Untangling Noncommutativity with Operator Integrals 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 function of this operator 푓(퐴) is given by the ods and techniques that enable analysis of functions with operator integral noncommuting arguments. Such functions arise, in par- ticular, in various problems of applied matrix analysis, 푓(퐴) = ∫ 푓(휆) 푑ℰ퐴(휆). mathematical physics, noncommutative geometry, 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 calculus. For instance, a self-adjoint operator 퐴 densely defined in a separable Hilbert space 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 norm 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 Banach space 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 sequences 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 spectral theory. 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 derivative 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- derivatives 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.