Math. Scand. Vol. 79, No. 2, 1996, (263-282) Preprint Ser. No. 5, 1994, Math. Inst. Aarhus
Randomly Weighted Series of Contractions in Hilbert Spaces G. PESKIR, D. SCHNEIDER, M. WEBER Conditions are given for the convergence of randomly weighted series of contractions 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 Zk k1 defined on ( ; ; P ) satisfying E Z1 2 < , and > 1=2 , there exists a (universal) P -null set N3 such that the series:
2F
j j
1
f g
X Z (!) T 1
k=1 converges in operator norm for all ! and T is a contraction in H .
k
k
F
k
2 n N 3 , whenever
H
is a Hilbert space
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)
X1 W (!) T k
pk
k=1 where fWk gk1 is a sequence of independent mean zero square-integrable random variables defined on the probability space ( ; F ; P ) , and T is a (linear) contraction in the Hilbert space H , while fpk gk1 is a non-decreasing sequence of non-negative integers, and ! 2 . Our main aim 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 H and 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 E jWk j2 for k 1 , under which there exists a (universal) P -null set N 3 2 F such that the series in (1.1) converges in operator norm for all ! 2 n N 3 , whenever H is a Hilbert space and T is a contraction in H . (This is the 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, L´evy’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 trigonometrical series.
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1
for k 1 and > 0 , where fZk gk1 is a sequence of independent and identically distributed mean zero random variables satisfying E jZ1 j2 < 1 (see Theorem 3.4), and in particular we obtain that for > 1=2 the series:
1 Z (! ) k k (1.2) T k k=1 converges in operator norm for all ! 2 n N 3 , whenever H is a Hilbert space and T is a contraction in H (see Corollary 3.5). In the end (see Remark 3.6) it is shown that the conditions X
obtained 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: N X ip k (1.3) sup Wk e 0
