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Untangling Noncommutativity with

Anna Skripka

Emergence of Operator Integration where ℰ퐴 is the projection-valued spectral measure of 퐴, Operator integration (OI) is a collection of powerful meth- and the of this operator 푓(퐴) is given by the ods and techniques that enable analysis of functions with operator noncommuting arguments. Such functions arise, in par- ticular, in various problems of applied analysis, 푓(퐴) = ∫ 푓(휆) 푑ℰ퐴(휆). , noncommutative , and ℝ statistical estimation. When 퐴 is a finite matrix, the above integral degenerates Single operator integrals are basic tools in the classical to a finite sum functional . For instance, a self-adjoint operator 퐴 densely defined in a separable Hilbert admits the 푓(퐴) = ∑ 푓(휆푘)ℰ퐴(휆푘). operator integral decomposition 푘

퐴 = ∫ 휆 푑ℰ퐴(휆), This spectral integral (or sum) decomposition induces a ℝ straightforward estimate of an operator function in terms of the scalar function Anna Skripka is an associate professor of mathematics at the University of New Mexico. Her email address is [email protected]. ‖푓(퐴)‖ ≤ ‖푓‖∞. The author’s work is partially supported by NSF CAREER grant DMS-1554456. Communicated by Notices Associate Editor Daniela De Silva. In approximation problems one might need to estimate For permission to reprint this article, please contact: the difference 푓(퐴) − 푓(퐵) of functions of self-adjoint op- [email protected]. erators 퐴 and 퐵 in terms of some norms of 퐴 − 퐵 and DOI: https://doi.org/10.1090/noti2008 푓. When 퐴 and 퐵 are bounded and commuting, we can

JANUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 45 represent this difference as the integral note. We will demonstrate applicability of OI to ques- tions arising in the study of smoothness properties of 푓(퐴) − 푓(퐵) operator functions, spectral shift, spectral flow, quantum 푓(휆) − 푓(휇) differentiability, and smoothness of noncommutative 퐿푝- = ∬ 푑ℰ퐴+푖퐵(휆, 휇)(퐴 − 퐵) ℝ2 휆 − 휇 norms. Technical details will be omitted, but an interested with respect to the spectral measure of the normal operator reader is invited to find them along with a continued dis- 퐴 + 푖퐵. By submultiplicativity of the operator and cussion in the recent book [19]. Due to the restriction on properties of the spectral integral, we deduce the bound the allowed number of references many important contri- ‖푓(퐴) − 푓(퐵)‖ ≤ ‖푓‖Lip(ℝ)‖퐴 − 퐵‖. butions in the field will not be cited here, but can befound in [19]. Similar bounds for noncommuting 퐴 and 퐵 are deep re- sults grounded on double or iterated operator integral de- Operator Integrals on Finite Matrices compositions Methods of operator integration have been actively used 푓(퐴) − 푓(퐵) and rediscovered in matrix analysis, often with many par- ticular cases treated separately without appeal to a general 푓(휆) − 푓(휇) = ∫ ∫ 푑ℰ (휆)(퐴 − 퐵)푑ℰ (휇). theory. In this section we give an overview of general re- 휆 − 휇 퐴 퐵 ℝ ℝ sults supplied by the OI approach along with types of prob- The right-hand side above can be interpreted as the value lems where they can be applied. 퐴,퐵 of a linear transformation 푇휑 on 퐴 − 퐵, where 휑(휆, 휇) = 2 푓(휆)−푓(휇) Linear case. Let ℓ푑 denote the 푑-dimensional Hilbert , and, thus, 휆−휇 space equipped with the Euclidean inner product and let 2 퐴,퐵 ℬ(ℓ푑) denote the of linear operators (or 푓(퐴) − 푓(퐵) = 푇휑 (퐴 − 퐵). 2 푑 × 푑 matrices) on ℓ푑 equipped with the operator (spec- 2 More generally, the OI approach reduces analysis of dif- tral) norm. Let 퐴, 퐵 ∈ ℬ(ℓ푑) be self-adjoint matrices, let 푑 푑 ferent noncommutative expressions to the analysis of a {푔푗}푗=1, {ℎ푘}푘=1 be complete systems of orthonormal eigen- 퐴1,…,퐴푛+1 푑 푑 multilinear transformation 푇휑 acting on a Carte- vectors, and let {휆푗}푗=1, {휇푘}푘=1 be of the corre- sian product of matrices or infinite-dimensional opera- sponding eigenvalues of 퐴 and 퐵, respectively. Let 푃ℎ de- 2 tors. The parameters 휑, 퐴1, … , 퐴푛+1 are determined by note the orthogonal projection onto the vector ℎ ∈ ℓ푑 and the model in question. The values of the transforma- let 휑 ∶ ℝ2 → ℂ be a function. 퐴1,…,퐴푛+1 tion 푇휑 are often decomposed into integrals with The double operator integral constructed from the spec- operator-valued integrands or measures. Properties of tral data of 퐴, 퐵 and the symbol 휑 is the bounded linear 퐴1,…,퐴푛+1 transformation 푇휑 depend on the space where it acts, on the type 퐴,퐵 2 2 of symbol 휑, and sometimes on the spectral properties of 푇휑 ∶ ℬ(ℓ푑) → ℬ(ℓ푑) the operators 퐴1, … , 퐴푛+1. given by Organization 푑 푑 퐴,퐵 푇휑 (푋) = ∑ ∑ 휑(휆 , 휇 )푃 푋푃 (1) Our first acquaintance with multiple operator integration 푗 푘 푔푗 ℎ푘 푗=1 푘=1 in this note will occur in the finite-dimensional setting 퐴1,…,퐴푛+1 2 퐴,퐵 since 푇휑 on tuples of matrices admits a finite sum for 푋 ∈ ℬ(ℓ푑). In other words, 푇휑 acts on 푋 as the entry- 푑 representation. We will see a nonchronological summary wise multiplier of the matrix of 푋 in the bases {푔푗}푗=1 and of major results and glimpses of fundamental ideas along 푑 {ℎ }푑 by the matrix (휑(휆 , 휇 )) : with questions approachable by the OI method. 푘 푘=1 푗 푘 푗,푘=1 Then we will become familiar with OI in the general set- 푑 푑 (푥푗푘) ↦ (휑(휆푗, 휇푘) 푥푗푘) . (2) ting of noncommutative 퐿푝-spaces, where in addition to 푗,푘=1 푗,푘=1 퐴,퐵 puzzles of noncommutativity one deals with convergence Due to this interpretation, the transformation 푇휑 is also issues. We will touch upon several constructions known called a Schur multiplier. under the name “multiple operator integral,” each one Given a differentiable function 푓 ∶ ℝ → ℂ and the ma- supplying a particular type of estimate for noncommuta- trix functions 푓(퐴) and 푓(퐵) defined by the functional cal- tive expressions arising in different setups. culus, From the beginning of its development, the theory of 푑 푑 operator integration has been motivated and guided by 푓(퐴) = ∑ 푓(휆푗)푃푔푗 and 푓(퐵) = ∑ 푓(휇푘)푃ℎ푘 , applications, and this synergy will be reflected in this 푗=1 푘=1

46 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 1 2 we have the representation for 푋1, 푋2, … , 푋푛 ∈ ℬ(ℓ푑). It is also called an 푛-linear Schur 퐴,퐵 multiplier. 푓(퐴) − 푓(퐵) = 푇 (퐴 − 퐵). 푓[1] (3) 퐴1,…,퐴푛+1 The transformation 푇휑 generalizes both the en- Here 푓[1] is the divided difference of 푓 given by trywise product (2) (when 푛 = 1) and the usual matrix 푓(휆)−푓(휇) product 푋1 ⋯ 푋푛 (when 휑 ≡ 1). In the bases of eigenvec- if 휆 ≠ 휇, 퐴,퐴,퐴 푓[1](휆, 휇) = { 휆−휇 tors of 퐴, the transformation 푇휑 acts on the pair (푋, 푌) 푓′(휆) if 휆 = 휇. by 푑 푑 The representation (3) was derived by K. Löwner in 1934 ((푥푖푗)푖,푗=1, (푦푗푘)푗,푘=1) in his work on characterization of matrix monotone func- 푑 푑 tions by means of basic . From (3) and the ↦ ( ∑ 휑(휆푖, 휆푗, 휆푘) 푥푖푗 푦푗푘) . 퐴,퐵 푗=1 푖,푘=1 Schur multiplier property (2) of 푇푓[1] we conclude that if 푑 (푓[1](휆 , 휇 )) This transformation appears in the representation for the matrix 푗 푘 푗,푘=1 is positive definite, then the function 푓 is matrix monotone, that is, 퐴 ≤ 퐵 implies the Taylor remainder 푓(퐴) ≤ 푓(퐵) 푛−1 . 1 푑푘 The representation (3) along with (1) can be used to 푓(퐴) − 푓(퐵) − ∑ 푓(퐵 + 푡(퐴 − 퐵))| (7) 푘! 푑푡푘 푡=0 prove existence and the Schur multiplier property (2) of 푘=1 the matrix 퐴,퐵,…,퐵 = 푇푓[푛] (퐴 − 퐵, … , 퐴 − 퐵). 푑 푓(퐵 + 푡(퐴 − 퐵))| = 푇퐵,퐵(퐴 − 퐵). (4) Here 푓 is an 푛 times differentiable function whose deriva- 푑푡 푡=0 푓[1] tive 푓(푛) is bounded on a segment that contains the spectra The double operator integral also calculates the quasicom- of 퐴 and 퐵, and 푓[푛] is the 푛th order divided difference of mutator 푓, which is defined recursively by 퐴,퐵 푓(퐴)푋 − 푋푓(퐵) = 푇푓[1] (퐴푋 − 푋퐵). (5) 푓[푚+1] = (푓[푚])[1]. The representations (3) – (5) indicate that increments and The relation (7) reduces questions on Taylor approxima- of operator functions as well as quasicommuta- tion and convexity of matrix functions to the analysis of tors can be studied and estimated in a unified way in the 퐴,퐵,…,퐵 푇푓[푛] . double operator integration framework. We will consider A multilinear Schur multiplier also calculates the a variety of such estimates in different norms. higher-order Fr´echetmatrix derivative 1 푑푘 Multilinear case. Multilinear operator integrals arise as 푓(퐵 + 푡(퐴 − 퐵))| (8) natural extensions of double ones in higher-order pertur- 푘! 푑푡푘 푡=0 퐵,퐵,…,퐵 bation problems. = 푇푓[푘] (퐴 − 퐵, … , 퐴 − 퐵), 2 Let 푛 be a natural number, let 퐴1, … , 퐴푛+1 ∈ ℬ(ℓ ) be 푑 where 푓 is 푘 times continuously differentiable. The idea {푔(푗)}푑 self-adjoint matrices, let 푖 푖=1 be an orthonormal ba- of involving OI in higher-order differentiation of oper- (푗) 푑 sis of eigenvectors, and let {휆푖 }푖=1 be the corresponding ator functions was introduced by Yu. L. Daletskii and of eigenvalues of 퐴푗 for 푗 = 1, … , 푛 + 1. Let S. G. Krein in 1956, while the best up-to-date results in 휑 ∶ ℝ푛+1 → ℂ be a function. this area are consequences of modern approaches to OI. The multilinear operator integral constructed from the spectral data of 퐴1, … , 퐴푛+1 and the symbol 휑 is the bound- Norm estimates for Schur multipliers. One of the main ed 푛-linear transformation objectives in the study of multiple operator integrals is 퐴1,…,퐴푛+1 2 2 2 finding useful estimates for their norms. While dimension 푇휑 ∶ ℬ(ℓ⏟⎵⎵⎵⎵⎵⏟⎵⎵⎵⎵⎵⏟푑) × ⋯ × ℬ(ℓ푑) → ℬ(ℓ푑) dependent bounds for multilinear Schur multipliers fol- 푛 times low directly from their definitions, delicate dimension in- given by dependent bounds stand on a decades-long development 퐴1,퐴2,…,퐴푛+1 of a comprehensive theory. 푇휑 (푋1, 푋2, … , 푋푛) (6) The best dimension independent bound is 푑 (1) (2) (푛+1) 퐴 ,…,퐴 = ∑ 휑(휆푟 , 휆푟 , … , 휆푟 ) ‖ 1 푛+1 2 2 2 ‖ 1 2 푛+1 ‖푇휑 ∶ 풮푑 × ⋯ × 풮푑 → 풮푑‖ (9) 푟1,푟2,…,푟푛+1=1 (1) (푛+1) ⋅ 푃 (1) 푋 푃 (2) 푋 ⋯ 푋 푃 (푛+1) = max |휑(휆푟1 , … , 휆푟푛+1 )|, 푔 1 푔 2 푛 푔 푟1 푟2 푟푛+1 1≤푟1,…,푟푛+1≤푑

JANUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 47 2 2 [푛] where 풮푑 is the space ℬ(ℓ푑) equipped with the Hilbert– tensor product norm of 푓 . This norm is generally greater Schmidt (Frobenius) norm. The result of (9) is also the than the norm of 푓 appearing in (10): simplest one to derive in the collection of dimension in- 1 dependent bounds. Indeed, the inequality ≤ in (9) for (푛) ‖ [푛]‖ ‖ [푛]‖ ‖푓 ‖∞ = ‖푓 ‖ ≤ ‖푓 ‖ . 푛 = 1 quickly follows from the entrywise multiplier prop- 푛! ∞ ⊗ 퐴,퐵 erty of 푇휑 and for 푛 ≥ 2 from the repeated application The inequality (12) is based on a factorization of the [푛] of Hölder’s inequality. function 푓 (휆1, … , 휆푛+1) separating the variables 휆1,…, 푝 2 The analog of (9) for 풮푑, which is the space ℬ(ℓ푑) 휆푛+1, equipped with the 푝th Schatten–von Neumann norm ‖⋅‖푝, [푛] takes the form 푓 (휆1, … , 휆푛+1) 퐴 ,…,퐴 푝 푝 푝 ‖푇 1 푛+1 ∶ 풮 1 × ⋯ × 풮 푛 → 풮 ‖ (10) ‖ 푓[푛] 푑 푑 푑‖ = ∫ 푎1(휆1, 휔) ⋯ 푎푛+1(휆푛+1, 휔) 푑휈(휔), (푛) Ω ≤ 푐푝1,…,푝푛 ‖푓 ‖∞. which was implemented in [11]and [2]. This factorization Here is in the spirit of the celebrated A. Grothendieck’s character- 2 1 < 푝, 푝푗 < ∞, 푗 = 1, … , 푛, ization of bounded linear Schur multipliers on ℬ(ℓ ). In 1 1 1 the next section, we will see alternatives to the bound (12) = + ⋯ + , 푝 푝1 푝푛 that reveal dependence on smoothness and decay proper- ties of the function 푓. 푓 is an 푛 times differentiable function with essentially From the estimates (9) – (12) and representations (7) bounded 푓(푛), and ‖ ⋅ ‖ denotes the sup norm. A simi- ∞ and (8) we immediately deduce analogous bounds for lar bound holds for the seminorm |Tr (⋅)|, which is smaller derivatives and Taylor remainders of matrix functions. than the norm ‖ ⋅ ‖ , of the multilinear Schur multiplier 1 We note that the Lipschitz-type dimension independent | 퐴,…,퐴 | |Tr (푇푓[푛] (푋1, … , 푋푛))| bound ≤ 푐 ‖푓(푛)‖ ‖푋 ‖ … ‖푋 ‖ , 푛 ∞ 1 푛 푛 푛 (11) ‖푓(퐴) − 푓(퐵)‖푝 ≤ 푐푝 ‖푓‖Lip(ℝ) ‖퐴 − 퐵‖푝, (13) where Tr is the canonical matrix trace and 1 < 푝 < ∞, which follows for every Lipschitz function 푓 by 푛 1/푛 ‖푋‖푛 = (Tr (|푋| )) the same method as (10), does not extend to the trace class 푝 = 1 is the 푛th Schatten–von Neumann norm of 푋. norm ( ) or . In particular, E. B. Davies 푑 ∈ ℕ The result of (10) is based on harmonic analysis of showed in 1988 that given , there exist self-adjoint 퐴, 퐵 ∈ ℬ(ℓ2 ) Banach spaces with unconditional martingale differences 2푑 such that (UMD) and an intricate recursive procedure reducing the ‖|퐴| − |퐵|‖1 ≥ const log(푑) ‖퐴 − 퐵‖1. order yet preserving the nature of the symbol 푓[푛]. The respective approach for 푛 = 1 was introduced and imple- There are also continuously differentiable functions 푓 with mented in [16], its higher-order equivalent in a more tech- bounded derivative for which the supremum of ‖푓(퐴) − nical setting of 푛 ≥ 2 in [12]. The estimate (11) follows 2 푓(퐵)‖1/‖퐴 − 퐵‖1 over distinct self-adjoint 퐴, 퐵 ∈ ℬ(ℓ2푑) [푛] from a suitable generalization of (10) from 푓 to more grows like √log(푑). Higher-order matrix Taylor remainders general symbols given by polynomial integral momenta possess a similar behavior with respect to the trace class and from Hölder’s inequality. norm [15]. Different lower bounds derived in [20] provide There is a more universal approach to estimating a theoretical limitation on the accuracy of matrix Taylor 퐴1,…,퐴푛+1 푇푓[푛] , which results in bounds for more general approximations. norms but with a coarser dependence on 푓[푛] and a smaller Bounds supplied by OI can be used to predict sensitiv- set of admissible functions 푓. Namely, ity of computational problems to small perturbations or rounding errors as well as the accuracy of estimators by ‖ 퐴1,…,퐴푛+1 ‖ ‖푇푓[푛] ∶ ℐ × ⋯ × ℐ → ℐ‖ (12) sample data. For instance, the norm of a matrix Fr´echetde- rivative calculates a condition number for problems with ≤ ‖푓[푛]‖ , ‖ ‖⊗ matrix functions in numerical analysis. For another exam- ple, if 퐴 is an unknown matrix whose function where ℐ is the space ℬ(ℓ2 ) equipped with a unitarily in- 푑 or functional 푓(퐴) is to be estimated and 퐵 is a sample co- variant norm, 푓 is a function with smoothness properties matrix, then the bounds for matrix derivatives and stronger than 푛-times but weaker than (푛+1)-times contin- Taylor remainders can help to gain insight into the quality uous differentiability, and ‖푓[푛]‖ is the integral projective ‖ ‖⊗ of the estimator 푓(퐵).

48 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 1 Operator Integrals on Noncommutative integral 퐿푝-spaces 퐴,퐵 In the noncommutative analysis involving functions of 푇휑 (푋) = ∬ 휑(휆, 휇) 푑ℰ(휆, 휇)(푋), (14) infinite-dimensional operators with continuous spectra, ℝ2 one works with suitable replacements of the transforma- where 휑 ∶ ℝ2 → ℂ is a bounded Borel function. The trans- 퐴,퐵 tions (1) and (6) that still satisfy properties like (3) – (5) formation 푇휑 inherits all the nice properties of a spectral and (7) – (12). Nowadays we have a rather comprehensive integral, including the inequality theory of multilinear operator integration as a cumulative 퐴,퐵 2 2 product of many groundbreaking results. ‖푇휑 ∶ 풮 → 풮 ‖ ≤ ‖휑‖∞, (15) Before proceeding, we briefly recall noncommutative 퐿푝-spaces where operator integrals are defined. Let ℳ be a which is an analog of (9). M. S. Birman and M. Z. semifinite von Neumann algebra of bounded linear oper- Solomyak showed that the transformation given by (14) 2 ators acting on a separable Hilbert space ℋ and let 휏 be a satisfies (3) for 퐴 − 퐵 ∈ 풮 and 푓 ∈ Lip(ℝ). The defini- 2 normal faithful semifinite trace on ℳ. The noncommuta- tion (14) and property (15) extend from 풮 to a general 2 2 tive 퐿푝-space 퐿푝(ℳ, 휏), 1 ≤ 푝 < ∞, associated with (ℳ, 휏) noncommutative 퐿 -space 퐿 (ℳ, 휏). 퐴,퐵 consists of operators affiliated with ℳ and satisfying If the transformation 푇휑 extends from a dense sub- set 퐿2(ℳ, 휏) ∩ 퐿푝(ℳ, 휏) of the noncommutative 퐿푝-space 푝 1/푝 ‖푋‖푝 = (휏(|푋| )) < ∞. 퐿푝(ℳ, 휏), 1 ≤ 푝 < ∞, to a bounded transformation on 푝 The space 퐿∞(ℳ, 휏) is identified with the algebra ℳ. 퐿 (ℳ, 휏), then it inherits properties of a double operator When ℳ equals the algebra ℬ(ℋ) of bounded linear integral, as was realized by B. de Pagter, H. Witvliet, and 1 ∗ operators on ℋ, the noncommutative 퐿푝-space coincides F. Sukochev in [6]. By the duality (풮 ) = ℬ(ℋ), the 퐴,퐵 1 with the Schatten–von Neumann ideal 풮푝. The latter con- transformation 푇휑 also extends from 풮 to ℬ(ℋ), as was sists of compact operators on ℋ whose sequences of singu- noted by M. S. Birman and M. Z. Solomyak in the 1960s. lar values belong to ℓ푝, and the 풮푝-norm is defined to be Sufficient conditions for existence of double operator 푝 1/푝 integrals on non-Hilbert spaces and useful estimates for ‖푋‖푝 = (Tr (|푋| )) . The reader unfamiliar with general noncommutative 퐿푝-spaces can assume the particular case their norms build on analysis of their symbols. One ap- of 풮푝 throughout the note. proach relies on existence of a factorization of the symbol 휑(휆, 휇) that separates its variables 휆 and 휇. It was first sug- gested by M. S. Birman and M. Z. Solomyak in 1966 and Linear case. The first transformation receiving the name then generalized by V. V. Peller in 1985 to characterize “double operator integral” was introduced by Yu. L. Dalet- 푇퐴,퐵 ℬ(ℋ) skii and S. G. Krein in 1956, following Löwner’s utiliza- bounded 휑 on . Later, the separation of variables tion of what nowadays is called double operator integra- approach to operator integrals was implemented on many ℐ tion (which was mentioned in the previous section). They symmetrically normed ideals of a semifinite von Neu- 푑 mann algebra by N. A. Azamov, A. L. Carey, P. G. Dodds, proved existence of the derivative 푓(퐴 + 푡푋)| and esti- 푑푡 푡=0 and F. Sukochev in [2]. These ideals possess a so-called mated its operator norm based on the representation (4), property (F) and include 퐿푝(ℳ, 휏)∩ℳ and 퐿푝,∞(ℳ, 휏)∩ℳ, 퐴,퐴 where 푇푓[1] is an iterated Riemann–Stieltjes integral with 1 < 푝 < ∞. This method is explained below. respect to the spectral family of 퐴 and where the class of Let 휑 ∶ ℝ2 → ℂ admit the representation scalar functions 푓 in their approach is assumed to be more

restrictive than turned out to be necessary for existence of 휑(휆, 휇) = ∫ 푎1(휆, 휔) 푎2(휇, 휔) 푑휈(휔) (16) the operator derivative. Later in the 1960s, M. S. Birman Ω 퐴,퐵 and M. Z. Solomyak developed several approaches to 푇휑 with some finite measure space (Ω, 휈) and bounded Borel and substantially extended the range of applicability of the functions 푎푖(⋅, ⋅) ∶ ℝ × Ω → ℂ, 푖 = 1, 2, satisfying double operator integration method, which we sketch be- low along with more recent approaches. ∫ ‖푎1(⋅, 휔)‖∞‖푎2(⋅, 휔)‖∞ 푑|휈|(휔) < ∞. Let 퐴, 퐵 be self-adjoint operators densely defined in ℋ Ω and let ℰ퐴,ℰ퐵 be their spectral measures. The product ℰ of 2 푇퐴,퐵 ℰ퐴 and ℰ퐵 defined on the rectangular sets 휎1 × 휎2 of ℝ by The double operator integral 휑 is then affixed to the factorization (16) via ℰ(휎1 × 휎2)(푋) = ℰ퐴(휎1)푋ℰ퐵(휎2) 퐴,퐵 푇휑 (푋)(푦) (17) for every 푋 in the Hilbert space 풮2 of Hilbert–Schmidt op- erators on ℋ is the spectral measure itself. The double = ∫ (푎1(퐴, 휔) 푋 푎2(퐵, 휔))(푦) 푑휈(휔) 퐴,퐵 2 operator integral 푇휑 on 풮 is then defined as the spectral Ω

JANUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 49 for every 푦 ∈ ℋ. For an ideal ℐ as above, we obtain taking limits to return to the initial operators. More pre- cisely, ‖푇퐴,퐵 ∶ ℐ → ℐ‖ ≤ ‖휑‖ ‖ 휑 ‖ ⊗ 퐴,퐵 푇휑 (푋) (18) ℐ ‖ ⋅ ‖ by the symmetric property of . Here the norm ⊗ is 푁 defined by 푙1 푙2 = lim lim ∑ 휑 ( , ) ℰ퐴,푙 ,푚푋ℰ퐵,푙 ,푚, 푚→∞ 푁→∞ 푚 푚 1 2 푙1,푙2=−푁 ‖휑‖⊗ = inf ∫ ‖푎1(⋅, 휔)‖∞‖푎2(⋅, 휔)‖∞ 푑|휈|(휔) 푙 푙+1 Ω where ℰ퐴,푙,푚 = ℰ퐴 ([ , )) and ℰ퐴 is the spectral mea- 푚 푚 with the infimum taken over all possible representations sure of 퐴. The transformation given by (18) exists for (16). We note that the transformation given by (17) satis- 휑 = 푓[1], where 푓 ∈ Lip(ℝ), and satisfies fies (3) for 퐴 − 퐵 ∈ ℬ(ℋ) and 푓 ∈ Lip(ℝ) with 휑 = 푓[1] ‖푇퐴,퐵 ∶ 퐿푝(ℳ, 휏) → 퐿푝(ℳ, 휏)‖ admitting the representation (16). The decomposition 푓[1] (19) (17) is also convenient for passing to the limit with respect ≤ 푐푝 ‖푓‖Lip(ℝ) to the parameters 퐴, 퐵, which was used to prove differen- for 1 < 푝 < ∞. The constant in (19) is known to behave tiability results for operator functions. like The existence of the representation (16) for 휑 = 푓[1] 푝2 푐 ∼ , is determined by smoothness and decay properties of the 푝 푝 − 1 function 푓. Namely, if 푓 belongs to the Besov space as established by M. Caspers, S. Montgomery-Smith, 1 [1] 퐵∞1(ℝ), then 푓 satisfies (16) and D. Potapov, and F. Sukochev in 2014. The transformations [1] ‖ [1]‖ given by (17) and (18) coincide on 휑 = 푓 for a large set ‖푓‖ ≤ 푓 ≤ const ‖푓‖ 1 , Lip(ℝ) ‖ ‖ 퐵∞1(ℝ) 1 ⊗ of functions 푓, including the space 퐵∞1(ℝ). as established by means of the Fourier analysis in There are several proofs of the estimate (19), each one V. V. Peller’s work in 1985. To see how this analysis works using deep results from harmonic analysis. To demon- in a nontechnical case, we assume that both 푓 and its strate a few ideas from the original proof, let 휓 be a 푡 푓̂ are integrable. By the Fourier inver- Schwartz function satisfying 휓(푡) = 푒 for 푡 ∈ [log 훼, log 훽], sion, where [훼, 훽] ⊂ (0, ∞). Then, the Fourier inversion implies 푓(휆) − 푓(휇) that 푓[1](휆, 휇) = 1 휆 − 휇 푥 = 휓(푡) = ∫ 휓(푠)̂ 푒푖푠푡푑푠 1 1 √2휋 ℝ = ∫ (푒푖휆푡 − 푒푖휇푡)푓(푡)̂ 푑푡 휆 − 휇 1 √2휋 ℝ = ∫ 휓(푠)̂ 푥푖푠푑푠, 푥 ∈ [훼, 훽]. 푡 √2휋 ℝ 푖 푖(휆푠+휇(푡−푠)) ̂ = ∫ ( ∫ 푒 푑푠)푓(푡) 푑푡 Hence, √2휋 ℝ 0 1 푓[1](휆, 휇) = ∫ 휓(푠)|푓(휆)̂ − 푓(휇)|푖푠|휆 − 휇|−푖푠 푑푠 푖 푖휆ᵆ 푖휇푣 ̂ = ∫ 푒 푒 푓(푢 + 푣) 푑푢 푑푣. √2휋 ℝ √2휋 ℝ2 [1] 2 for those 푓, 휆, 휇 for which 푓 (휆, 휇) ∈ [훼, 훽]. The latter de- The latter integral is in the form (16), where Ω = ℝ , 퐴,퐴 composition of 푓[1] induces a representation of 푇 as the 푖 ̂ 푖휆ᵆ 푓[1] 푑휈(푢, 푣) = 푓(푢 + 푣) 푑푢 푑푣, 푎1((푢, 푣), 휆) = 푒 , and √2휋 integral of compositions of the double operator integrals 푖휇푣 퐴,퐴 −푖푠 푖푠 푎2((푢, 푣), 휇) = 푒 for 푢, 푣 ∈ ℝ. The above calculation 푇휑 with 휑(휆, 휇) = |휆 − 휇| and 휑(휆, 휇) = |푓(휆) − 푓(휇)| . and further Fourier analysis induce the bounds Such transformations are bounded by the Marcinkiewicz– ‖ [1]‖ ˆ′ Mihlin multiplier theory. The respective bounds along 푓 ≤ ‖푓 ‖퐿1(ℝ) ‖ ‖⊗ with integrability of the moments of 휓 ultimately imply 퐴,퐴 ′ ″ the boundedness of 푇 [1] and estimate (19). ≤ √2 (‖푓 ‖퐿2(ℝ) + ‖푓 ‖퐿2(ℝ)). 푓 Sharpening dependence on the symbol 푓[1] for double Multilinear case. Attempts to extend linear double opera- operator integrals on UMD spaces was crucial for resolu- tor integrals to the multilinear case and derive analogous tion of several open problems. It was implemented by useful bounds were driven by applications and made con- D. Potapov and F. Sukochev in [16] by taking a novel ap- currently with development of the double OI theory. proach and carrying out a different type of harmonic anal- The first practical multiple operator integral construc- [1] 퐴,퐵 ysis on 푓 . The respective transformation 푇휑 is con- tion was implemented on the product of Hilbert spaces in structed by discretizing the spectra of 퐴, 퐵, then following B. S. Pavlov’s work in 1969. If 퐴1, … , 퐴푛+1 are self-adjoint the pattern of the finite-dimensional case (1), and finally operators densely defined in ℋ with spectral measures

50 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 1 2 퐴1,…,퐴푛+1 ℰ퐴1 ,…,ℰ퐴푛+1 , respectively, and 푋1, … , 푋푛 ∈ 풮 , then the A transformation 푇휑 that captures nice proper- 풮2-valued set function 푚 defined on the rectangular sets of ties of UMD spaces and sharpens dependence on the sym- ℝ푛+1 by bol 휑 was constructed by D. Potapov, A. Skripka, and F. Sukochev in [12] to solve open problems in higher-order 푚(훿1 × 훿2 × ⋯ × 훿푛+1) noncommutative analysis. This transformation is given by = ℰ (훿 )푋 ℰ (훿 )푋 ⋯ 푋 ℰ (훿 ) 퐴1 1 1 퐴2 2 2 푛 퐴푛+1 푛+1 a limit generalizing (18), and it satisfies the bounds (10) admits extension to a countably additive measure with and (11) as well as their variants in noncommutative 퐿푝- semivariation bounded by spaces. We will discuss the importance of (10) and (11) in the next section. ‖푚‖ ≤ ‖푋1‖2 ⋯ ‖푋푛‖2. The estimate (10) is established inductively through an The multiple operator integral defined by 퐴1,…,퐴푛+1 elaborate reduction of the order of 푇푓[푛] that pre- 퐴1,…,퐴푛+1 serves major features of its symbol. For instance, a frag- 푇휑 (푋1, … , 푋푛) = ∫ 휑(휔) 푑푚(휔) (20) ℝ푛+1 ment of this reduction for 푛 = 2 and 휆 < 휇 < 휈 involves for a bounded function 휑 that is measurable with re- the decomposition spect to the product of scalar-valued spectral measures of [2] 2 푓 (휆, 휇, 휈) 퐴1, … , 퐴푛+1 and 푋1, … , 푋푛 ∈ 풮 enjoys the bound 퐴1,…,퐴푛+1 2 2 2 푖푠 −푖푠 ̂ ‖푇휑 ∶ 풮 × ⋯ × 풮 → 풮 ‖ ≤ ‖휑‖∞. = ∫(휇 − 휆) 휑푓″ (휆, 휇)(휈 − 휆) 휓(푠) 푑푠 ℝ The definition (20) does not extend beyond the 풮2 setting, 푖푠 −푖푠 ̂ as confirmed by K. Dykema and A. Skripka in 2009. + ∫(휈 − 휇) 휑푓″ (휈, 휇)(휈 − 휆) 휓(푠) 푑푠, There is a more recent approach to the transformation ℝ (20) by C. Coine, C. Le Merdy, and F. Sukochev in [5], where 휓 is a Schwartz function and which adds new technical opportunities. They realized ∗ 1 this transformation as a 푤 -continuous contractive ″ 퐴1,…,퐴푛+1 휑푓″ (휆, 휇) = ∫ 푡 푓 (휆 + (휇 − 휆) 푡) 푑푡. 휑 → 푇휑 acting on elementary tensors by 0

퐴1,…,퐴푛+1 푇 (푋 , … , 푋 ) [2] 푓1⊗⋯⊗푓푛+1 1 푛 The decomposition of the symbol 푓 given above leads = 푓 (퐴 )푋 ⋯ 푋 푓 (퐴 ). 퐴,퐴,퐴 1 1 1 푛 푛+1 푛+1 to the representation of the triple operator integral 푇푓[2] , 퐴 ,…,퐴 subject to the restriction 휆 < 휇 < 휈, as the integral of This realization makes 푇 1 푛+1 amiable to approxima- 푓1⊗⋯⊗푓푛+1 compositions of the double operator integrals 푇퐴,퐴 with tion arguments with respect to the parameters 휑, 퐴 ,…, 휑 1 휑(휆, 휇) = (휆 − 휇)±푖푠 (discussed in the previous subsection) 퐴푛+1. It was recently used to substantially extend results ′ [1] 휑(휆, 휇) = 휑 ″ (휆, 휇) (푓 ) (휆, 휇) on differentiability of operator functions and obtain the and 푓 , which is similar to representation (7) for a broad set of of functions. (also discussed in the previous subsection). The sets like 휆 < 휇 < 휈 The double operator integral (17) was extended to the are carved out by triangular truncation opera- multilinear transformation tors.

퐴1,…,퐴푛+1 푇휑 ∶ ℬ(ℋ) × ⋯ × ℬ(ℋ) → ℬ(ℋ) Below we consider several important problems that for 휑 admitting a separation of variables analogous to (16) were solved by methods of multilinear operator integra- and to the transformation tion implemented in different technical setups, each one 퐴1,…,퐴푛+1 푇휑 ∶ ℐ × ⋯ × ℐ → ℐ, determined by the intrinsic nature of the problem. where ℐ is a symmetrically normed ideal with property (F), Smoothness of Operator Functions by V. V. Peller in [11] and by N. A. Azamov, A. L. Carey, In the 1960s, M. G. Krein posed a question on Lipschitz P. G. Dodds, and F. Sukochev in [2], respectively. This continuity of operator functions that opened a new direc- transformation satisfies the bound tion of research in noncommutative analysis. Although ex- 퐴1,…,퐴푛+1 ‖푇휑 ∶ℐ×⋯×ℐ→ℐ‖≤푐ℐ ‖휑‖⊗. isting results of K. Löwner, Yu. L. Daletskii, and S. G. Krein 푛 suggested that OI would be a natural tool in questions If 푓 ∈ 퐵∞1(ℝ), then [푛] on operator smoothness, this idea was successfully imple- ‖푓 ‖ ≤ const ‖푓‖ 푛 . ⊗ 퐵∞1(ℝ) mented only in this century after development of a deep As in the linear case, this OI construction is convenient for and beautiful OI theory. passing to the limit with respect to the parameters 퐴1,…, More precisely, M. G. Krein asked for which functions 퐴푛+1. 푓 ∶ ℝ → ℝ the respective operator function is Lipschitz in

JANUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 51 푝 풮 , 1 ≤ 푝 < ∞, that is, where Sym푘 is the group of all permutations of the set {1, … , 푘}. ‖푓(퐴) − 푓(퐵)‖ ≤ 푐 ‖퐴 − 퐵‖ 푝 푓,푝 푝 When 퐴 is unbounded or when the direction of differ- for all self-adjoint 퐴, 퐵 such that 퐴 − 퐵 ∈ 풮푝. The answer entiation belongs to another ideal, we do not have a full to M. G. Krein’s question is twofold. The set of scalar func- characterization of Fr´echetdifferentiability. Nonetheless, tions giving rise to operator functions that are Lipschitz we have sufficient conditions for existence of the 푛th-order in the 풮푝-norm, 1 < 푝 < ∞, coincides with Lip(ℝ), and Fr´echetderivatives stated in terms of smoothness and de- the bound (13) holds, as established by D. Potapov and cay properties of 푓 and also a characterization of the 푛th- F. Sukochev in [16]. However, not every 푓 ∈ Lip(ℝ) is Lip- order Gâteaux 풮푝-differentiability, 1 < 푝 < ∞. The respec- schitz with respect to the 풮1-norm or operator norm, as tive 푘th-order Gâteaux derivative is given by (8). noted by Yu. B. Farforovskaya in the 1970s. Examples of Thanks to the representation (7) in the infinite-dimen- 퐴, 퐵 with 퐴 − 퐵 ∈ 풮1 and continuously differentiable 푓 ∈ sional case, the estimates for operator integrals apply to Lip(ℝ) for which 푓(퐴) − 푓(퐵) ∉ 풮1 can be constructed by operator Taylor remainders taking direct sums of matrices 퐴푑, 퐵푑 of increasing dimen- 푅푛,푓,퐵(퐴 − 퐵) sion 푑 for which the quotient ‖푓(퐴푑)−푓(퐵푑)‖1/‖퐴푑 −퐵푑‖1 푛−1 grows logarithmically with 푑, as discussed in the section 1 푑푘 = 푓(퐴) − 푓(퐵) − ∑ 푓(퐵 + 푡(퐴 − 퐵))| . on finite-dimensional OI. 푘! 푑푡푘 푡=0 The set of 풮1-Lipschitz functions is known to contain 푘=1 [1] those 푓 ∈ Lip(ℝ) for which 푓 admits the decomposition 푑푘 In particular, when the derivatives 푓(퐵 + 푡(퐴 − 퐵)) are (16). It follows from the infinite-dimensional version of 푑푡푘 (3) and the bound for the respective double operator inte- evaluated in the operator norm, we have gral that ‖푅푛,푓,퐵(퐴 − 퐵)‖푝 (21) [1] ‖푓(퐴) − 푓(퐵)‖1 ≤ ‖푓 ‖⊗‖퐴 − 퐵‖1. (푛) 푛 ≤ 푐푝,푛‖푓 ‖∞‖퐴 − 퐵‖푝푛, 1 < 푝 < ∞, We also have the bound in the weak noncommutative 퐿1 for a self-adjoint bounded 퐵 and 푓 ∈ 퐶푛(ℝ). When the quasi-norm operator derivatives are calculated in the 풮푝-norm, 1 < ‖푓(퐴) − 푓(퐵)‖퐿1,∞(ℳ,휏) 푝 < ∞, the bound (21) extends to an unbounded 퐵 and 푛- times differentiable 푓 with bounded derivatives. We also ≤ const ‖푓‖Lip(ℝ) ‖퐴 − 퐵‖퐿1(ℳ,휏) have for every 푓 ∈ Lip(ℝ), as established by M. Caspers, ‖ ‖ 푛 D. Potapov, F. Sukochev, and D. Zanin in [4]. 푅푛,푓,퐵(퐴 − 퐵) ≤ 푐푛‖푓‖퐵푛 (ℝ)‖퐴 − 퐵‖푛 (22) ‖ ‖1 ∞1 As distinct from Lipschitzness, the operator functions 1 푛 inherit their Hölder property with 0 < 훼 < 1 in the opera- for 푓 ∈ 퐵∞1(ℝ) ∩ 퐵∞1(ℝ). The bound (22) follows from tor norm from scalar functions, according to the result of the work of V. V. Peller [11], and it does not extend to all A. B. Aleksandrov and V. V. Peller [1]. That is, 푓 ∈ 퐶푛(ℝ) by the counterexamples constructed in the work of D. Potapov, A. Skripka, F. Sukochev, and A. Tomskova ‖푓(퐴) − 푓(퐵)‖ ≤ 푐 (1 − 훼)−1 ‖푓‖ ‖퐴 − 퐵‖훼 Λ훼(ℝ) [15]. for all self-adjoint 퐴, 퐵 with bounded 퐴−퐵 and scalar func- Trace Formulas in Mathematical Physics tions 푓 in the Hölder class Λ훼(ℝ). Another aspect of smoothness is differentiability. Trace formulas relate spectral properties of two operators, A function 푓 is 푛 times continuously Fr´echet 풮푝- where one operator is well understood and the other is differentiable, 1 < 푝 < ∞, at every bounded self-adjoint viewed as its perturbation whose spectral properties are un- operator 퐴 on ℋ if and only if 푓 ∈ 퐶푛(ℝ). This result known. This line of research originates from M. G. Krein’s was obtained by E. Kissin, D. Potapov, V. Shulman, and seminal 1953 work addressing I. M. Lifshits’s findings and F. Sukochev in [8] and by C. Le Merdy and A. Skripka in questions in physics in the 1940s. Its development and [9] for 푛 = 1 and 푛 ≥ 2, respectively. The proof for 푛 ≥ 2 several breakthroughs benefited from the method of mul- combines advantages of two approaches to OI introduced tilinear operator integration. in [12] and [5]. The respective 푘th Fr´echetdifferential is Let 퐻 and 푉 be self-adjoint operators, 퐻 possibly un- expressed in terms of the multilinear operator integral bounded. If 푉 belongs to the Schatten–von Neumann 푛 푘 ideal 풮 for some 푛 ∈ ℕ, then there exists a unique real- 퐷푝 푓(퐴)(푋1, … , 푋푘) 1 valued function 휂푛 ∈ 퐿 (ℝ) depending only on 푛, 퐻, 푉 퐴,…,퐴 such that = ∑ 푇푓[푘] (푋휍(1), … , 푋휍(푘)), 휍∈Sym 푛 푘 ‖휂푛‖1 ≤ 푐푛‖푉‖푛

52 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 1 and and F. Sukochev in [14] and 푛 ≥ 3 via a reduction to uni- 푛−1 taries by A. Skripka in [18]. The respective higher-order 1 푑푘 Tr (푓(퐻 + 푉) − ∑ 푓(퐻 + 푡푉)| ) (23) spectral shift theory for contractions and unitaries builds 푘! 푑푡푘 푡=0 푘=0 on OI methods developed in [13] and [18]. The formula (23) extends to more general traces and = ∫ 푓(푛)(푡) 휂 (푡) 푑푡 푛 unsummable perturbations in semifinite von Neumann ℝ algebras that arise in noncommutative geometry. In par- for sufficiently nice 푓. This result was established by ticular, the trace formula with locally integrable 휂 holds M. G. Krein in 1953, L. S. Koplienko in 1984, and 푛 for an arbitrary bounded self-adjoint 푉 and unbounded D. Potapov, A. Skripka, and F. Sukochev in [12] in the cases self-adjoint 퐻 whose resolvent (퐻 − 푖퐼)−1 is 휏-compact. 푛 = 1, 푛 = 2, and 푛 ≥ 3, respectively. The function 휂 , 푛 This property is satisfied, for instance, by differential op- called the spectral shift function, plays a fundamental role erators on compact Riemannian manifolds. The trace in perturbation theory, but little is known about properties formulas also hold for perturbations in symmetrically of 휂 when 푛 ≥ 3 due to its complexity. 푛 normed ideals equipped with continuous, including sin- The proof of (23) for 푛 = 2 utilizes the double operator gular, traces. They were established for general ideals and integrals introduced by M. S. Birman and M. Z. Solomyak 푛 = 1, 2 by K. Dykema and A. Skripka in [7] and for the to carry out an approximation of 푉 ∈ 풮2 by a sequence of dual Macaev ideal and 푛 ≥ 3 by D. Potapov, A. Usachev, finite rank 푉 , for which 휂 can be calculated explicitly in 푘 2 F. Sukochev, and D. Zanin in 2015. terms of 휂1. The trace formula for 푛 ≥ 3 crucially relies on the estimate (11). Indeed, by the integral representation Spectral Flow for the remainder and OI representation for the operator The concept of the spectral flow stems from the celebrated derivative (8), work of M. F. Atiyah, V. K. Patodi, and I. M. Singer in the 푛−1 1 푑푘 1970s, where it was introduced primarily in a topological 푓(퐻 + 푉) − ∑ 푓(퐻 + 푡푉)| sense. At the International Congress of Mathematicians in 푘! 푑푡푘 푡=0 푘=0 1974, I. M. Singer communicated that the spectral flow can 1 be expressed as an integral of 1-form. After J. Phillips in- 1 푛−1 퐻푡,…,퐻푡 = ∫ (1 − 푡) 푇푓[푛] (푉, … , 푉) 푑푡, troduced an analytic approach to the spectral flow in the (푛 − 1)! 0 context of von Neumann algebras in the 1990s, the imple- where 퐻푡 = 퐻 + 푡푉. Then the estimate (11) implies ex- mentation of I. M. Singer’s suggestion was pursued in the istence of the measure 휇푛 such that the left-hand side of framework of noncommutative geometry. This represen- (푛) (23) equals ∫ℝ 푓 (푡) 푑휇푛(푡) and the total variation of 휇푛 tation via the integral of a 1-form was confirmed in a gen- 푛 is bounded by 푐푛‖푉‖푛. Finally, an approximation argu- eral setting without summability restrictions by incorpo- ment involving OI and reduction to the lower-order case rating double operator integration techniques in the work confirms absolute continuity of 휇푛 or, equivalently, exis- of N. A. Azamov, A. L. Carey, and F. Sukochev [3]. tence of 휂푛. Let 퐷0, 퐷1 be self-adjoint operators affiliated with ℳ While the condition 푉 ∈ 풮푛 can hold for perturbations whose resolvents are 휏-compact so that 푉 = 퐷0 − 퐷1 of discrete Schrödinger operators, it does not hold for typ- is bounded. If 퐷0 and 퐷1 are unitarily equivalent, then ical perturbations of differential operators. Instead, when the spectral flow sf(퐷0, 퐷1) from 퐷0 to 퐷1 along the path 퐻 is a continuous Schrödinger or Dirac operator, its per- 푟 → 퐷0 + 푟푉 can be calculated by turbation 푉 can possibly satisfy 1 −1 −1 푛 1 (퐻 + 푉 − 푖퐼) − (퐻 − 푖퐼) ∈ 풮 . sf(퐷0, 퐷1) = ∫ 휏(푉푓(퐷0 + 푟푉)) 푑푟 (24) ‖푓‖퐿1(ℝ) 0 Analogs of the trace formula (23) were obtained in this 2 extended setting with modified right- and, in some cases, for every nonnegative 푓 ∈ 퐶푐 (ℝ). The expression also left-hand sides and the respective 휂푛 integrable with 1 a weight. These results are based on the theory of spectral ∫ 휏(푉푓(퐷0 + 푟푉)) 푑푟 shift functions for unitary and contractive operators with 0 perturbations in 풮푛, a change of variables in multiple op- 푓 is the result of integration of the closed exact 1-form 푑휃퐷 erator integrals, and creation of summable weights within given by the OI framework. The condition (퐻 + 푉 − 푖퐼)−1 − (퐻 − −1 푛 푓 푖퐼) ∈ 풮 in the cases 푛 = 1 and 푛 = 2 was handled via 푑휃퐷(푋) a reduction to unitaries by M. G. Krein and H. Neidhardt 1 푑 | in 1962 and 1988, respectively. The case 푛 ≥ 2 was treated = (∫ 휏((푉 + 푠푋)푓(퐷0 + 푟푉 + 푠푟푋)) 푑푟) | . 푑푠 푠=0 via a reduction to contractions by D. Potapov, A. Skripka, 0

JANUARY 2020 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 53 The above conclusion builds on OI methods for func- Smoothness of Banach Norms tions of operators with 휏-compact resolvents developed by The study of smoothness properties in function Banach analogy with OI for summable perturbations discussed in spaces was naturally lifted to a similar investigation in the previous section. 푝 푝 the noncommutative Haagerup 퐿 -spaces 퐿퐻푎푎푔(ℳ) asso- When 퐷0 and 퐷1 are not unitarily equivalent, the rep- ciated with an arbitrary von Neumann algebra ℳ. This resentation (24) is modified by the truncated 휂-invariants general setting includes the following cases: when ℳ is 푝 and 휏-dimensions of the kernels of 퐷0 and 퐷1. The spec- semifinite, the space 퐿퐻푎푎푔(ℳ) is isometric to the classical tral flow sf(퐷0 −휆퐼, 퐷1 −휆퐼) coincides with the spectral shift noncommutative 퐿푝-space associated with ℳ, 1 ≤ 푝 < ∞; 푝 function 휂1(휆) discussed above up to the kernel correction when ℳ is a type I von Neumann algebra, then 퐿퐻푎푎푔(ℳ) terms. contains an isometric copy of the ℓ푝; when 푝 Quantum Differentiability ℳ is not of type I, 퐿퐻푎푎푔(ℳ) contains an isometric copy of the function space 퐿푝(0, 1). A quantized differential was introduced by A. Connes in The first-order Fr´echet differentiability of the norm the 1980s to replace the differential calculus in noncom- 푝 power ‖ ⋅ ‖ 푝 , 1 < 푝 < ∞, was essentially established mutative differential geometry by an operator theoretic no- 퐿퐻푎푎푔 tion involving a commutator. The asymptotic behavior of by H. Kosaki in 1984 and its higher-order Fr´echetdiffer- the singular values of the quantized derivative determines entiability by D. Potapov, F. Sukochev, A. Tomskova, and 푝 the dimension of an infinitesimal in the quantized calcu- D. Zanin in [17]. Namely, we have that ‖ ⋅ ‖ 푝 is 퐿퐻푎푎푔 lus. (i) infinitely many times Taylor–Fr´echet differentiable ∞ 푑 Let 푓 ∈ 퐿 (ℝ ), 푑 ∈ ℕ, and let 푀푓 be the operator of whenever 푝 is an even integer; pointwise multiplication by 푓. If 퐷 is the Dirac operator (ii) (푝 − 1)-times Taylor–Fr´echetdifferentiable whenever 푝 푑 ⌊ ⌋ 2 푑 densely defined in ℂ 2 ⊗퐿 (ℝ ) and sgn(퐷) is its sign given is an odd integer; by the , then the quantized derivative of (iii) ⌊푝⌋-times Taylor–Fr´echetdifferentiable whenever 푝 is 푓 is defined to be the commutator not an integer. 푝 More specifically, for self-adjoint 퐴, 푋 ∈ 퐿퐻푎푎푔(ℳ), 푑̄푓 ∶= 푖[sgn(퐷), 퐼 ⊗ 푀푓]. 푝 ‖퐴 + 푋‖ 푝 퐿 It follows from work of S. Janson and T. H. Wolff in 1982 퐻푎푎푔 that 푑̄푓 ∈ 풮푑 if and only if 푓 is a constant, while the prop- ⌈푝−1⌉ 푝 퐴 푝 erty of the quantized derivative to be in the weak Schatten– = ‖퐴‖ 푝 + ∑ 훿푘 (푋, … , 푋) + 풪(‖푋‖ 푝 ), 퐿퐻푎푎푔 퐿퐻푎푎푔 von Neumann ideal 풮푑,∞ is achievable in nontrivial cases. 푘=1 ∞ 푑 푑,∞ In particular, if 푑 > 1 and 푓 ∈ 퐿 (ℝ ), then 푑̄푓 ∈ 풮 퐴 푑 푑 푑 where 훿푘 is a symmetric 푘-linear bounded functional on if and only if ∇푓 ∈ 퐿 (ℝ ,ℂ ). Moreover, there exist con- 푝 푝 푝 퐿퐻푎푎푔(ℳ)×⋯×퐿퐻푎푎푔(ℳ). When 푝 is even, 풪(‖푋‖ 푝 ) = stants 푐푑, 퐶푑 > 0 such that 퐿퐻푎푎푔 퐴 퐴 훿푝 (푋, … , 푋). The differentials 훿푘 are given in terms of mul- 푐푑 ‖∇푓‖퐿푑(ℝ푑,ℂ푑) ≤ ‖푑̄푓‖풮푑,∞ ≤ 퐶푑 ‖∇푓‖퐿푑(ℝ푑,ℂ푑). tilinear operator integrals. 푝 푝 Since ‖푋‖ 푝 = 휓(|푋| ), where 휓 is a normalized The latter result was proved by S. Lord, E. McDonald, 퐿퐻푎푎푔 F. Sukochev, and D. Zanin in [10] using the method of trace on the weak noncommutative 퐿1-space 퐿1,∞(풩) asso- double operator integration. The appearance of OI in this ciated with a certain semifinite crossed product von Neu- problem is suggested by the formula (5), which extends to mann algebra 풩, the analysis of smoothness properties of 푝 the infinite-dimensional case under appropriate assump- ‖ ⋅ ‖ 푝 reduces to the analysis of the operator function 퐿퐻푎푎푔 tions. This variant of (5) gives arising from the scalar function 푓(푡) = |푡|푝. The latter 1 analysis is fulfilled by means of multilinear operator in- 2 − 퐷,퐷 [퐷(퐼 + 퐷 ) 2 , 퐼 ⊗ 푀 ] = 푇 [퐷, 퐼 ⊗ 푀 ] , 푝 푓 푔[1] ( 푓 ) tegration developed on weak noncommutative 퐿 -spaces 푝 퐿푝,∞(풩), which contain 퐿 (ℳ) as a closed subspace. 푔(푡) = 푡(1 + 푡2)−1/2 퐻푎푎푔 where is the regularized sign function. One of the features specific to this setting is involvement of Results for double operator integrals allow estimation of 푑 Calder´on-typeoperators in derivation of Hölder-type esti- 퐷,퐷 ⌊ ⌋ 1 2 2 푑 푇푔[1] on 풮 and on ℬ(ℂ ⊗ 퐿 (ℝ )). Then interpola- mates for OI. tion transfers this estimate to 푇퐷,퐷 on 풮푑,∞. Along with 푔[1] Closing Remarks a suitable analysis of the commutators [퐷, 퐼 ⊗ 푀푓] and 1 In this brief journey to operator integration we considered 2 − [sgn(퐷)−퐷(퐼+퐷 ) 2 , 퐼⊗푀푓], it ultimately gives the stated only the case of self-adjoint operators 퐴1, … , 퐴푛+1. Multi- upper bound. linear operator integration has also been developed in the

54 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 67, NUMBER 1 cases when 퐴1, … , 퐴푛+1 are unitary, contractive, dissipative, [13] Potapov D, Skripka A, Sukochev F. Higher-order spec- or unbounded normal operators. The theory has also been tral shift for contractions, Proc. Lond. Math. Soc. (3), no. 2 extended to noncommutative weak 퐿푝-spaces and, in the (108):327–349, 2014. MR3166355 linear case, to general Banach spaces. The details can be [14] Potapov D, Skripka A, Sukochev F. Trace formulas for re- solvent comparable operators, Adv. Math. (272):630–651, found in [19]. 2015, DOI 10.1016/j.aim.2014.12.016. MR3303244 While we demonstrated usefulness of operator inte- [15] Potapov D, Skripka A, Sukochev F, Tomskova A. Multi- gration on selected representative applications, the OI linear Schur multipliers and Schatten properties of opera- method has more to offer. Further general results based on tor Taylor remainders, Adv. Math. (320):1063–1098, 2017, this method along with their variants designed for specific DOI 10.1016/j.aim.2017.09.012. MR3709129 models of mathematical physics and noncommutative ge- [16] Potapov D, Sukochev F. Operator-Lipschitz functions ometry can be found in [19], thematic surveys, and recent in Schatten-von Neumann classes, Acta Math., no. 2 and upcoming articles. (207):375–389, 2011. MR2892613 [17] Potapov D, Sukochev F, Tomskova A, Zanin D. Fr´echet References differentiability of the norm of 퐿푝-spaces associated with [1] Aleksandrov AB, Peller VV. Operator Hölder-Zygmund arbitrary von Neumann algebras, Trans. Amer. Math. Soc., functions, Adv. Math., no. 3 (224):910–966, 2010. no. 11 (371):7493–7532, 2019, DOI 10.1090/tran/7215. MR2628799 MR3955526 [2] Azamov NA, Carey AL, Dodds PG, Sukochev FA. Operator [18] Skripka A. Estimates and trace formulas for unitary and integrals, spectral shift, and spectral flow, Canad. J. Math., resolvent comparable perturbations, Adv. Math. (311):481– no. 2 (61):241–263, 2009, DOI 10.4153/CJM-2009-012-0. 509, 2017, DOI 10.1016/j.aim.2017.02.026. MR3628221 MR2504014 [19] Skripka A, Tomskova A. Multilinear Operator Integrals: [3] Azamov NA, Carey AL, Sukochev FA. The spectral Theory and Applications, Lecture Notes in Mathematics, shift function and spectral flow, Comm. Math. Phys., vol. 2250, 2019. no. 1 (276):51–91, 2007, DOI 10.1007/s00220-007-0329- [20] Skripka A, Zinchenko M. Stability and uniqueness 9. MR2342288 properties of Taylor approximations of matrix func- [4] Caspers M, Potapov D, Sukochev F, Zanin D. Weak type tions, Linear Algebra Appl. (582):218–236, 2019, DOI commutator and Lipschitz estimates: resolution of the 10.1016/j.laa.2019.07.037. MR3992426 Nazarov-Peller conjecture, Amer. J. Math., no. 3 (141):593– 610, 2019, DOI 10.1353/ajm.2019.0019. MR3956516 [5] Coine C, Le Merdy C, Sukochev F. When do triple operator integrals take value in the trace class?, arXiv:1706.01662. https://arxiv.org/abs/1706 .01662 [6] de Pagter B, Witvliet H, Sukochev FA. Double operator integrals, J. Funct. Anal., no. 1 (192):52–111, 2002, DOI 10.1006/jfan.2001.3898. MR1918492 [7] Dykema K, Skripka A. Perturbation formulas for traces on normed ideals, Comm. Math. Phys., no. 3 (325):1107–1138, 2014. MR3152748 Anna Skripka [8] Kissin E, Potapov D, Shulman V, Sukochev F. Oper- ator smoothness in Schatten norms for functions of Credits several variables: Lipschitz conditions, differentiability Opener image is courtesy of Getty. and unbounded derivations, Proc. Lond. Math. Soc. (3), Author photo is courtesy of the author. no. 4 (105):661–702, 2012, DOI 10.1112/plms/pds014. MR2989800 [9] Le Merdy C, Skripka A. Higher order differentiability of operator functions in Schatten norms, J. Inst. Math. Jussieu. https://doi.org/10.1017/S1474748019000033 [10] Lord S, McDonald E, Sukochev F, Zanin D. Quantum differentiability of essentially bounded functions on Eu- clidean space, J. Funct. Anal., no. 7 (273):2353–2387, 2017, DOI 10.1016/j.jfa.2017.06.020. MR3677828 [11] Peller VV. Multiple operator integrals and higher opera- tor derivatives, J. Funct. Anal., no. 2 (233):515–544, 2006, DOI 10.1016/j.jfa.2005.09.003. MR2214586 [12] Potapov D, Skripka A, Sukochev F. Spectral shift func- tion of higher order, Invent. Math., no. 3 (193):501–538, 2013, DOI 10.1007/s00222-012-0431-2. MR3091975

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