The existence of unitary dilations makes it possible to study arbitrary contractions on a Hilbert space using the tools of harmonic analysis. The first edition of this book was an account of the progress done in this direction in 1950-70. Since then, this work has influenced many other areas of mathematics, most notably interpolation theory and control theory. This second edition, in addition to revising and amending the original text, focuses on further developments of the theory, including the study of two operator classes: operators whose powers do not converge strongly to zero, and operators whose functional calculus (as introduced in Chapter III) is not injective. For both of these classes, a wealth of material on structure, classification and invariant subspaces is included in Chapters IX and X. Several chapters conclude with a sketch of other developments related with (and developing) the material of the first edition.

generalization of the notion of dilation for systems of operators. Let A = {Aω }ω ∈Ω be a commutative system of bounded operators on the space H. A system B = {Bω }ω ∈Ω of bounded operators on a space K is called a dilation of the system A , if (i) H is a subspace of K, (ii) the system B is commutative, and (iii) Anω11 · · · Anωrr = pr Bnω11 · · · Bnωrr (ni ≥ 0; i = 1, . . . , r) for every finite set of subscripts ωi ∈ Ω . The dilation B is said to be isometric, unitary, and so on, when it

T (0) = I and T (−n) = T (n)∗ . According to the general definition, T (n) is positive definite on Z if ∞ ∞ (8.1) ∑ ∑ (T (n − m)hn, hm ) ≥ 0 n=−∞ m=−∞ for every two-way sequence {hn }∞ −∞ of elements of H, which is finitely nonzero, that is, such that hn = 0 for a finite set of subscripts only. For such a sequence we can choose an integer a so that the sequence {h′ν }∞ −∞ defined by h′ν = hν +a satisfies h′ν = 0 for ν < 0. When ν = n − a and µ = m − a we have ν − µ = n − m, and therefore (8.1)

p and, if 0 < p < ∞, it is also the strong limit of u(reit ): 2π 0 |u(eit ) − u(reit )| p dt → 0 (r → 1 − 0); (1.4) if p ≥ 1 this implies weak convergence, that is, 2π 0 f (t)u(reit ) dt → 2π 0 f (t)u(eit ) dt (r → 1 − 0) (1.5) for an arbitrary f ∈ Lq ((1/p) + (1/q) = 1). In particular, the Cauchy and Poisson formulas hold for the limit functions; thus for u ∈ H p (p ≥ 1) we have 1 2π and 1 2π 2π 0 2π 0 u(0) (n = 0), 0 (n = 1, 2, . . .) eint u(eit ) dt = P(ρ , τ − t)u(eit ) dt =

circle |λ | = 12 . Thus (1.29) is satisfied if f (t) = eiν t and consequently also if f (t) is an arbitrary trigonometric polynomial. Every function f ∈ L1 can be approximated in the metric of L1 , as closely as we wish, by trigonometric polynomials, therefore we conclude that (1.29) is satisfied for every f ∈ L1 . Proposition 1.5. Let {uα } be a finite or infinite system of inner functions and let v be their largest common inner divisor. For every function f ∈ L1 such that uα (eit ) f (t) ∈ L1+0

is, m2 (λ ) is an inner function. Now from Theorem II.2.1 it follows that H 2 ⊖ H2 = M+ (L) = m2 H 2 . But H 2 ⊖ H2 = H1 ⊕ mH 2 , so mH 2 ⊂ m2 H 2 . Consequently m = m2 m1 where m1 ∈ H 2 . Because m and m2 are inner so must be m1 . We have thus obtained that H1 = m2 H 2 ⊖ mH 2 = m2 (H 2 ⊖ m1 H 2 ), concluding the proof. Further important examples of contractions of class C0 are studied later, particularly in Chaps. VIII and X. 2. We now state an important result. Proposition 4.4. For every