In this article we describe properties of unbounded operators related to evolutionary problems. It is a survey article which also contains several new results. For instance we give a characterization of cosine functions in terms of mild well-posedness of the Cauchy problem of order 2, and we show that the property of having a bounded H°°-calculus is stable under rank-1 perturbations whereas the property of being associated with a closed form and the property of generating a cosine function are not. Introduction. Many second-order elliptic differential operators can be realized on Lspaces by means of closed quadratic forms (see [Ev, Chapter 6]). Typically the space is L(0) where O is an open subset of R^, and the domain V of the form is a Sobolev space such as H(£L) or HQ(SI). The domain of the associated operator A is more difficult to identify but it often happens that the domain of the square root of A coincides with the form domain V. Kato [Kat] initiated a study of closed forms and the associated operators as an abstract approach to such differential operators. If one takes a fixed inner product, one can characterize the operators which are associated with forms by means of a condition that the numerical range of the operators should be contained in a suitable sector. If one allows changes of the inner product the class of operators associated with forms becomes much wider, so it is useful to study their properties modulo similarity. The notion of a bounded i7°°-calculus of a sectorial operator was introduced by Mcintosh [McI3] in work on singular integral operators but it has subsequently proved to be very important for questions of maximal regularity in evolution equations (see [KW] for an extended survey). Not every sectorial operator on Hilbert space has such a calculus, 2000 Mathematics Subject Classification: Primary 47A60; Secondary 35L90, 47A07, 47D09. The paper is in final form and no version of it will be published elsewhere. [17] © Instytut Matematyczny PAN, 2007 18 W. ARENDT AND C. J. K. BATTY and remarkably it turns out that the class of operators on Hilbert space which have a bounded H°°-calculus on a sector of angle less than n/2 is exactly the same as the class of operators associated with forms, modulo similarity (Theorem 2.5). Cosine functions were first studied by Fattorini (see [Fa3]) and Kisyriski [Ki] as the second-order analogues of Co-semigroups. Indeed the second-order Cauchy problem u"(t) = Au(t) (t > 0) is well-posed if and only if A generates a cosine function (see Theorem 5.3). A remarkable recent result of Haase [Hal] and Crouzeix [Cr] is that generators of cosine functions on Hilbert space can be characterized by a condition that the numerical range, with respect to some inner product, is contained inside a parabola (Theorem 5.11). Perturbation theory is an important tool for studying differential operators, where more complicated operators may be regarded as perturbations of simpler operators. Abstract perturbation theory may then allow a more general case to be reduced to a simpler case. It is standard to regard the lower-order terms of a differential operator as a perturbation of the principal part A which is relatively bounded with respect to a fractional power of A. Here we are interested more in A-bounded perturbations which are of finite rank or relatively compact. In this article we describe some of the connections between these topics. The emphasis is on Hilbert spaces but we state results for Banach spaces where appropriate. The article is mostly a survey of some known results but it includes some new results. For example, we show that bounded iT°°-calculus is stable under A-bounded perturbations of finite rank (Theorem 4.1), but association with a form, for a fixed scalar product, is not (Theorem 3.8). Generation of a cosine function is also not stable under these perturbations (Theorem 5.9) but we refer to [AB] for the proof. We do not attempt to give a complete survey of any of the individual topics or to give a full historical account, and broader recent surveys may be found in [Ar] and [KW]. 1. Forms. Let H, V be complex Hilbert spaces such that V^H, i.e., V is continuously d embedded into H with dense image. Let a : V xV C b e a continuous sesquilinear form which is closed, i.e., (1.1) Rea(u,U) + (J(U\U)H > ®\\u\\y (u € V) holds for some a > 0, u G M. Here ( | )# denotes the scalar product of H. We call V the domain of the form. We can associate with a an operator A on H by D(A) = {u e V : there exists v £ H such that a(u, (p) = (v \ cp)n for all tp e V}, An — v. We write A ~ a and say that A is associated with a. More precisely, we may write A ~ a on (iJ, ( | )H)> In this situation it is always the case that D(A) is dense in H and —A generates a holomorphic Co-semigroup T on H. If to = 0, i.e., if the form a is coercive, then the semigroup T is exponentially stable. We refer to [Kat, Chapter VI] for the general theory of closed forms and the associated operators. FORMS, FUNCTIONAL CALCULUS, COSINE FUNCTIONS 19 E X A M P L E 1 . 1 (The Laplacian with Dirichlet boundary conditions). Let Q C be open, if = L(Q) with the usual inner product, V = HQ(CI) and .a(u, v) = fn Vn • Vv dx. Then a is a closed form. Let A ~ a. It is not difficult to see that where Au is understood in the sense of distributions [ABHN, Theorem 7.2.1]. D E F I N I T I O N 1 . 2 . Let A be an operator on (if, ( | )#). We say that is associated with a form if there exists a closed form a such that A ~ a. It is easy to characterize those operators which are associated with a form on (if, (| )#). For 0 0, |a| 0. Consider the equivalent scalar product on £(fl). Then the operator mA is associated with a on ( | )i). Here we use m also to denote the bounded operator of multiplication by the function rait is natural to ask which operators A on H are associated with a closed form on (if, ( | )i) whatever the equivalent scalar product is. This occurs if A is bounded because we may take the form a(ii, v) = (Au | v)i. Matolcsi has recently answered the question by showing that only bounded operators have this property. T H E O R E M 1.5 ([Mat]). Let A be an operator on i f . Assume that for each equivalent scalar product ( | )i on if there exists a closed form a on H such that A ~ a on (if, ( | )i). Then A is bounded. Thus, given an unbounded operator A, we can always find a bad scalar product so that W(A) (with respect to this scalar product) is not contained in any sector. Another natural and more interesting question is which operators are associated with a closed form with respect to some equivalent scalar product. D E F I N I T I O N 1.6. A densely defined operator A on if is called form-similar if there exists an equivalent scalar product ( | )i on_H and a closed form a on H such that A ~ a on D(A) = {ue HQ ( O ) : Au G L 2 ( F T ) } , Au = A u , W(A) = {(AX,X)H : Z G D(A)9 \\X\\H = 1}.