Math. Scand. Vol. 79, No. 2, 1996, (263-282)

Preprint Ser. No. 5, 1994, Math. Inst. Aarhus

Randomly Weighted Series of Contractionsin Hilbert Spaces

G. PESKIR, D. SCHNEIDER, M. WEBER

Conditions are given for the convergence of randomly weighted series of contrac-

tions in Hilbert spaces. It is shown that under these conditions the series converges

in operator norm outside of a (universal) null set simultaneously for all Hilbert

spaces and all contractions of them. The conditions obtained are moreover shown

to be as optimal as possible. The method of proof relies upon the spectral lemma

for Hilbert space contractions (which allows us to imbed the initial problem into a

setting of Fourier analysis), the standard Gaussian randomization (which allows us

further to transfer the problem into the theory of Gaussian processes), and finally

an inequality due to Fernique [3] (which gives an estimate of the expectation of the

supremum of the Gaussian (stationary) process over a finite interval in terms of the

spectral measure associated with the process by means of the Bochner theorem). As

a consequence of the main result we obtain: Given a sequence of independent and

identically distributed mean zero random variables fZkgk�1 defined on (;F ; P )satisfying EjZ1j2 < 1 , and � > 1=2 , there exists a (universal) P -null setN� 2 F such that the series:

1Xk=1

Zk(!)

k�T k

converges in operator norm for all ! 2 nN� , whenever H is a Hilbert spaceand T is a contraction in H .

1. Introduction

The purpose of the paper is to investigate and establish conditions for the convergence in

operator norm of the randomly weighted series of contractions in Hilbert spaces:

(1.1)

1Xk=1

Wk(!) Tpk

where fWkgk�1 is a sequence of independent mean zero square-integrable random variablesdefined on the probability space (;F ; P ) , and T is a (linear) contraction in the Hilbert spaceH , while fpkgk�1 is a non-decreasing sequence of non-negative integers, and ! 2 . Our mainaim is to find sufficient conditions (and in this context to prove that they are as optimal as possible)

for the convergence of the series in (1.1) which is valid simultaneously for all Hilbert spaces Hand all contractions T in H . More precisely, we find conditions (see (3.1) in Theorem 3.1 and(3.1’) in Remark 3.2) in terms of the numbers pk and EjWkj2 for k � 1 , under which thereexists a (universal) P -null set N� 2 F such that the series in (1.1) converges in operator normfor all ! 2 n N� , whenever H is a Hilbert space and T is a contraction in H . (This isthe main result of the paper.) Then we specialize and investigate this result when Wk = Zk=k

�

AMS 1980 subject classifications. Primary 40A05, 40A30, 47A60, 60F25. Secondary 42A20, 42A24, 46C10, 60G15, 60G50.Key words and phrases: Randomly weighted series, independent, contraction, Hilbert space, the spectral lemma for Hilbert space contractions,Lévy’s inequality, Gaussian randomization, Gaussian process, separable, continuous in quadratic mean, stationary, the spectral measure,the spectral representation theorem, orthogonal stochastic measure, the covariance function, the Bochner theorem, the Herglotz theorem,nonnegative definite, the dilation theorem of Sz.-Nagy, Kolmogorov’s three series theorem, Szidon’s theorem on lacunary trigonometricalseries. goran@imf.au.dk

1

for k � 1 and � > 0 , where fZkgk�1 is a sequence of independent and identically distributedmean zero random variables satisfying EjZ1j2 < 1 (see Theorem 3.4), and in particular weobtain that for � > 1=2 the series:

(1.2)

1Xk=1

Zk(!)

k�T k

converges in operator norm for all ! 2 n N� , whenever H is a Hilbert space and T is acontraction in H (see Corollary 3.5). In the end (see Remark 3.6) it is shown that the conditionsobtained throughout are as optimal as possible.

The method of proof may be described as follows. First, we use the spectral lemma for Hilbert

space contractions (Lemma 2.1), and in this way imbed the initial problem about the series in (1.1)

into a setting of Fourier analysis (see [4] and [7]), which concerns expressions of the form:

(1.3) sup��

We begin by recalling a useful fact (called the spectral lemma for Hilbert space contractions)

which allows us to transfer the norm operator problems into the setting of Fourier analysis. For

this, we should first clarify that a linear operator T ( defined in a (complex) Hilbert space H ) iscalled a contraction, if kT (f)k � kfk for all f 2 H (see [2]). Given a contraction T in H andf 2 H , denote Pn(f) =

T n(f); f

�for n � 0 , and put Pn(f) = P�n(f) for n < 0 . Then

the sequence fPn(f)gn2Z is non-negative definite�P

k

Pl zk�zlPk�l(f) � 0 , for all zk 2 C

with jkj � N and N � 1 �, see [5] (p.94-95). Thus by the Herglotz theorem [5] there exists afinite positive measure �f on B(]� �; �]) ( called the spectral measure of f ) such that:

(2.1)

T n(f); f

�=

Z ���

ein��f (d�)

for all n � 0 . From this fact, by induction in degree of the polynomial, one might deduce thefollowing remarkable inequality. (This proof was communicated to us by M: Wierdl.)

Lemma 2.1 (The spectral lemma)

If T is a contraction in a Hilbert space H , and f is an element from H with the spectralmeasure �f , then the inequality is satisfied:

(2.2)

P (T )(f)

� �Z �

��

��P (ei�)��2 �f (d�)�1=2whenever P (z) =

PNk=0 akz

k is a complex polynomial of degree N � 0 .Proof. The proof is carried out by induction on the degree N of the polynomial P (z) . If

N = 0, then we have equality in (2.2), since it follows from (2.1) that kfk2 = �f (] � �; �]) .Suppose that the inequality (2.2) is true for N � 1 � 0 . Denote Q(z) = PNk=1 ak zk andR(z) =

PNk=1 ak z

k�1 . Then by (2.1) we find:

(2.3) kP (T )(f)k2 = k a0f + Q(T )(f) k2 = ja0j2kfk2 +

a0f;Q(T )(f)

�+

Q(T )(f); a0f

�+ kQ(T )(f)k2 = ja0j2kfk2 +

R ��� a0Q(ei�) �f (d�)

+R ��� a0 Q(e

i�) �f (d�) + kQ(T )(f)k2 .Since T is a contraction, then by the assumption we get:

(2.4) kQ(T )(f)k2 = kT �R(T )�(f)k2 � kR(T )(f)k2� R ��� jR(ei�)j2 �f (d�) = R ��� jQ(ei�)j2 �f (d�) .

From (2.3) and (2.4) we conclude:

kP (T )(f)k2 � R ��� �ja0j2 + a0Q(ei�) + a0Q(ei�) + jQ(ei�)j2� �f (d�)=

R ��� ja0 + Q(ei�)j2 �f (d�) =

R ��� jP (ei�)j2 �f (d�) .

It should be noted from (2.4) in Lemma 2.1 that if T is an isometry in H ( kT (f)k = kfkfor f 2 H ), then we have equality in (2.2). (It also follows more directly by (2.1).) It allows us to

3

deduce the inequality (2.2) in a straightforward way by using the (oldest and best known) dilation

theorem of Sz.-Nagy (see Theorem 1 in [9] (p.2)): If T is a contraction in a Hilbert space H ,then there exists a Hilbert space K and a unitary operator U in K such that H is a subspaceof K and Tn(f) = Z(Un)(f) for all n � 0 and all f 2 H , where Z is the orthogonalprojection from K into H . ( It remains to apply the equality in (2.2) to the isometry U , andthen use the preceding identity and the fact that kZk � 1 . )

However, if kTk < 1 , then the error appearing in the estimate (2.2) may be noticeably large.To see this more explicitly, take for instance P (z) = zn with n � 1 . Then the left-hand sidein (2.2) equals kT n(f)k which is bounded by kTknkfk , and thus arbitrarily small when nincreases. On the other hand, the right-hand side in (2.2) equals kfk , thus being arbitrarily largeas f runs through H . We will come back to this sort of analysis later on in section 4.

Our next aim is to display an inequality due to Fernique [3] which is shown to be at the basis

of our results in the next section. For this reason we shall first recall the (wide sense) stationary

setting upon which this result relies (see [1]).

Let G = fG�g�2R be a (wide sense) stationary process defined on the probability space(;F ; P ) and taking values in the set of complex numbers C . Thus, we have:(2.5) EjG�j2 < 1(2.6) E(G�) = E(G0)

(2.7) Cov(G�+h; G�+h) = Cov(G�; G�)

for all � , � , h 2 R . As a matter of convenience, we suppose:(2.8) E(G�) = 0

for all � 2 R . Thus the covariance function of G is given by:(2.9) R(�) = E(G�G0)

for all � 2 R. By the Bochner theorem there exists a finite positive measure � on B(R) such that:

(2.10) R(�) =

Z 1�1

ei�x�(dx)

for all � 2 R . The measure � is called the spectral measure of G . The spectral representationtheorem states if R is continuous (which is equivalent to the fact that G is continuous in quadraticmean), then there exists an orthogonal stochastic measure Z on � B(R) such that:

(2.11) G� =

Z 1�1

ei�xZ(dx)

for all � 2 R . The fundamental identity in this context is as follows:

(2.12) E

���� Z 1�1 '(x) Z(dx)����2 = Z 1�1 ��'(x)��2�(dx)

whenever ' : R ! C belongs to L2(�) . Hence we find:

4

(2.13) E��G��G���2 = 2Z 1

�1

�1�cos �(���)x�� �(dx)

for all � , � 2 R . The result may be now stated as follows.

Lemma 2.2 (Fernique)

Let G = fG�g�2R be a stationary mean zero Gaussian process with values in R and thespectral measure � . Suppose that G is separable and continuous in quadratic mean. Then thereexists a (universal) constant K > 0 ( which does not depend on the Gaussian process G itself ),such that the inequality is satisfied:

(2.14) E�

sup0���1

G�

�� K

����Z 1�1

(x2^ 1) �(dx)���1=2 + Z 1

0

����]ex2;1[)��1=2dx� .Proof. It follows from (0.4.5) in Theorem 0.4 in [3] upon taking c1 = b1 = 1 and ck = bk = 0

in (0.4.4) for k � 2 .

In order to make use of the inequality (2.14) we shall denote Ik = [k; k+1] for k 2 Z . LetI � R be any bounded interval, then I � [k2AIk for some finite A � Z . Let KI denote thenumber of elements in A . Then by the separability, stationarity, and symmetry of the Gaussianprocess G = fG�g�2R , we easily obtain:(2.15) E

�sup�2I

jG�j�= E

�maxk2A

sup�2Ik

jG�j��Xk2A

E�sup�2Ik

jG�j�= KI E

�sup

0���1jG�j

�� KI

�EjG0j + 2E

�sup

0���1G�

��.

We will apply Lemma 2.2 with (2.15) in the proof of Theorem 3.1 (next section) to the following

Gaussian process (obtained by the randomization procedure described below):

(2.16) G�(!0; !00) =

NXk=1

Wk

�g0k(!

0) cos(pk�) + g00k(!00) sin(pk�)

�with � 2 R , (!0; !00) 2 0g

00g , W1; . . . ;WN 2 R , and 0 � p1 � . . . � pN with N � 1 . Hereg0 = fg0kgk�1 and g00 = fg00kgk�1 are (mutually independent) sequences of independent standardGaussian ( � N(0; 1) ) random variables defined on the probability spaces (0g;F 0g; P 0g) and(00g ;F 00g ; P 00g ) respectively. It is easily verified that the orthogonal stochastic measure associatedwith G = fG�g�2R is given by:

(2.17) Z�(!0; !00);�

�=

NXk=1

Wk g0k(!

0)��f�pkg(�)+�fpkg(�)

�=2

+ iNXk=1

Wk g00k(!

00)��f�pkg(�)��fpkg(�)

�=2

for (!0; !00) 2 0g

00g and � 2 B(R) . From (2.12) and (2.17) it follows easily by independencethat the spectral measure of G is given by:

5

(2.18) �(�) =NXk=1

jWkj2��f�pkg(�)+�fpkg(�)

�=2

for � 2 B(R) . Having this expression in mind, one might observe that the right-hand side ofinequality (2.14) (applied to the Gaussian process (2.16)) gets a rather explicit form which is easily

estimated in a rigorous way. This will be demonstrated in more detail in the proof of Theorem 3.1.

In the remaining part of this section we describe the procedure of Gaussian randomization. It

allows us to imbed the problem under consideration (which involves expressions (1.3) and appears

by applications of (2.2)) into the theory of Gaussian processes. It will be used below in the proof

of our main result (Theorem 3.1). It should be observed that the Gaussian process in (2.16) appears

precisely after performing Gaussian randomization as described in the next lemma.

Lemma 2.3 (Gaussian randomization)

Let W = fWkgk�1 be a sequence of independent mean zero random variables defined on theprobability space (;F ; P ) , let g0 = fg0kgk�1 and g00 = fg00kgk�1 be (mutually independent andindependent from W ) sequences of independent standard Gaussian ( �N(0; 1) ) random variablesdefined on the probability spaces (0g;F 0g; P 0g) and (00g ;F 00g ; P 00g ) respectively, and let pk � 1 benon-negative integers for k � 1 . Then the inequality is satisfied:

(2.19) E

sup

��

understood to be independent from both W and W 0 . Then f"k(Wk�W 0k)gk�1 and fWk�W 0kgk�1are identically distributed for any given and fixed choice of signs "k = �1 with k � 1 , andsince Wk’s are of mean zero, we get:

(2.21) E

sup

��

(2.25) E

�max1�k�n

kSkkp�� 2E

�kSnkp

�for all t > 0 , all 0 < p < 1 , and all n � 1 .

3. The main results

In this section we present the main results of the paper. We begin by the next fundamental

theorem which should be read together with Remark 3.2 following it. In the subsequent part of

this section we pass to a more elaborate analysis of this result.

Theorem 3.1

Let W = fWkgk�1 be a sequence of independent mean zero random variables defined onthe probability space (;F ; P ) , and let fpkgk�1 be a non-decreasing sequence of non-negativeintegers ( with p1 > 1 ). Suppose that there exist integers 0 := n0 < n1 < n2 < . . . , such thatthe following condition is satisfied:

(3.1)

1Xi=0

qlog(pni+1)

� ni+1Xk=ni+1

E��Wk��2�1=2

!< 1 .

Then there exists a (universal) sequence of P -integrable random variables M = fMNgN�1 definedon (;F ; P ) which converges to zero P -a.s. and in P -mean such that for any Hilbert space Hand any contraction T in H we have:

(3.2) supR>nN

RXk=nN+1

Wk(!)Tpk

� MN (!)

for all ! 2 and all N � 1 . In particular, there exists a (universal) P -null set N� 2 Fsuch that the series:

(3.3)

1Xk=1

Wk(!) Tpk

converges in operator norm for all ! 2 n N� , whenever H is a Hilbert space and T is acontraction in H .

Proof. Given R > nN � 1 for some N � 1 , there exists (a unique) lR � 0 such that:(3.4) nN+lR < R � nN+lR+1 .Let f 2 H be given and fixed, and let �f be the spectral measure of f . Then by the triangleinequality and the spectral lemma (see (2.2) in Lemma 2.1) we get:

(3.5)

RXk=nN+1

Wk(!) Tpk(f)

� N+lR�1Xi=N

ni+1Xk=ni+1

Wk(!) Tpk(f)

+

RXk=nN+lR+1

Wk(!) Tpk(f)

� 1Xi=N

ni+1Xk=ni+1

Wk(!) Tpk(f)

8

+1Xj=0

supnN+j

+1Xj=0

sup

nN+j

for the given and fixed ! 2 and i � 0 . From (3.10), (3.11) and (3.12) we obtain the estimate:

(3.13) E

sup

��

explicitly, we take rk = k� with � > 0 , but other choices are possible as well. The result of

Theorem 3.1 may be then refined as follows.

Theorem 3.4

Let Z = fZkgk�1 be a sequence of independent and identically distributed mean zero randomvariables defined on the probability space (;F ; P ) such that EjZ1j2 1 ) satisfying the following condition:

(3.17)

1Xi=1

qlog (pni+1)

(ni) ��1=2< 1

for some integers 1 � n1 < n2 < . . . , and some � > 1=2 . Then there exists a (universal) sequenceof P -integrable random variables M = fMNgN�1 defined on (;F ; P ) which converges to zeroP -a.s. and in P -mean such that for any Hilbert space H and any contraction T in H we have:

(3.18) supR>nN

RXk=nN+1

Zk(!)

k�T pk

� MN (!)

for all ! 2 and all N � 1 . In particular, there exists a (universal) P -null set N� 2 Fsuch that the series:

(3.19)

1Xk=1

Zk(!)

k�T pk

converges in operator norm for all ! 2 n N� , whenever H is a Hilbert space and T is acontraction in H .

Proof. We apply Theorem 3.1 with Wk = Zk=k� for k � 1 . By the identical distribution

of Zk’s we see that in order to verify condition (3.1) we have to estimate the expression:

(3.20)

1Xi=0

qlog(pni+1)

� ni+1Xk=ni+1

E��Wk��2�1=2

!

=�EjZ1j2

�1=2 1Xi=0

qlog(pni+1)

� ni+1Xk=ni+1

1

k2�

�1=2 !.

For this, a simple integral comparison shows that:

(3.21)

ni+1Xk=ni+1

1

k2��Z ni+1ni

1

x2�dx =

1

2��1�

1

(ni)2��1� 1

(ni+1)2��1

�� 1

2��1 �1

(ni)2��1

for all i � 1 . Inserting (3.21) into (3.20), and using (3.17), we get:

(3.22)

1Xi=1

qlog(pni+1)

� ni+1Xk=ni+1

E��Wk��2�1=2

!

12

��EjZ1j22��1

�1=2 1Xi=1

plog (pni+1)

(ni) ��1=2< 1 .

Thus (3.1) is satisfied, and (3.18) and (3.19) follow from (3.2) and (3.3). This completes the proof.

Corollary 3.5

Let Z = fZkgk�1 be a sequence of independent and identically distributed mean zero randomvariables defined on the probability space (;F ; P ) such that EjZ1j2 1=2 bea given number. Then there exists a (universal) P -null set N� 2 F such that the series:

(3.23)

1Xk=1

Zk(!)

k�T k

converges in operator norm for all ! 2 n N� , whenever H is a Hilbert space and T is acontraction in H .

Proof. We apply Theorem 3.4 with pk = k and ni = 2i . For this, note that we have:

1Xi=1

plog (pni+1)

(ni) ��1=2=plog 2

1Xi=1

pi+1

2 i(��1=2)< 1 .

Thus condition (3.17) is fulfilled, and (3.23) follows from (3.19). This completes the proof.

Remark 3.6

The conclusion of Theorem 3.4 (and Corollary 3.5) fails for 0 < � � 1=2 , unless Zk = 0P -a.s. for all k � 1 . Indeed, if the conclusion would be true with � = 1=2 for instance, thenupon taking T = I we would have:

1X

k=1

Zk(!)pk

T pk(f)

= ���� 1X

k=1

Zk(!)pk

���� � kfk < 1for P -a.s. ! 2 . Thus by Kolmogorov’s three series theorem we would obtain:

1Xk=1

Var

�Zkpk1fjZkj�C

pkg

�=

1Xk=1

1

kVar�Zk1fjZkj�C

pkg�< 1

for all C > 0 . Hence (with C = 1) we would easily get:

Var�Z11fjZ1j�

pkg�! 0

as k !1 , so that we could conclude Z1=0 P -a.s. The case 0 < � < 1=2 is treated in exactlythe same manner. This completes the proof of the claim.

It should also be observed by triangle inequality that the result of Theorem 3.4 (and Corollary

3.5) follows easily either for � > 1 or kTk < 1 ( provided that � > 0 and pk’s do not tendto infinity too slowly when k ! 1 ). This shows that Theorem 3.4 (with Corollary 3.5) treatsessentially the cases (up to the simultaneity) where kTk = 1 and 1=2 < � � 1 .

13

4. Some remarks on the proof

In the remaining part of the paper we shortly analyze optimality of the inequalities used in

the proof of our main result in Theorem 3.1. Our motivation for this direction relies upon two

facts. Firstly, we feel that the method of proof presented is instructive and can be modified to treat

similar questions. Secondly, we see from Remark 3.6 that the conditions obtained throughout are

as optimal as possible, but up to the fact that the assertions implied are valid simultaneously for all

Hilbert spaces and all contractions of them. Thus the information to be given below might clarify

the scope and lead to an improvement when something similar (in such a generality) is not required.

The first inequality in (3.5) is a consequence of the triangle inequality and reflects our basic

idea of passing to the blocks of random elements under consideration. Since the length of the

blocks is not fixed and may vary, the error appearing in (3.5) can be made arbitrarily small.

The third inequality in (3.5) may contain a noticeable large error ( when kTk�for k � 1 with some �> 1 ( taken to be stricly less than p1 > 1 ). Then pk >�k , and thuslog(pk)>k log(�) for all k � 1 . Hence from (4.2) we easily get:

(4.3)

ni+1Xk=ni+1

EjWkj � 1p�

qlog(pni+1)

� ni+1Xk=ni+1

EjWkj2�1=2

.

The inequality (4.3) shows that condition (3.1) in Theorem 3.1 implies that:

(4.4) E

� 1Xk=1

jWkj�

=1Xk=1

EjWkj < 1 .

This clearly indicates that the result of Theorem 3.1 (and Theorem 3.4 with Corollary 3.5) does

not go beyond a triangle inequality argument in the case when the sequence fpkgk�1 is lacunary.We think that this fact is by itself of theoretical interest.

To explore the Gaussian randomization inequality (3.9) (see Lemma 2.3), we could note that by

14

(3.27)+(3.31) in [8] (together with Lévy’s inequality and a passage to Wk�W 0k , where fW 0kgk�1is an independent copy of fWkgk�1 , independent of fg0kgk�1 as well) we get the inequality:

(4.5) E

sup

��