1 5 n 5 N), then IIAel,II < (&/2)'12 (1 5 n 5 no). Set 61 E 2-N62. If g E C ( X ) , 0 5 g 5 f, and S g dp c 6 1 , then

61 > C;=p=12-"(g(T)en 1 en) = C,"==, 2 - n ( g 1 / 2 ( ~ ) e n , g ' /2(T)en)

= C,"=l 2-nIlg'/2(T)en [ I 2 , so llg'/2(T)enll c 6 (1 5 n 5 N). Hence llg1/2(T)elII c (&/2)'12 (1 5 n 5 no) and therefore

j - 9 dP' = C,"=l 2-nll~1/2(T)e1112 = E::' 2-nl19'/2(T)e1112 + C;=no+l 2-nl19'/2(T)el 1 1 2 < C:L1 2-%/2 + C;=n,tl 2-"

< E/2+&/2 = E . 0

C o r o l l a r y 1. Under the hypotheses of Proposition 1, every p-full set is p'-full, and vice versa; Lm(p) = L,(p'); and a complemented set is p-integrable if and only if it is p'-integrable.

P r o o f . By [l, Chapter 7, (3.28)], every p-full set is p'-full, L,(p) c Lm(p') , and every p-integrable set is $-integrable. Interchanging the r6les of p and p' completes the proof. 0

We are now in a position to prove T h e o r e m 3. Let T E (Tn)F=l be a sequence of commuting selfadjoint operators,

each with bound b > 0, on a separable Hilbert space H . Then any two functional calculus measures p and p' give rise to the same class of functions f for which f ( T ) is defined; and for any given f in that class, the value of f ( T ) is independent of the functional calculus measure used in its construction.

P r o o f . The first of these conclusions is established in Corollary 1. To deal with the second, for clarity let f H f ( T ; p ) denote the functional calculus constructed using the measure p , and f H f (T;p ' ) the functional calculus constructed using p'. Given f E Lm(p), choose a uniformly bounded sequence (pn) of general polynomial functions on X G n;='.[-b, b] such that s I f - pnl dp -, 0; then (pn) converges to f


almost everywhere relative to p , and (p,(T)) converges strongly to f ( T ; p ) . Since p’ is absolutely continuous relative to p , it is a simple consequence of Corollary 1 that ( p , ) converges to f almost everywhere relative to p’. It follows from the dominated convergence theorem (see [l, Chapter 6, (8.8)]) that I f - Pnl dp’ -+ 0; so (p,(T)) converges strongly5) to f ( T ; p‘ ) , which therefore equals f ( T ; p ) .

References

[l] BISHOP, E., and D. BRIDGES, Constructive Analysis. Springer-Verlag, Berlin-Heidelberg-

[2] BRIDGES, D., and 0. DEMUTH, On the Lebesgue measurability of continuous functions

[3] BRIDGES, D., and F. Richman, Varieties of Constructive Mathematics. London Math.

[4] DINCULEANU, N., Integration on Locally Compact Spaces. Noordhoff, Leyden 1974.

New York 1985.

in constructive analysis. Bull. Amer. Math. SOC. 24 (1991), 259 - 276.

SOC. Lecture Notes 97, Cambridge University Press, London 1987.

(Received: July 15, 1993; Revised: January 31, 1994)

5)Note that the convergence of (p,(T)) to j ( T ; p ’ ) can also established using integal estimates like those in the proof of Proposition 1.

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Absolute Continuity and the Uniqueness of the Constructive Functional Calculus