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THE LIFSHITS–KREIN TRACE FORMULA AND OPERATOR LIPSCHITZ FUNCTIONS V.V.PELLER 6 1 0 Abstract. Wedescribethemaximalclassoffunctionsf ontherealline,forwhichthe 2 Lifshitz–Krein trace formula trace(f(A)−f(B)) = R f′(s)ξ(s)ds holds for arbitrary R n self-adjoint operators A and B with A−B in the trace class S1. We prove that this a class of functions coincide with theclass of operator Lipschitz functions. J 4 ] A Contents F 1. Introduction 1 . h 2. Double operator integrals and Schur multipliers 3 t a 3. Differentiability in the Hilbert–Schmidt norm 5 m 4. Schur multipliers and Haagerup tensor products 5 [ 5. The trace of double operator integrals 6 1 6. The Lifshits–Krein trace formula for v arbitrary operator Lipschitz functions 7 0 References 8 9 4 0 0 . 1 0 6 1. Introduction 1 : v i The purpose of this paper is to describe the class functions, for which the Lifshitz– X Krein trace formula holds. The Lifshitz–Krein trace formula plays a significant role in r a perturbation theory. It was discovered by Lifshits [L] in a special case and by Krein [Kr] in the general case. This formula allows one to compute the trace of the difference f(A)−f(B) of a function f of an unperturbed self-adjoint operator A and a perturbed self-adjoint operator B provided the perturbation B−A belongs to trace class S . M.G. 1 Krein proved that for each such pair there exists a unique function ξ in L1(R) such that for every function f whose derivative is the Fourier transform of and L1 function, the operator f(A)−f(B) belongs to S and the following trace formula holds: 1 trace f(A)−f(B) = f′(s)ξ(s)ds (1.1) R Z (cid:0) (cid:1) theauthor is partially supported by NSFgrant DMS 1300924. 1 (see [Kr]). The function ξ is called the spectral shift function associated with the pair (A,B). Clearly, the right-hand side of (1.1) makes sense for arbitrary Lipschitz function f. In this connection Krein asked the question of whether it is true that for an arbitrary Lipschitz function f, the operator f(A)−f(B) is in S and trace formula (1.1) holds. It 1 turns out that this is false. In[F]Farforovskaya gave anexampleof self-adjoint operators A and B with A−B ∈ S and a Lipschitz function f such that f(A)−f(B)6∈ S . 1 1 Later it was shown in [Pe2] and [Pe3] that formula (1.1) holds, whenever A and B are self-adjoint operators with A−B ∈ S and f belongs to the Besov space B1 (R) 1 ∞,1 (we refer the reader to [Pee] and [Pe4] for an introduction to Besov classes). Necessary conditions are also obtained in [Pe2] and [Pe3]. In particular, it was shown in [Pe2] and [Pe3] that if f(A)−f(B) ∈ S whenever A and B are self-adjoint operators with 1 A−B ∈ S ,thenf locally belongstotheBesovspaceB1 (R). Notethatthosenecessary 1 1,1 conditions were deduced from the description of trace class Hankel operators [Pe1] (see also [Pe4]). Themain objective of this paperis to describetheclass of functions f, for which trace formula (1.1) holds for arbitrary self-adjoint operators A and B with A−B ∈ S . 1 It is well known (see e.g. [AP2]) that for a function f on R, the following properties are equivalent: (i) there exists a positive number C such that kf(A)−f(B)k≤ CkA−Bk (1.2) for all bounded self-adjoint operators A and B; (ii) there exists a positive number C such that inequality (1.2) holds, whenever A and B are (not necessarily bounded) self-adjoint operators such that A−B is bounded; (iii) there exists a positive number C such that kf(A)−f(B)kS ≤ CkA−BkS (1.3) 1 1 for all bounded self-adjoint operators A and B with A−B ∈ S ; 1 (iv) there exists a positive number C such that inequality (1.3) holds,, whenever A and B are (not necessarily bounded) self-adjoint operators such that A−B ∈ S ; 1 (v) f(A)−f(B) ∈ S , whenever A and B are (not necessarily bounded) self-adjoint 1 operators such that A−B ∈S . 1 Note that the minimal value of the constant C is the same in (i)–(iv). Functions satisfying (i) are called operator Lipschitz. We denote by OL(R) the space of operator Lipschitz functions on R. For f ∈ OL(R), we define its quasi norm kfk as OL the infimum of all constants C, for which inequality (1.2) holds. In other words, kf(A)−f(B)kS kfk = sup 1 : A and B are self-adjoint, A−B is bounded OL kA−BkS (cid:26) 1 (cid:27) kf(A)−f(B)k = sup : A and B are self-adjoint, A−B ∈ S . 1 kA−Bk (cid:26) (cid:27) It was shown in [JW] that operator Lipschitz functions are differentiable everywhere on R. Note that this implies that the function x 7→ |x| is not operator Lipschitz, the fact established earlier in [Mc] and [Ka]. On the other hand, an operator Lipschitz function 2 doesnothavetobecontinuously differentiable; inparticular, thefunctionx 7→ x2sinx−1 is operator Lipschitz, see [KS]. For a differentiable function f on R, we consider the divided difference Df defined by f(x)−f(y), x 6= y (Df)(x,y) = x−y (1.4) ( f′(x), x = y. It turns out (see e.g., [AP2]) that a differentiable function on R is operator Lipschitz if and only if the divided difference Df is a Schur multiplier (see §2) for the definition. Themain purposeof this paper is to prove that the condition f ∈ OL(R) is not only a necessary condition for the Lifshits–Krein trace formula (1.1) to hold for arbitrary self- adjoint operators A and B with A−B ∈ S , but is also sufficient. This will be proved 1 in §6. In § 2 We define double operator integrals and Schur multipliers. In § 3 we state a result of [KPSS] on the differentiability of the function t 7→ f(A+tK)−f(A) in the Hilbert–Schmidt norm. We state a characterization of the space of Schur multipliers in terms of Haagerup tensor products in § 4. Finally, in § 5 we obtain a formula for the trace of double operator integrals. 2. Double operator integrals and Schur multipliers Double operator integrals appeared in the paper [DK] by Daletskii and S.G. Krein. Later the beautiful theory of double operator integrals was created by Birman and Solomyak in [BS1], [BS2], and [BS4]. Let (X,E ) and (Y ,E ) be spaces with spectral measures E and E on a Hilbert 1 2 1 2 space H and let Φ be a bounded measurable function on X × Y . Double operator integrals are expressions of the form Φ(x,y)dE (x)T dE (y). (2.1) 1 2 XZ YZ Birman and Solomyak starting point is the case when T belongs to the Hilbert–Schmidt class S . For a bounded measurable function on Φ on X ×Y and an operator T of 2 class S , consider the spectral measure E whose values are orthogonal projections on 2 the Hilbert space S , which is defined by 2 E(L ×∆)T = E (L )TE (∆), T ∈ S , 1 2 2 L and ∆ being measurable subsets of X and Y . It was shown in [BS5] that E extends to a spectral measure on X ×Y . For a bounded measurable function Φ on X ×Y , the double operator integral (2.1) is defined by Φ(x,y)dE (x)T dE (y) d=ef ΦdE T. 1 2   XZ YZ XZ×Y 3   Clearly, (cid:13) Φ(x,y)dE1(x)T dE2(y)(cid:13) ≤ kΦkL∞kTkS2. (cid:13)(cid:13)XZ YZ (cid:13)(cid:13)S2 (cid:13) (cid:13) If (cid:13) (cid:13) (cid:13) (cid:13) Φ(x,y)dE (x)T dE (y) ∈ S 1 2 1 XZ YZ for every T ∈ S , we say that Φ is a Schur multiplier of S associated with the spectral 1 1 measures E and E . We denote by M(E ,E ) the space of Schur multipliers of S 1 2 1 2 1 with respect to E1 and E2. The norm kΦkM(E1,E2) of Φ in the space M(E1,E2) is, by definition, the norm of the linear transformer T 7→ Φ(x,y)dE (x)T dE (y) 1 2 XZ YZ on the class S . 1 If Φ ∈ M(E ,E ), one can define by duality double operator integrals of the form 1 2 (2.1) for an arbitrary bounded linear operator T. However, we do not need this in this paper. We are going to discuss briefly in § 4 characterizations of Schur multipliers. Birman and Solomyak proved in [BS4] that if f is a Lipschitz function and A and B are not necessarily bounded self-adjoint operators with A−B ∈ S , then 2 f(x)−f(y) f(A)−f(B)= dE (x)(A−B)dE (y). (2.2) A B x−y RZ×ZR Note that for an arbitrary Lischitz function f, the divided difference Df is not always naturally defined on the diagonal. However, we can define Df on the diagonal by an arbitrary bounded measurable function and the right-hand side of (2.2) does not depend on the values on the diagonal. It follows from (2.2) that kf(A)−f(B)kS2 ≤ kfkLipkA−BkS2, where the Lipschitz (semi)norm kfk of f is, by definition, Lip |f(x)−f(y)| sup : x, y ∈ R, x 6= y . |x−y| (cid:26) (cid:27) Onthe other hand, if Aand B are notnecessarily boundedself-adjoint operators with A−B ∈ S and f is an operator Lipschitz function, then 1 f(A)−f(B)= Df (x,y)dE (x)(A−B)dE (y). (2.3) A B RZ×ZR (cid:0) (cid:1) (see [BS4]). Here the divided difference Df is defined by (1.4). It follows from (2.3) that kf(A)−f(B)kS1 ≤ kfkOLkA−BkS1. 4 3. Differentiability in the Hilbert–Schmidt norm Suppose that A and B are not necessarily bounded self-adjoint operators on Hilbert space such that A−B ∈ S . Consider the parametric family A , 0 ≤ t ≤ 1, defined by 2 t def A = A+tK, whereK = B−A. We need the following result of [KPSS], Theorem 7.18: t Suppose that f is a Lipschitz function on R that is differentiable at every point of R. Then the function s 7→ f(A )− f(A) is differentiable on [0,1] in the Hilbert–Schmidt s norm and d f(A )−f(A) = (Df)(x,y)dE (x)KdE (y). s t t ds s=t (cid:0) (cid:1)(cid:12)(cid:12) RZ×ZR (cid:12) 4. Schur multipliers and Haagerup tensor products Let (X,E ) and (Y ,E ) be spaces with spectral measures E and E on Hilbert 1 2 1 2 space. There are several characterizations of the class M(E ,E ) of Schur multipliers, 1 2 see [Pe2], [Pi], [AP2]. We need the following characterization in terms of the Haagerup tensor product of L∞ spaces: Let Φ be a measurable function on X ×Y . Then Φ ∈ M(E ,E ) if and only if Φ 1 2 belongs to the Haagerup tensor product L∞(E )⊗ L∞(E ), i.e., Φ admits a representaion 1 h 2 Φ(x,y) = ϕ (x)ψ (y), n n n X where ϕ ∈ L∞(E ), ψ ∈ L∞(E ), and n 1 n 2 |ϕ |2 ∈ L∞(E ) and |ψ |2 ∈ L∞(E ). n 1 n 2 n n X X Supposenow that E and E are Borel spectral measures on locally compact topolog- 1 2 ical spaces X and Y. In this case the following result holds (see [AP2]): Let Φ be a function on X × Y that is continuous in each variable. Then Φ ∈ M(E ,E ) if and only if it belongs to the Haagerup tensor product C (X)⊗ C (Y ) 1 2 b h b of the spaces of bounded continuous functions on X and Y , i.e., Φ admits a representation Φ(x,y) = ϕ (x)ψ (y), n n n X where ϕ ∈ C (X), ψ ∈ C (Y ) and the functions n b n b |ϕ |2 and |ψ |2 n n n n X X are bounded. 5 5. The trace of double operator integrals Let T be a trace class operator on Hilbert space and let E be a spectral measure on a σ-algebra of subsets of a set X. If Φ is a Schur multiplier, i.e., Φ ∈ M(E,E), then the double operator integral Φ(x,y)dE(x)T dE(y) ZZ belongs to S . Let us compute its trace. In [BS4] the following trace formula was found: 1 trace Φ(x,y)dE(x)T dE(y) = Φ(x,x)dµ(x), (5.1) (cid:18)ZZ (cid:19) Z where µ is the complex measure defined by µ(∆) =trace TE(∆) . The problem is how we can interpret the(cid:0)function(cid:1)x 7→ Φ(x,x) for functions Φ in M(E,E). In [Pe5] the following justification of formula (5.1) was given. We can define the trace T Φ of a function Φ in M(E,E) on the diagonal by the formula (T Φ)(x) d=ef ϕ (x)ψ (x), n n n X where Φ(x,y) = ϕ (x)ψ (y) (5.2) n n n X is a representation of Φ as an element of the Haagerup tensor productL∞(E)⊗ L∞(E), h i.e., |ϕ |2 ∈ L∞(E) and |ψ |2 ∈L∞(E). (5.3) n n m m X X Clearly, the trace of Φ ∈ M(E,E) on the diagonal belongs to L∞(E). Then formula (5.1) holds if Φ(x,x) is understood as (T Φ)(x), see [Pe5], § 1.1. Suppose now that E is a Borel spectral measure on a locally compact topological space X and Φ is a function on X ×X that is continuous in each variable. As we have mentioned in § 2, Φ admits a representation of the form (5.2), in which the functions ϕ n and ψ satisfy (5.3) and are continuous functions on X. It is easy to see that in this n case (T Φ)(x) = Φ(x,x), x ∈ X. In other words, the following theorem holds: Theorem 5.1. Let E be a spectral measure on a locally compact topological space X. Suppose that Φ is a function of class M(E,E). If Φ is continuous in each variable, then formula (5.1) holds for an arbitrary trace class operator T. 6 6. The Lifshits–Krein trace formula for arbitrary operator Lipschitz functions SupposethatAandB areself-adjointoperatorsonHilbertspacesuchthatA−B ∈S . 1 Letξbethespectralshiftfunctionassociatedwiththepair(A,B). Aswehavementioned in § 2, for an arbitrary operator Lipschitz function f on R, the operator f(A)−f(B) belongs to trace class. The following theorem is the main result of the paper. Theorem 6.1. Let f ∈ OL(R). Then trace f(A)−f(B) = f′(s)ξ(s)ds. R Z (cid:0) (cid:1) To prove the theorem, we are going to use an approach of Birman and Solomyak in [BS3] to the Lifshits–Krein trace formula. In [BS3] they used their approach under more restrictive assumptions on f. Proof. Obviously, f is a Lipschitz function. As we have mentioned in the introduction, f is a differentiable function at every point of R (but not necessarily contin- def uously differentiable!). Put K = B − A. Consider the parametric family {A } , t 0≤t≤1 def A = A+tK. Then A = A and A = B. The operator K obviously belongs to the t 0 1 Hilbert–Schmidt class S . As we have mentioned in § 3, the function t 7→ f(A )−f(A) 2 t is differentiable in the Hilbert–Schmidt norm and d f(x)−f(y) def Q = f(A )−f(A) = dE (x)KdE (y) ∈S , t s t t 2 ds s=t x−y (cid:0) (cid:1)(cid:12)(cid:12) RZ×ZR where E is the spectral measure(cid:12)of A . t t On the other hand, since the divided difference Df is a Schur multiplier of of S (see 1 the introduction), it follows that Qt ∈ S1, 0≤ t ≤1, and sup kQtkS1 < ∞. t∈[0,1] We have 1 f(A)−f(B)= − Q dt, t Z0 where the integral on the right is understood in the sense of Bochner in the space S . It 1 follows that 1 trace f(A)−f(B) = − trace(Q )dt. t Z0 Since the function f is diff(cid:0)erentiable eve(cid:1)rywhere, the divided difference Df is continuous in each variable. By Theorem 5.1, traceQ = f′(x)dν (x), t t R Z where the signed measure ν is defined by t ν (∆)d=ef trace E (∆)K) for a Borel subset ∆ of R. t t 7 (cid:0) We identify here the space of complex Borel measures on R with the dual space to the Banach space of continuous functions on R with zero limit at infinity. Then the function t 7→ ν is continuous in the weak-∗ topology on the space of complex Borel measures. t Indeed, if h is continuous on R and lim h(x) = 0, then |x|→∞ hdν = trace h(A )K . t t Z The function t 7→ h(A ) is a continuous funct(cid:0)ion on [(cid:1)0,1] in the operator norm; this t follows from the fact that h is an operator continuous function (see [AP1], § 8). Thus the function t 7→ trace h(A )K is continuous. t Therefore we can define the signed Borel measure ν on R by (cid:0) (cid:1) 1 ν = ν dt. t Z0 It follows that trace f(A)−f(B) = − f′dν. R Z On the other hand, for smooth(cid:0) functions g w(cid:1)ith compact support, trace g(A)−g(B) = g′ξdm, R Z where ξ is the spectral shift func(cid:0)tion. (cid:1) It follows that ν is absolutely continuous with respect to Lebesgue measure and dν = −ξdm. (cid:4) Theorem 6.1 implies the following result: Theorem 6.2. Let f be an operator Lipschitz function and let A and B be self-adjoint operators such that A−B ∈S . Then the function 1 t 7→ trace f(A−tI)−f(B−tI) , t ∈ R, is continuous on R. (cid:0) (cid:1) Proof. Consider the function f , t ∈ R, defined by f (x) d=ef f(x−t). Let ξ be the t t spectral shift function associated with (A,B). It is easy to see that trace f(A+tI)−f(B+tI) = trace f (A)−f (B) = f′(x−t)ξ(x)dm(x), t t R Z which dep(cid:0)ends on t continuously(cid:1). (cid:4) (cid:0) (cid:1) References [AP1] A.B. AleksandrovandV.V. Peller,Operator Ho¨lder–Zygmund functions,AdvancesinMath. 224 (2010), 910-966. [AP2] A.B. Aleksandrov and V.V. 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[Pe2] V.V.Peller,Hankeloperators inthetheoryofperturbations ofunitaryandself-adjointoperators, Funktsional. Anal. i Prilozhen. 19:2 (1985), 37–51 (Russian). English transl.: Funct. Anal. Appl. 19 (1985) , 111–123. [Pe3] V.V. Peller, Hankel operators in the perturbation theory of of unbounded self-adjoint operators. Analysis and partial differential equations, 529–544, Lecture Notes in Pure and Appl. Math., 122, Dekker,New York,1990. [Pe4] V.V. Peller, Hankel operators and their applications, Springer-Verlag, New York,2003. [Pe5] V.V. Peller, Multiple operator integrals in perturbation theory, Bull. Math. Sci., [Pi] G. Pisier, Similarity problems and completely bounded maps, Second, expanded edition. Includes thesolutionto“TheHalmosproblem”.LectureNotesinMathematics,1618.Springer-Verlag,Berlin, 2001. V.V.Peller Department of Mathematics Michigan StateUniversity East Lansing Michigan 48824 9