Higher derivatives of operator functions in ideals of von Neumann algebras

Let $\mathscr{M}$ be a von Neumann algebra and $a$ be a self-adjoint operator affiliated with $\mathscr{M}$. We define the notion of an"integral symmetrically normed ideal"of $\mathscr{M}$ and introduce a space $OC^{[k]}(\mathbb{R}) \subseteq C^k(\mathbb{R})$ of functions $\mathbb{R} \to \mathbb{C}$ such that the following result holds: for any integral symmetrically normed ideal $\mathscr{I}$ of $\mathscr{M}$ and any $f \in OC^{[k]}(\mathbb{R})$, the operator function $\mathscr{I}_{\mathrm{sa}} \ni b \mapsto f(a+b)-f(a) \in \mathscr{I}$ is $k$-times continuously Fr\'{e}chet differentiable, and the formula for its derivatives may be written in terms of multiple operator integrals. Moreover, we prove that if $f \in \dot{B}_1^{1,\infty}(\mathbb{R}) \cap \dot{B}_1^{k,\infty}(\mathbb{R})$ and $f'$ is bounded, then $f \in OC^{[k]}(\mathbb{R})$. Finally, we prove that all of the following ideals are integral symmetrically normed: $\mathscr{M}$ itself, separable symmetrically normed ideals, Schatten $p$-ideals, the ideal of compact operators, and -- when $\mathscr{M}$ is semifinite -- ideals induced by fully symmetric spaces of measurable operators.


Introduction
Notation.Let X be a topological space, V, W be normed vector spaces, and H be a complex Hilbert space.
(a) B X is the Borel σ-algebra on X.
(c) If A is a (possibly unbounded) self-adjoint operator on H, then P A : B σ(A) → B(H) is its projectionvalued spectral measure.If f : R → C is Borel measurable, then we define f (A) := σ(A) f (λ) P A (dλ).

Known results
Let H be a complex Hilbert space.Given an appropriately regular scalar function f : R → C, one of the goals of perturbation theory is to "Taylor expand" the operator function that takes a self-adjoint operator A on H and maps it to the operator f (A).This delicate problem has its beginnings in the work of Yu.L. Daletskii and S. G. Krein.In their seminal paper [9], they proved that if f : R → C is 2k-times continuously differentiable and A, B ∈ B(H) sa , then the curve R ∋ t → f (A + tB) ∈ B(H) is k-times differentiable in the operator norm and where f [k] : R k+1 → C is the k th divided difference (Section 4.2) of f , defined recursively as for (λ 1 , . . ., λ k+1 ) ∈ R k+1 .The reader might be (rightly) puzzled by the multiple integral in (1), since standard projection-valued measure theory only allows for the integration of scalar-valued functions.Indeed, while the innermost integral σ(A) f [k] (λ 1 , • • • , λ k+1 ) P A (dλ 1 ) makes sense using standard theory, it is already unclear how to integrate the map f [k] (λ 1 , λ 2 , . . ., λ k+1 ) P A (dλ 1 ) B with respect to P A .Daletskii and Krein dealt with this by using a Riemann-Stieltjes-type construction to define t s Φ(r) P A (dr) for certain operator-valued functions Φ : [s, t] → B(H), where σ(A) ⊆ [s, t].This approach, which requires rather stringent regularity assumptions on Φ, allows one to make sense of the right hand side of (1) as an iterated operator-valued integral -in other words, a multiple operator integral.Now, for j ∈ {1, . . ., k+1}, let (Ω j , F j ) be a measurable space and P j : F j → B(H) be a projection-valued measure.Emerging naturally from the formula (1) is the general problem of making sense of for certain functions ϕ : Ω 1 × • • • × Ω k+1 → C and operators b 1 , . . ., b k ∈ B(H).An object successfully doing so is called a multiple operator integral (MOI).Under the assumption that H is separable, these have been studied and applied to various branches of noncommutative analysis extensively.Please see A. Skripka and A. Tomskova's book [32] for an excellent survey of the MOI literature and its applications.
In this paper, we shall make use of the "separation of variables" approach to defining (3).For separable H, this approach was developed by V. V. Peller [29,30] and N. A. Azamov, A. L. Carey, P. G. Dodds, and F. A. Sukochev [2] in order to differentiate operator functions at unbounded operators.The present author extended the approach to the case of a non-separable Hilbert space in [26].We review the relevant definitions and results in Sections 2.1 and 4.1.Henceforth, any MOI expression we write or reference is to be interpreted in accordance with Section 4.1 (specifically, Theorem 4.1.2).Now, we quote the best known general results on higher derivatives of operator functions.If Ḃs,p q (R m ) is the homogeneous Besov space (Definition 4.3.10),then we write for the k th Peller-Besov space.It turns out that P B 1 (R) ∩ P B k (R) = P B 1 (R) ∩ Ḃk,∞ 1 (R).(Please see the paragraph containing Equation (25) at the end of Section A.2.) Theorem 1.1.1(Peller [29]).Let H be a separable complex Hilbert space, A be a self-adjoint operator on H, and B ∈ B(H) sa .If f ∈ P B 1 (R) ∩ P B k (R), then the map R ∋ t → f (A + tB) − f (A) ∈ B(H) is k-times differentiable in the operator norm, and the formula (1) holds.This is Theorem 5.6 in [29].To quote the relevant result from [2], we need some additional terminology.First, recall that if H is a complex Hilbert space, then M ⊆ B(H) is a von Neumann algebra if it is a unital * -subalgebra that is closed in the weak operator topology (WOT).Second, suppose I ⊆ M is a * -ideal with another norm • I on it.We call I an invariant operator ideal if (I, • I ) is a Banach space, r ≤ r I = r * I for r ∈ I, and I is symmetrically normed, i.e., arb I ≤ a r I b for r ∈ I and a, b ∈ M. Third, an invariant operator ideal I has property (F) if whenever (a j ) j∈J is a net in I such that sup j∈J a j I < ∞ and a j → a ∈ M in the strong * operator topology (S * OT), we get a ∈ I and a I ≤ sup j∈J a j I .Finally, we write W k (R) for the k th Wiener space (Definition 4.3.5) of functions f : R → C that are Fourier transforms of complex measures with finite k th moment.
Theorem 1.1.2(Azamov-Carey-Dodds-Sukochev [2]).Let H be a separable complex Hilbert space, M ⊆ B(H) be a von Neumann algebra, and a be a self-adjoint operator on H affiliated with M (Definition 2. This is Theorem 5.7 in [2].As is noted in [2], the motivating example of an invariant operator ideal with property (F) comes from the theory of symmetric operator spaces.Indeed, if (E, • E ) is a symmetric Banach function space with the Fatou property, (M, τ ) is a semifinite von Neumann algebra (Definition 2.1.8),and (E(τ ), • E(τ ) ) is the symmetric space of τ -measurable operators induced by E, then (I, is an invariant operator ideal with property (F).(Please see Section 2.3 for the meanings of the preceding terms.)Though Theorem 1.1.2applies to this interesting general setting, the result demands much more regularity of the scalar function f than Theorem 1.1.1.(Indeed, W k (R) P B 1 (R) ∩ P B k (R).)It has remained an open problem (Problem 5.3.22 in [32]) to find less restrictive conditions for higher Fréchet differentiability of operator functions in the symmetric operator space ideals described above.The present paper makes substantial progress on this problem: a corollary of our main results is that if E is fully symmetric (a weaker condition than the Fatou property), then the result of Theorem 1.1.2holds for (I, • I ) as in (5) with f ∈ P B 1 (R) ∩ P B k (R).In other words, we are able to close the regularity gap between Theorems 1.1.1 and 1.1.2in the (fully) symmetric operator space context.Moreover, we are able, for the first time in the literature on higher derivatives of operator functions, to remove the separability assumption on H by using the MOI development from [26].
Remark 1.1.3(Other related work).The Schatten p-ideals have property (F), so Theorem 1.1.2applies to them when the underlying Hilbert space is separable.There are, however, much sharper results known about differentiability of operator functions in the Schatten p-ideals (again, when the underlying Hilbert space is separable).Please see [23,22].Also, there is a seminal paper of de Pagter and Sukochev [11] that studies once Gateaux differentiability of operator functions in certain symmetric operator spaces at measurable operators; we discuss its relation to the results in this paper in Remark 4.4.7.

Main results
Let H be a complex Hilbert space, M ⊆ B(H) be a von Neumann algebra, and a be a self-adjoint operator affiliated with M (Definition 2.2.7).We recall from the previous section that our goal is to differentiate the operator function I sa ∋ b → f (a + b) − f (a) ∈ I, where (I, • I ) is some normed ideal of M and f ∈ P B 1 (R) ∩ P B k (R).(Please see (4).)The ideals we consider are the integral symmetrically normed ideals (ISNIs).The definition of integral symmetrically normed is an "integrated" version of the symmetrically normed condition arb I ≤ a r I b .Loosely speaking, (I, • I ) is integral symmetrically normed if (iii) The ideal (K(H), • ) of compact operators is an integral symmetrically normed ideal of B(H).With these in mind, we now state our second main result.Recall that f [k] : R k+1 → C is the k th divided difference of f : R → C (please see (2)), S k is the symmetric group on k letters, and all multiple operator integral (MOI) expressions as in (3) are to be interpreted in accordance with Section 4.1.
Theorem 1.2.2 (Derivatives of operator functions in ISNIs).Let H be an arbitrary (not-necessarilyseparable) complex Hilbert space, M ⊆ B(H) be a von Neumann algebra, and a be a self-adjoint operator on H affiliated with M. If (I, • I ) is an integral symmetrically normed ideal of M and f ∈ P B 1 (R)∩P B k (R), then the operator function Proof.Combine Theorem 4.4.6 and Corollary 4.3.14.
Theorems 1.2.2 and 1.2.1(iv) generalize the best known results, from [23], on differentiability of operator functions in the ideal (I, • I ) = (S p (H), • Sp ) to the non-separable case when p = 1.We do not, however, fully recover the optimal regularity on f , established in [22], when p ∈ (1, ∞).Also, to the author's knowledge, the present paper's result on the ideal of compact operators (i.e., Theorems 1.2.2 and 1.2.1(iii)) is new even when H is separable.Finally, as promised at the end of the previous section, Theorems 1.2.2 and 1.2.1(v) (together with Fact 2.3.2) make substantial progress on the open problem (Problem 5.3.22 in [32]) of finding general conditions for higher Fréchet differentiability of operator functions in ideals of semifinite von Neumann algebras induced by (fully) symmetric Banach function spaces.

Preliminaries
For Section 2, fix a complex Hilbert space (H, •, • ) and a von Neumann algebra M ⊆ B(H).Recall also that S ′ := {b ∈ B(H) : ab = ba, for all a ∈ S} is the commutant of a set S ⊆ B(H) and that a unital * -subalgebra N ⊆ B(H) is a von Neumann algebra if and only if N = N ′′ := (N ′ ) ′ .This is the well-known (von Neumann) Bicommutant Theorem.

Weak * integrals in von Neumann algebras
Following parts of Section 3.3 of [26], we review some basics of operator-valued integrals.We shall write Also, for the duration of this section, fix a measure space (Σ, H , ρ).
Definition 2.1.1 (Weak * measurability and integrability).A map for all (h n ) n∈N , (k n ) n∈N ∈ ℓ 2 (N; H).We say that F is weak * integrable if for all S ∈ H , there exists (necessarily unique) In this case, we call I S the weak * integral (over S) of F with respect to ρ, and we write S F dρ = S F (σ) ρ(dσ) := I S .
Remark 2.1.2.Let M * := {σ-WOT-continuous linear functionals M → C} be the predual of M. Part of Theorem 3.3.6 in [26] says that F is weak * measurable (respectively, integrable) in the sense of Definition 2.1.1 if and only if F is weakly measurable (respectively, integrable) in the weak * topology on M induced by the usual identification M ∼ = M * * .The above terminology is therefore justified.It turns out (Theorem 3.3.6(iii) in [26]) that ( 6) is actually already enough to guarantee that F is weak * integrable.Since we do not need this level of generality, we shall prove a weaker statement from scratch.Definition 2.1.3(Upper and lower integrals).For a function h : Σ → [0, ∞] that is not necessarily measurable, we define to be, respectively, the upper and lower integral of h with respect to ρ. (Please see Section 3.1 of [26].) This is called the (operator norm) integral triangle inequality.
Remark 2.1.5.First, the operator norm of a weak * measurable map is not necessarily measurable when H is not separable.This is why we need the lower integral above.Second, one can also prove that weak * integrals are independent of the representation of M. Please see Theorem 3.3.6(iv) in [26].
Proof.Fix S ∈ H , and define B S : Then B S is linear in the first argument and conjugate-linear in the second argument.Also, for all h, k ∈ H.In particular, B S is bounded.By the Riesz Representation Theorem, there exists unique If we can show ρ S (F ) = S F dρ, then we are done.
To this end, let a ∈ M ′ and h, k by the Cauchy-Schwarz Inequality (twice).Therefore, by the Dominated Convergence Theorem, Other than basic algebraic properties of the weak * integral, which are usually clear from the definition, the most important fact about weak * integrals that we shall use is an operator-valued Dominated Convergence Theorem, which we prove from scratch for the convenience of the reader.(But we remark that it follows from Proposition 3.2.3(v)and Theorem 3.3.6 in [26].)First, we note that one can do better than the operator norm triangle inequality.Indeed, retaining the notation from the proof above, we have Proof.By definition of the upper integral, there is some measurable g : Σ → [0, ∞] such that Σ g dρ < ∞ and sup n∈N g n ≤ g ρ-almost everywhere.Now, if n ∈ N, then, by definition of the lower integral, there exists a measurable gn : Σ → [0, ∞] such that 0 ≤ gn ≤ g n ρ-almost everywhere and Since g n → 0 ρ-almost everywhere, and 0 ≤ gn ≤ g n ρ-almost everywhere, we also have gn → 0 ρ-almost everywhere as n → ∞.Also, gn ≤ g n ≤ g ρ-almost everywhere, for all n ∈ N. Therefore, by the Dominated Convergence Theorem, gn dρ = 0, as desired.
Proposition 2.1.7(Operator-valued Dominated Convergence Theorem).Let (F n ) n∈N be a sequence of weak * integrable maps Σ → M, and suppose F : Σ → M is such that F n → F pointwise in the weak, strong, or strong * operator topology as n → ∞.
then F : Σ → M is weak * integrable and Σ F n dρ → Σ F dρ in, respectively, the weak, strong, or strong * operator topology as n → ∞.
Proof.Fix h, k ∈ H.In all cases, F n → F pointwise in the WOT as n → ∞.Therefore, F (•)h, k is the pointwise limit of ( F n (•)h, k ) n∈N .Consequently, F is weak * measurable.Also, F ≤ sup n∈N F n , so F is weak * integrable by ( 8) and Proposition 2.1.4.Now, (8) also gives Therefore, by the Dominated Convergence Theorem, as n → ∞.Thus Σ F n dρ → Σ F dρ in the WOT as n → ∞.Assume now that F n → F pointwise in the strong operator topology (SOT) as n → ∞, and write T n := Σ F n dρ and T := Σ F dρ. Then as n → ∞, by the integral triangle inequality observed above and Lemma 2.1.6,which applies because of ( 8) and the fact that sup n∈N (F n − F )h ≤ 2 h sup n∈N F n .Finally, the S * OT case follows from the SOT case because (F * n ) n∈N and F * satisfy the same hypotheses as (F n ) n∈N and F , and the adjoint is easily seen to commute with the weak * integral.
We end this section by stating an additional property of weak * integrals when M is semifinite.Before stating the result, we recall some notation and terminology.For a, b ∈ B(H), we write a ≤ b or b ≥ a to mean (b − a)h, h ≥ 0, for all h ∈ H. Also, we write It is easy to see that M + is closed in the WOT.Also, if (a j ) j∈J is a net in M + that is bounded (there exists b ∈ B(H) such that a j ≤ b, for all j ∈ J) and increasing (j 1 ≤ j 2 ⇒ a j1 ≤ a j2 ), then sup j∈J a j exists in B(H) + and belongs to M + .(Please see Proposition 43.1 in [7].)This is often known as Vigier's Theorem.If τ is a normal, faithful, semifinite trace on M, then (M, τ ) is called a semifinite von Neumann algebra.Remark 2.1.9.In the presence of (a) and (b), condition (c) is equivalent to τ (u * au) = τ (a) for all a ∈ M + and all unitaries u belonging to M. This is Corollary 1 in Section I.6.1 of [13].
In particular, if the right hand side is finite, then Σ F dρ ∈ L p (τ ).This is Theorem 3.4.7 in [26], and the motivation for its name is the classical Minkowski Inequality for Integrals (e.g., 6.19 in [18]).In view of the name of ( 7), an equally sensible name for Theorem 2.1.11is the "noncommutative L p -norm integral triangle inequality."

Unbounded operators and spectral theory
In this section, we provide the information about unbounded operators, projection-valued measures, and the Spectral Theorem that is necessary for this paper.Please see Chapters 3-6 of [4] or Chapters IX and X of [6] for more information and proofs of the facts we state (without proof) in this section.
An (unbounded linear) operator A on H is a linear subspace dom(A) ⊆ H (the domain of A) and a linear map A : dom(A) → H.We may identify A with its graph Γ(A) = {(h, Ah) : h ∈ dom(A)} ⊆ H × H.We say A is densely defined if dom(A) ⊆ H is dense, closable if the closure of Γ(A) in H × H is the graph of some operator A (called the closure of A) on H, and closed if Γ(A) ⊆ H × H is closed (i.e., A = A).If B is another unbounded operator on H, then the sum A + B has domain dom(A + B) := dom(A) ∩ dom(B), and the product AB has domain dom(AB) := {h ∈ dom(B) : Bh ∈ dom(A)} = B −1 (dom(A)).In addition, we write A ⊆ B if Γ(A) ⊆ Γ(B), i.e., dom(A) ⊆ dom(B) and Ah = Bh, for all h ∈ dom(A); and A = B if A ⊆ B and B ⊆ A, i.e., dom(A) = dom(B) and Ah = Bh for h in this common domain.
Given a densely defined operator A on H, we may form the adjoint A * of A as follows.First, let Then, for k ∈ dom(A * ), let A * k ∈ H be the unique vector in H such that Ah, k = h, A * k , for all h ∈ dom(A).One can show that if A is densely defined and closed, then so is A * , and Notation 2.2.1.Write C(H) for the set of closed, densely defined linear operators on H. Also, write for the set of normal, self-adjoint, and positive operators on H, respectively.
Next, we recall basic definitions and facts about integration with respect to a projection-valued measure.
Definition 2.2.2 (Projection-valued measure).Let (Ω, F ) be a measurable space and P : F → B(H).We call P a projection-valued measure if P (Ω) = id H , P (G) 2 = P (G) = P (G) * whenever G ∈ F , and whenever (G n ) n∈N ∈ F N is a sequence of disjoint measurable sets.In this case, we call the quadruple (Ω, F , H, P ) a projection-valued measure space.
Remark 2.2.3.To be clear, it is implicitly required in (9) that the series on the right hand side converges in the WOT.It actually follows from the above definition that P (∅) = 0 and P (G 1 ∩ G 2 ) = P (G 1 ) P (G 2 ) whenever G 1 , G 2 ∈ F .(Please see Theorem 1 in Section 5.1.1 of [4].)These are often added to the definition of a projection-valued measure because they guarantee that the series in (9) converges in the SOT.
Finally, if µ is a complex measure on (Ω, F ), then we write |µ| for the total variation measure of µ and µ var := |µ|(Ω) for the total variation norm of µ.
The reason projection-valued measures are relevant for us is the Spectral Theorem, which we now recall.If A ∈ C(H), then the resolvent set ρ(A) ⊆ C of A is the set of λ ∈ C such that A − λ id H : dom(A) → H is a bijection with bounded inverse.The spectrum σ(A) ⊆ C of A is the complement of ρ(A).The resolvent set ρ(A) is open in C, and the spectrum σ(A) is closed in C. Also, a normal operator A ∈ C(H) ν is self-adjoint if and only if σ(A) ⊆ R. Finally, A ∈ C(H) + if and only if A ∈ C(H) sa and σ(A) ⊆ [0, ∞).Theorem 2.2.6 (Spectral Theorem for normal operators).If A ∈ C(H) ν , then there exists a unique projection-valued measure P A : B σ(A) → B(H) such that A = σ(A) λ P A (dλ).We call P A the projectionvalued spectral measure of A. We shall frequently abuse notation and consider P A to be a projection-valued measure defined on B C or, when A ∈ C(H) sa , on B R .

The Spectral Theorem leads to the usual definition of functional calculus. If
This definition enjoys the property that if f, g : We end this section with a review of the concept of an operator affiliated with a von Neumann algebra.

Definition 2.2.7 (Affiliated operators
).An operator a ∈ C(H) is said to be affiliated with M if u * au = a, for all unitaries u belonging to M ′ .In this case, we write a η M. If in addition a is normal (respectively, self-adjoint), then we write a η M ν (respectively, a η M sa ).
Here are some properties of affiliated operators.
(iv) If a ∈ C(H) and a = u|a| is its polar decomposition, then a η M if and only if u ∈ M and We sketch the proofs for the reader's convenience.As we shall see, the first three properties follow without much difficulty from the definitions, the Bicommutant Theorem, and the Spectral Theorem.For the difficult part of item (iv), please see also Lemma 4.4.1 in [24].Sketch of proof.We take each item in turn.
(i) Let a ∈ B(H) and u ∈ M ′ be a unitary.If a ∈ M, then of course u * au = u * ua = a.Now, if a η M, then au = uu * au = ua.Since all C * -algebras are spanned by their unitaries, we conclude that ab = ba, for all b ∈ M ′ .Thus a ∈ M ′′ = M by the Bicommutant Theorem.
(ii) Suppose that P (G) ∈ M, for all G ∈ F .If h, k ∈ H and u ∈ M ′ is a unitary, then it is easy to see that P uh,uk = P h,k .Unraveling the definition of P (ϕ) then gives u * P (ϕ)u = P (ϕ).Thus P (ϕ) η M. It is worth mentioning that one can prove much more directly -without knowing anything about unbounded operators or the Bicommutant Theorem -that if P (G) ∈ M, for all G ∈ F , then P (ϕ) ∈ M, for all ϕ ∈ ℓ ∞ (Ω, F ). Please see Lemma 4.2.16 in [26]. (iii , then a = σ(a) λ P a (dλ) η M by the previous item.Now, suppose that a η M ν , and let u ∈ M ′ be a unitary.Note that Q a := u * P a (•)u : B σ(a) → B(H) is a projection-valued measure, and it is easy to see from the Spectral Theorem and definition of Q a that u * au = σ(a) λ Q a (dλ).But u * au = a by assumption, so the uniqueness part of the Spectral Theorem forces P a = Q a = u * P a (•)u.In other words, P a (G) η M and thus, by item (i), P a (G) ∈ M, for all G ∈ B σ(a) .
(iv) Let a ∈ C(H), a = u|a| be the polar decomposition of a, and v ∈ M ′ be a unitary.If P |a| (G) ∈ M whenever G ∈ B σ(|a|) , then |a| η M by the previous item.If in addition u ∈ M, then we have that Thus |a| η M, and, by the previous item, P |a| (G) ∈ M whenever G ∈ B σ(|a|) .Next, notice that v * uv is a partial isometry, and a = v * av = v * u|a|v = v * uv|a| by what just proved.Finally, |a| η M implies that v * uv has initial space im |a|, and a η M implies that v * uv has final space im a.We conclude that v * uv = u by the uniqueness of the polar decomposition.Thus u η M and so, by item (i), u ∈ M.

Symmetric operator spaces
In Section 3.3, we shall make use of the theory of symmetric operator spaces.In the present section, we review the notation, terminology, and results from this theory that are necessary for our purposes.We refer the reader to [14] for extra exposition, examples, and a thorough list of references.(The reader who is uninterested in Section 3.3 may safely skip at this point to Section 3.1.)For the duration of this section, suppose (M, τ ) is a semifinite von Neumann algebra.
Write Proj(M) := {p ∈ M : p 2 = p = p * } for the lattice of (orthogonal) projections in M.An operator a η M is called τ -measurable if there exists some s ≥ 0 such that τ and let a, b ∈ S(τ ).Then a + b is closable, and a + b ∈ S(τ ); ab is closable, and ab ∈ S(τ ); and a * , |a| ∈ S(τ ).Moreover, S(τ ) is a * -algebra under the adjoint, strong sum (closure of sum), and strong product (closure of product) operations; we shall therefore omit the closures from strong sums and products in the future.For the preceding facts (and more) about τ -measurable operators, please see [25,34].
Fix a ∈ S(τ ).For s ≥ 0, define By definition of τ -measurability, d s (a) < ∞ for sufficiently large s.The function The function µ(a) = µ • (a) is called the (generalized) singular value function or (noncommutative) decreasing rearrangement of a, and µ(a) is decreasing and right-continuous.For properties of d(a) and µ(a), please see [17].Now, let S(τ We therefore extend τ to S(τ ) + via the formula above; this extension is still notated τ : S(τ In this case, we say that a is submajorized by b or that b submarjorizes a (in the "noncommutative" sense of Hardy-Littlewood-Pólya).We now define symmetric operator spaces.
) is a (strongly, fully) symmetric space of τ -measurable operators.
For the strongly/fully symmetric cases, please see Section 9.1 of [14].For the (highly nontrivial) case of an arbitrary symmetric space, please see [20].When 1 ≤ p ≤ ∞ and E = L p := L p ((0, ∞), m), then L p (τ ) as defined using the construction in Fact 2.3.2 is a concrete description of the abstract (completion-based) definition from Section 2.1.When p = ∞, this follows from Lemma 2.5(i) in [17]; when p < ∞, it follows from Lemma 2.5(iv) in [17] and Proposition 2.8 in [15].Moreover, we have 1 Beware: This has nothing to do with the notion of a (Riemannian) symmetric space from geometry.
with equality of norms (if we give L p (τ ) the norm max{ with equality of norms.(This follows from Proposition 2.5 in [15].)To be clear, if Z is a vector space and X, Y ⊆ Z are normed linear subspaces with respective norms • X and • Y , then the subspace In general, if (E, • E ) is a strongly symmetric space of τ -measurable operators, then E ⊆ L 1 (τ ) + M with continuous inclusion, and c E = 1 ⇐⇒ L 1 (τ ) ∩ M ⊆ E with continuous inclusion.This is Lemma 25 in [14] (combined with the last paragraph of the proof of Lemma 3.4.6 in [26]).By Theorem 4.1 in Section II. 4 Finally, we discuss Köthe duals.For a symmetric space (E, • E ), define Of course, a E × could be infinite.
Remark 2.3.4.In the classical case of symmetric Banach function spaces, the Köthe dual of E is called the associate space of E or the space associated with E.
For a proof of this fact, please see Section 5 of [15] or Sections 5.2 and 6 of [14].Now, let (E, • E ) be a strongly symmetric space of τ -measurable operators with c E = 1.Since E × is fully symmetric and c E × = 1, we can consider the Köthe bidual (E ×× , then we call E Köthe reflexive.(This term is not standard; a more common term is maximal.)Note that, by Fact 2.3.3, if E is Köthe reflexive, then E is automatically fully symmetric.
The following is a celebrated equivalent characterization of Köthe reflexivity.It is stated and proven as Proposition 5.14 in [15] and Theorem 32 in [14].
Theorem 2.3.5 (Noncommutative Lorentz-Luxemburg).Let (E, • E ) be a strongly symmetric space of τ -measurable operators with c E = 1.Then E is Köthe reflexive if and only if E has the Fatou property: whenever (a j ) j∈J is an increasing net (j The definition of the Fatou property involves rather arbitrary nets.It is therefore reasonable to be concerned that verifying the Fatou property in classical situations might be quite difficult.However, as we explain shortly, the sequence formulation of the Fatou property is equivalent in classical situations.Let (E ⊆ L 0 ((0, ∞), m), • E ) be a symmetric Banach function space.We say that E has the classical Fatou property if whenever (f n ) n∈N is an increasing sequence of nonnegative functions in E such that sup n∈N f n E < ∞, we have sup n∈N f n ∈ E and sup n∈N f n E = sup n∈N f n E .It turns out (Theorem 4.6 in Section 2.4 of [3]) that if E has the classical Fatou property, then E is fully symmetric, so we may speak of its Köthe dual as a (fully) symmetric Banach function space when E is nonzero.The classical Lorentz-Luxemburg Theorem (e.g., Theorem 1 in Section 71 of [37]) says that a nonzero symmetric Banach function space has the classical Fatou property if and only if it is (strongly symmetric and) Köthe reflexive.In particular, by the Noncommutative Lorentz-Luxemburg Theorem, a symmetric Banach function space has the Fatou property if and only if it has the classical Fatou property.
Remark 2.3.7.Let E be a symmetric Banach function space.By Theorem 3 in Section 65 of [37], E has the classical Fatou property if and only if whenever (f n ) n∈N is a sequence of nonnegative functions in E with lim inf n→∞ f n E < ∞, we have lim inf n→∞ f n ∈ E and lim inf n→∞ f n E ≤ lim inf n→∞ f n E , i.e, Fatou's Lemma holds for • E .Hence the property's name.

Ideals of von Neumann algebras
For Section 3, fix a complex Hilbert space (H, •, • ) and a von Neumann algebra M ⊆ B(H).

Properties to request of ideals
In this section, we introduce some abstract properties of ideals of M that are useful in the study of MOIs and their applications to the differentiation of operator functions.In Section 3.2, we give several classes of examples that do not require the theory of symmetric operator spaces to understand.In Section 3.3, we give a large class of additional examples using the theory of symmetric operator spaces.Definition 3.1.1(Symmetrically normed ideals).Let A be a Banach algebra and J ⊆ A be an ideal, i.e., a linear subspace such that arb ∈ J whenever a, b ∈ A and r ∈ J ; in this case, we write J A. Suppose we have another norm If in addition a, b ∈ A and r ∈ J imply arb J ≤ a A r J b A , then we call (J , • J ) a symmetrically normed ideal of A and write (J , • J ) s A or J s A when confusion is unlikely.Remark 3.1.2.Beware: Definitions of a symmetrically normed ideal vary in the literature.Sometimes it is required that C J = 1.Sometimes A is required to be a von Neumann or C * -algebra and J is required to be a * -ideal with r * J = r J , for all r ∈ J .Sometimes even more requirements are imposed.We take the above minimal definition because it is all we need.We now define two additional properties one can demand of Banach or symmetrically normed ideals of a von Neumann algebra.Before doing so, however, we make an observation.Let (Σ, H , ρ) be a measure space, (I, • I ) M be a Banach ideal, and F : Σ → I ⊆ M be weak * measurable.By definition, In particular, if Σ F I dρ < ∞, then Proposition 2.1.4says that F : Σ → M is weak * integrable.Remark 3.1.4.First, the name for property (M) is inspired by Theorem 2.1.11.However, inequalities like the one required in (a) are called triangle inequalities in the theory of vector-valued integrals.Therefore, it would also be appropriate to name (a) the "integral triangle inequality property."However, this would lead naturally to the abbreviation "property (T)," which is already decidedly taken.Second, if H is separable, then one can show that the pointwise product of weak * measurable maps Σ → M is itself weak * measurable.
In particular, the requirement in (b By testing the definition on the one-point probability space, we see that an integral symmetrically normed ideal is symmetrically normed.We also have the following.
Thus I is integral symmetrically normed.

Examples of ideals I
In this section, we exhibit several examples of ideals with property (M), namely the trivial ideals, the noncommutative L p -ideals, separable ideals, and the ideal of compact operators.
by Noncommutative Hölder's Inequality (Théorème 6 in [12]) and the completeness of (L p (τ ), Notice that the ideal of compact operators is left out of the above examples.To include it in the mix, we first prove that separable ideals have property (M).Proof.To prove this, we make use of the basic theory of the Bochner integral; please see, for instance, Appendix E of [5] for the relevant background.
Let (Σ, H , ρ) be a measure space, F : Σ → I ⊆ M be weak * measurable, and h, k ∈ H. Now, define ℓ h,k : I → C by r → rh, k .Since the inclusion ι I : (I, measurable by assumption.Since the collection {ℓ h,k : h, k ∈ H} clearly separates points, we conclude from the (completeness and) separability of I and Proposition 1.10 in Chapter I of [36] that F : Σ → (I, • I ) is Borel measurable.Using again the separability of I, this implies F : Σ → (I, • I ) is strongly (or "Bochner") measurable.Therefore, if in addition Σ F I dρ = Σ F I dρ < ∞, then F : Σ → (I, • I ) is also Bochner integrable, and -by applying ℓ h,k to the Bochner integral -the Bochner and weak * integrals of F agree.Thus Σ F dρ ∈ I and Σ F dρ I ≤ Σ F I dρ, by the triangle inequality for Bochner integrals.This completes the proof.In particular, if H is separable, then the ideal K(H) s B(H) of compact operators H → H has property (M).Actually, this also implies the non-separable case by an argument suggested by J. Jeon.Lemma 3.2.4.For a closed linear subspace K ⊆ H, write ι K : K → H and π K : H → K for, respectively, the inclusion of and the orthogonal projection onto K. Fix A ∈ B(H).Then A ∈ K(H) if and only if Proof.The "only if" direction is clear.For the "if" direction, suppose that Of course, K is a separable, closed linear subspace of H that contains {h n : n ∈ N} and is invariant under Proof.Let (Σ, H , ρ) be a measure space and F : Σ → K(H) ⊆ B(H) be weak * measurable with Σ F dρ < ∞.
Since we already know the triangle inequality for the operator norm, it suffices to prove Σ F dρ ∈ K(H).To this end, let K ⊆ H be a closed, separable linear subspace.Then, in the notation of Lemma 3.2.4, we conclude from Lemma 3.2.4 that Σ F dρ ∈ K(H).Remark 3.2.6.In case one only wants to know K(H) is integral symmetrically normed, there is a different proof available that does not go through the separable case first.Indeed, let (Σ, H , ρ) be a measure space and A, B : Σ → B(H) be as in 3.1.3(b).To prove the claim, it suffices to show that if c ∈ K(H), then is arbitrary, then -using, for instance, the singular value decomposition -there is a sequence (c n ) n∈N of finite-rank linear operators H → H such that c n − c → 0 as n → ∞.But then, by the operator norm triangle inequality, Σ A(σ) c n B(σ) ρ(dσ) → Σ A(σ) c B(σ) ρ(dσ) in the operator norm topology as n → ∞.Since this exhibits Σ A(σ) c B(σ) ρ(dσ) as an operator norm limit of compact operators, we conclude it is compact, as desired.

Examples of ideals II
At this point, we shall make heavy use of the theory reviewed in Section 2.3.For the duration of this section, suppose (M, τ ) is a semifinite von Neumann algebra.To begin, we note that if (E, • E ) is a symmetric space of τ -measurable operators, then This follows from Proposition 17 in [14].We call E the ideal induced by E. In this section, we prove that ideals induced by fully symmetric spaces are integral symmetrically normed and that ideals induced by symmetric spaces with the Fatou property have property (M).
Proof.Let (Σ, H , ρ) be a measure space and A, B : Σ → M be as in 3.
By Proposition 4.1 in [15], this implies because E is fully symmetric; in other words, T restricts to a bounded linear map Thus E is integral symmetrically normed.If (Σ, H , ρ) is a measure space and F : Σ → M is weak * integrable, then Proof.Let a := Σ F dρ ∈ M. By Fact 2.3.3 (twice) and Theorem 2.1.11,we have In particular, by Fact 2.3.3, if the right hand side is finite, then Σ F dρ ∈ E ×× .
Proof.Applying Theorem 3.3.3 to the space E ×× = (E × ) × and using that

Comments about property (F)
A Banach ideal (I, • I ) M has (the sequential) property (F) if whenever a ∈ M and (a j ) j∈J is a net (sequence) in I such that sup j∈J a j I < ∞ and a j → a in the S * OT, we have a ∈ I and a I ≤ sup j∈J a j I .In [2], certain multiple operator integrals in invariant operator ideals with property (F) are considered.We now take some time to discuss the relationship between properties (M) and (F).First, there are certainly ideals with property (M) that do not have property (F), e.g., the ideal of compact operators (Proposition 3.2.5).Second, as mentioned in [2], the motivating example of an invariant operator ideal with property (F) is an ideal induced via Fact 2.3.2 by a (nonzero) symmetric Banach function space with the Fatou property.By Theorem 3.3.6 and Example 2.3.6,such ideals have property (M).Third, the author is unaware of an example of a symmetrically normed ideal with property (F) that does not have property (M).It would be interesting to know if such an ideal exists.
In this context, it is worth discussing a technical issue in [2] with its treatment of operator-valued integrals.For the rest of this section, assume H is separable.It is implicitly assumed in the proof of (the second sentence of) Lemma 4.6 in [2] that at least some form of the integral triangle inequality holds for the I-norm • I when I has property (F).Specifically, it seems to be assumed that if (Σ, H , ρ) is a finite measure space and F : Σ → I ⊆ M is • I -bounded and weak * measurable, then (ignoring that F I may not be measurable).Let us call this the finite property (M).Then we may rephrase the implicit claim as "property (F) implies the finite property (M)."As far as the author can tell, the arguments in [2] are only sufficient to prove Indeed, the authors of [2] prove that I has property (F) if and only if I 1 = {r ∈ I : r I ≤ 1} is a complete, separable metric space in the strong * operator topology and then apply Propositions 1.9-1.10 in Chapter I of [36] to approximate F by simple functions in the strong * operator topology.Crucially, Propositions 1.9-1.10 in [36] only guarantee the existence of a sequence (F n ) n∈N of simple functions Σ → I such that sup σ∈Σ F n (σ) I ≤ sup σ∈Σ F (σ) I , for all n ∈ N, and F n → F pointwise in the strong * operator topology as n → ∞.Now, by Proposition 2.1.7,Σ F n dρ → Σ F dρ in the strong * operator topology as n → ∞.Also, by the (obvious) triangle inequality for integrals of simple functions, if k ∈ N, then The definition of property (F) does not guarantee that F n (σ) I → F (σ) I as n → ∞, so we cannot evaluate the limit superior above much further without an upgraded version of property (F).(Interestingly, this does not damage the applications in [2], since it seems only the estimate ( 10) is used seriously.)It therefore seems that property (F) almost implies some weaker form of property (M) -but perhaps not quite.
Remark 3.4.1.Though we centered the discussion above on the "finite property (M)," it is worth pointing out that in order to prove Lemma 4.6 in [2], it would actually be sufficient to know the following "finite integral symmetrically normed" condition: for every finite measure space (Σ, H , ρ) and • -bounded, weak * measurable A, B : Σ → M, we have Σ A(σ) r B(σ) ρ(dσ) ∈ I and Σ A(σ) r B(σ) ρ(dσ) I ≤ r I Σ A B dρ, for all r ∈ I.As mentioned, in the presence of property (F), we would already know Σ A(σ) r B(σ) ρ(dσ) ∈ I, so -as was the case above -it is really only the integral triangle inequality that is potentially missing.
This follows from Theorem 1.1.3(and Equation (4.14)) in [26].We shall also need to know that, in general, if is weak * measurable, for all b 1 , . . ., b k ∈ M.This follows from a repeated application of Proposition 4.2.3 in [26], and we shall use it in the sequel without further comment.[26] for an explanation of the use of the # symbol above.Also, please be aware that T a1,...,a k+1 ϕ is a common alternative way to notate I a1,...,a k+1 ϕ.
The following are two algebraic properties of MOIs that we shall use.They are proven as part of Proposition 4.3.1 in [26].Proposition 4.1.5(Algebraic properties of MOIs).Suppose 1 ≤ m ≤ k.
. In this case, I P ϕ also restricts to a bounded k-linear map (I, This definition may seem contrived, but the following shows that all the examples of ideals from Sections 3.2 and 3.3 are MOI-friendly.Proof.Suppose that I is integral symmetrically normed and that we are in the setup of Theorem 4.1.2.

Divided differences and perturbation formulas
Our goal is to differentiate operator functions in integral symmetrically normed ideals.As is common practice, we begin by proving "perturbation formulas."To do so, we shall use a generalization of the argument from the proof of Theorem 1.2.3 in [30].First, we review divided differences.Definition 4.2.1 (Divided differences).Let f : R → C be a function.Define f [0] := f and, for k ∈ N and distinct λ 1 , . . ., λ k+1 ∈ R, recursively define We call f [k] the k th divided difference of f .
In particular, f [k] is symmetric in its arguments.As we shall see shortly, if in addition f ∈ C k (R), then f [k]  extends uniquely to a continuous function defined on all of R k+1 .
Explicitly, ρ m is the finite Borel measure on ∆ m such that The following is proven using the Fundamental Theorem of Calculus and induction.
where λ := (λ 1 , . . ., λ k+1 ) and • is the Euclidean dot product.In particular, f [k] extends uniquely to a symmetric continuous function on all of R k+1 .We shall use the same notation for this extended function.
Before stating and proving our perturbation formulas, we make an observation that we shall use repeatedly.If f : R → C is Lipschitz, then there are constants for all λ ∈ R. In particular, it follows from the definition of functional calculus and the Spectral Theorem that dom(a) ⊆ dom(f (a)), (11) for all a ∈ C(H) sa .
More precisely, f (a + c) − f (a) is densely defined and bounded, and f [1] (a + c, a)#c is its unique bounded linear extension.Now, suppose k for all j ∈ {1, . . ., k}.
Proof.We first make an important observation.
For general a ∈ C(H) sa , we begin by showing f (a + c) − f (a) is densely defined; specifically, we show dom(a) ⊆ dom(f (a + c) − f (a)).Indeed, since f [1] ∈ ℓ ∞ (R 2 , B R 2 ), f is Lipschitz on R. By (11), we have dom(a 0 ) ⊆ dom(f (a 0 )), for all a 0 ∈ C(H) sa .In particular, since dom(a) = dom(a + c), we get dom(a) = dom(a) ∩ dom(a + c) ⊆ dom(f (a + c)) ∩ dom(f (a)) = dom(f (a + c) − f (a)), as desired.Next, let p n , q n , a n , and d n be as in the first paragraph.If (Σ, ρ, ϕ 1 , ϕ 2 ) is a ℓ ∞ -IPD of f [1] , then the results of the previous two paragraphs and Lemma 4.2.4 give [1] (a + c, c)#[q n c p n ] → f [1] (a + c, a)#c in the SOT as n → ∞, since q n → 1 and p n → 1 in the SOT as n → ∞.But now, notice and similarly q n f (d n )p n = q n f (a + c)p n .(For the latter, we use that im p n ⊆ dom(a) ⊆ dom(f (a + c)).)It follows that if m ∈ N, h ∈ im p m , and n ≥ m, then [1] (a + c, c)#c)h for h ∈ im p m .Since m∈N im p m ⊆ H is a dense linear subspace, we are done with the first part.
Next, let k ≥ 2 and f ∈ C k (R) be such that f k+1) .By definition and symmetry of divided differences, if j ∈ {1, . . ., k} and λ, µ ∈ R, then for λ = (λ 1 , . . ., λ k−1 ) ∈ R k−1 .Now, suppose a, c ∈ B(H) sa again, and fix a = (a 1 , . . ., This allows us to apply I aj−,a+c,a, aj+ to (15), which may be rewritten If we do so and then plug (b j− , 1, b j+ ) into the result, then we get where in the last line we used Proposition 4.1.5(ii)and the definition of ψ.
Finally, for general a ∈ C(H) sa , let p n , q n , a n , and d n be as in the first paragraph.If 1 < j < k, then we also let b •,n := (b (j−1)− , b j−1 q n , p n b j , b (j+1)+ ).Since p n → 1 and q n → 1 in the SOT, Lemma 4.2.4 gives in the SOT as n → ∞, by the observation from the first paragraph and Lemma 4.2.4.(Above, empty products are declared to be 1.)Since we already know from the previous paragraph that for all n ∈ N, this completes the proof when 1 < j < k.For the cases j ∈ {1, k}, we redefine Then we use an argument similar to the one above to see that and Then we use Lemma 4.2.4 to take n → ∞.This completes the proof.Corollary 4.2.7.Let M ⊆ B(H) be a von Neumann algebra and a η M sa .Suppose (I, Proof.Since a η M sa and c ∈ I sa ⊆ M sa , it is easy to see a + c η M sa as well.In particular, the projectionvalued measures P a+c and P a take values in M. It then follows from (12) and the definition of MOI-friendly that One can show using essentially the same proofs that if a, b ∈ C(H) sa and q ∈ B(H) are such that aq − qb ∈ B(H) (i.e., aq − qb is densely defined and bounded), then f (a)q − qf (b) ∈ B(H) and As a result, we get a quasicommutator estimate in MOI-friendly ideals.Let M ⊆ B(H) be a von Neumann algebra, and suppose (I, ) aq − qb I .Such quasicommutator estimates are of interest in the study of operator Lipschitz functions.Please see [1] or [30] for more information.

4.3
The spaces BOC(R) ⊗i (k+1) and OC [k] (R) In the following section, we prove a general result about derivatives of operator functions.In this section, we introduce the functions whose operator functions we shall be differentiating.Then we use Peller's work from [29], which we review in detail in Appendix A, to give a large class of examples of such functions.Definition 4.3.1 (Operator continuity).Fix f ∈ ℓ 0 (R, B R ).We say that f is operator continuous if (a) for every complex Hilbert space H, a ∈ C(H) sa , and c ∈ B(H) sa , the operator f (a + c) − f (a) is densely defined and bounded; and (b) for every complex Hilbert space H and a ∈ C(H) sa , f (a + c) − f (a) → 0 in B(H) as c → 0 in B(H) sa .
(More precisely, for every a ∈ C(H) sa and ε > 0, there is some δ > 0 such that f (a + c) − f (a) < ε whenever c ∈ B(H) sa and c < δ.) In this case, we write f ∈ OC(R).If in addition f is bounded, then we write f ∈ BOC(R).
Taking H = C in the definition, it is clear that operator continuous functions are continuous.Also, we observe that if f, g ∈ BOC(R), H is a complex Hilbert space, a ∈ C(H) sa , and c ∈ B(H) sa , then In particular, σ → ψ(•, σ) ℓ ∞ (R) is measurable.Thus the following definition makes sense (without needing to use upper or lower integrals).
is a Banach * -algebra under pointwise operations.k+1) follows from the definitions and an application of the (standard) Dominated Convergence Theorem.The rest of the statement follows from the observation above that BOC(R) is a * -algebra and arguments similar to (but easier than) those in the proof of Proposition 4.1.4in [26].
We now introduce the space of functions to which the main result of the following section applies.
and let for r > 0, then it can be shown -using standard arguments and Proposition 4.3.3-that OC [k] (R) is a Fréchet space with the topology induced by the collection { • OC [k] ,r : r > 0} of seminorms.One can even show that OC [k] (R) is a * -algebra under pointwise operations, and that these operations are continuous.Since we shall not need these facts, we shall not dwell on them.Instead, we turn to examples.Definition 4.3.5 (Wiener space).If k ∈ N, then we define the k th Wiener space to be the set of functions f : R → C such that there is a (unique) Borel complex measure . We now prove by elementary means that W k (R) ⊆ OC [k] (R).Then we use Peller's work from [29] to generalize this substantially.Proof.Of course, f is bounded and continuous.Now, if λ, µ ∈ R, then f [1] (λ, µ) = e itλξ e i(1−t)ξµ dt by Proposition 4.2.3.This is clearly a ℓ ∞ -integral projective decomposition of f [1] that yields f [1]  ℓ ∞ (R,B R ) ⊗i ℓ ∞ (R,B R ) ≤ |ξ|.In particular, if a ∈ C(H) sa and c ∈ B(H) sa , then f (a + c) − f (a) ≤ |ξ| c by Corollary 4.2.7.It follows that f is operator continuous.
Remark 4.3.8.For similar reasons, if f ∈ C k (R) and, for all j ∈ {1, . . ., k}, f (j) and the Fourier transform of f (j) belong to L 1 (R), then f ∈ OC [k] (R).Now, we use more serious harmonic analysis done by Peller [29] to exhibit a large class -containing W k (R) strictly -of functions belonging to OC [k] (R).We begin by defining Besov spaces.Notation 4.3.9.If m ∈ N, then we write S (R m ) for the Fréchet space of Schwartz functions R m → C and S ′ (R m ) := S (R m ) * for the space of tempered distributions on R m .Also, the conventions we use for the Fourier transform and its inverse are, respectively, We call Ḃs,p q (R m ) := {f ∈ S ′ (R m ) : f Ḃs,p q < ∞} the homogeneous (s, p, q)-Besov space.
Remark 4.3.11.First, note that ϕ * f, ϕ j * f have compactly supported Fourier transforms and so are smooth by the Paley-Wiener Theorem; it therefore makes sense to apply the L p -norm to them.Second, since it is easy to show that f Ḃs,p q = 0 if and only if f is a polynomial, it is usually best to define Ḃs,p q (R m ) as a quotient space in which all polynomials are zero.The definition above is given in Chapter 3 of [27] and Sections 5.1.2and 5.1.3 of [35].The definition "modulo polynomials" is given in Section 2.4 of [31].(Please see Section 1.2.5.3 of [31] as well.)Finally, beware that the positions of p and q in Ḃs,p q (R m ) are far from consistent in the literature.
The case of interest is m = 1 and (s, p, q) = (k, ∞, 1) for k ∈ N. As we show in Section A.2, in this case it turns out Ḃk,∞ Therefore, if we are to prove sensible results about differentiating the operator function c → f (a + c) − f (a) when a is unbounded and f ∈ Ḃk,∞ 1 (R), it is necessary to impose additional restrictions that exclude (at least) polynomials of degree higher than two.We accomplish this with the following modified Besov spaces.Definition 4.3.12(Peller-Besov spaces).If k ∈ N, then we define The following result is a slight upgrade of Theorem 5.5 in [29] or Theorem 2.2.1 in [30].
Theorem 4.3.13(Peller [29]).If k ∈ N, then there is a constant c k < ∞ such that . The proof given in [29] is not very detailed and is only explicit in the cases k ∈ {1, 2}, so we present a full proof of this theorem in Appendix A. As a result, we obtain the following.
for all f ∈ P B 1 (R) ∩ P B k (R), where c 1 , . . ., c k are as in Theorem 4.3.13.
Since it is easy to show that W k (R) P B 1 (R) ∩ Ḃk,∞

Derivatives of operator functions in ideals
In this section, we finally differentiate operator functions in integral symmetrically normed ideals.For the duration of this section, fix a complex Hilbert space (H, •, • ), a von Neumann algebra M ⊆ B(H), and (I, • I ) M. Also, write As a consequence of the definition of a Banach ideal, I sa is a real Banach space when it is given (the restriction of) the I-norm • I .Now, before setting up the main result of this section, we prove a key technical lemma that is the main reason integral symmetrically normed ideals are considered in this paper.

But
Therefore, the definition of integral symmetrically normed gives T j,n (b 1 , . . ., b k ) ∈ I and Next, fix σ ∈ Σ.Since c j,n − c j ≤ C I c j,n − c j I → 0 as n → ∞, the operator continuity of ϕ j (•, σ) gives which is finite, we conclude from ( 16) and Lemma 2. Next, we recall the notion of Fréchet differentiability of maps between normed vector spaces and then define what it means for a scalar function to be I-differentiable.For these purposes, note that if We use this identification below.Definition 4.4.4(Fréchet differentiability).Let V and W be normed vector spaces, U ⊆ V be open, and F : U → W be a map.For p ∈ U , we say F is Fréchet differentiable at p if there exists (necessarily unique) DF (p) ∈ B(V ; W ) such that In this case, we write Concretely, if F : U → W is k-times Fréchet differentiable (in U ), then one can show by induction that for all p ∈ U and h 1 , . . ., h k ∈ V .In this case, we write Suppose that f : R → C is Lipschitz and f (a + c) − f (a) ∈ I, for all a η M sa and c ∈ I sa (i.e., f a,I : I sa → I is defined everywhere).We claim that if f is k-times I-differentiable, then f a,I is k-times Fréchet differentiable everywhere -not just at 0 ∈ I sa .Indeed, fix b, c ∈ I sa , and note that This is the case because ( 17) is immediate from the definition on which is dense in H. (Note that we used (11).)In other words, With this in mind, here is the main result of this section.Theorem 4.4.6 (Derivatives of operator functions in ISNIs).Suppose (I, • I ) M is integral symmetrically normed, and fix a η M sa .If f ∈ OC [k] (R), then f a,I : I sa → I is defined everywhere, and f a,I ∈ C k (I sa ; I).In particular, f is k-times I-differentiable.Moreover, for all (b 1 , . . ., b k ) ∈ I k sa .By the observation above, we therefore also have In particular, by Corollary 4.2.7,f a,I (c) = f (a + c) − f (a) ∈ I, for all c ∈ I sa .In addition, we observe that if f ∈ OC [k] (R), then the map is continuous by the Continuous Perturbation Lemma (Lemma 4.4.2).Therefore, the claimed k th derivative map is, in fact, continuous.Thus, to prove the theorem, it suffices to prove the claimed formula for D k I f (a).We do so by induction on k.
Therefore, by the Continuous Perturbation Lemma (Lemma 4.4.2), 0) − f [1] (a, a)#c I ≤ I a+c,a f [1] − I a,a f [1]  B(I) → 0 as c → 0 in I sa .This completes the proof when k = 1.Next, suppose k ≥ 2 and that we have proven the claimed derivative formula when f ∈ OC [k−1] (R).To prove the formula for f ∈ OC [k] (R), we set some notation and make some preliminary observations.If S is a set, s ∈ S, and m ∈ N 0 , then we write using Notation 4.2.5.Next, by the inductive hypothesis, Combining this inductive hypothesis with the expression for δ(b, c) above gives as claimed.This completes the proof.
Remark 4.4.7.Let H be a separable complex Hilbert space, (M ⊆ B(H), τ ) be a semifinite von Neumann algebra, and (E, • E ) be a separable symmetric Banach function space.In [11], it is proven (Theorem 5.16) that if f : R → R is a continuous function such that f [1] admits a decomposition as in Definition 4.3.2 with only ϕ 1 (•, σ), ϕ 2 (•, σ) ∈ BC(R) (i.e., these functions are not assumed to be operator continuous) and if a ∈ S(τ ) sa , then the map E(τ sa makes sense and is Gateaux differentiable at 0 with Gateaux derivatives expressible as double operator integrals involving f [1] .In particular, this result applies when E = L p with 1 ≤ p < ∞.It is noted, however, in the introduction of [11] that Fréchet differentiability does not in general hold in this setting.This is why we must work in the space (E(τ ), ) (e.g., L p (τ )) instead of the space (E(τ ), • E(τ ) ) (e.g., L p (τ )), to prove positive results about Fréchet differentiability in this setting.(Also, our method -particularly the extra assumption of operator continuity in our decompositions -allows us to assume only that a η M sa , i.e., we need not assume that a is τ -measurable.)In short, the results in [11] are, for good reason, of a different flavor than the results in the present paper.
Notice that Proposition A.1.2allows us to make sense of the expression (18) in the first place.By item (iv), the integrand in ( 18) is bounded, continuous, and vanishes when | u| > σ.Therefore, the integral above is really over { u ∈ R k + : | u| ≤ σ}, which has finite measure.This, together with the continuity part of item (ii) and the Dominated Convergence Theorem, also implies the right hand side of ( 18) is continuous in λ.
The expression ( 18) is proven, inspired by the sketch in [29], in the following steps.
Step 1. Use an approximation procedure (Lemma A.1.6) to reduce to the case f, f ∈ L 1 (R).
Step 2. Use an inductive argument to reduce to the case k = 1.
The approximation procedure in Step 1 will also help us to prove Proposition A.1.2(iv).Proof.We take each item in turn.
(ii) Of course, ω n ∈ S (R) ⊆ L 1 (R), so that We are now ready for the proof of Step 2.

Σ
A(σ) r B(σ) ρ(dσ) I ≤ r I Σ A B dρ, r ∈ I.The precise definition (Definition 3.1.3(b))is slightly technical, so we omit it from this section.Our first main result comes in the form of a list of interesting examples of ISNIs.Theorem 1.2.1 (Examples of ISNIs).Let H be an arbitrary (not-necessarily-separable) complex Hilbert space and M ⊆ B(H) be a von Neumann algebra.(i) The ideal (M, • ) is integral symmetrically normed.(ii)If (I, • I ) is a separable symmetrically normed ideal of M, then I is integral symmetrically normed.
Fréchet differentiable (Definition 4.4.4) in the I-norm • I , and

1 2
so we may define the absolute value |A| := (A * A) ∈ C(H) + of A via functional calculus.Also, there exists a unique partial isometry U ∈ B(H) with initial space im |A| = im(A * ) and final space im A such that A = U |A|.(In particular, dom(A) = dom(|A|).)This is called the polar decomposition of A. (Please see Section 8.1, particularly Theorems 2 and 3, of[4].) and µ(a) ≤ µ(b) imply a ∈ E and a E ≤ b E ; (b) a strongly symmetric space of τ -measurable operators -a strongly symmetric space for short -if it is a symmetric space, and a, b ∈ E and a ≺≺ b imply a E ≤ b E ; and (c) a fully symmetric space of τ -measurable operators -a fully symmetric space for shortif a ∈ S(τ ), b ∈ E, and a ≺≺ b imply a ∈ E and a E ≤ b E .If (E, • E ) is a symmetric space, then we define Proj(E) := E ∩ Proj(M) and c E := sup Proj(E) ∈ Proj(M) and call c E the carrier projection of E. Next, we describe a large class of examples of symmetric spaces.Let m be the Lebesgue measure on the positive halfline (0, ∞) and (N
Proposition 3.1.5.Let (I, • I ) s M. If I has property (M), then I is integral symmetrically normed.Proof.Suppose that I s M has property (M).Let A, B : Σ → M be as in 3.1.3(b),and fix r ∈ I. Since I is symmetrically normed, A(σ) r B(σ) I ≤ r I A(σ) B(σ) whenever σ ∈ Σ. Applying the definition of property (M) to

Remark 3 . 3 . 2 .
The argument above is inspired in part by Section 4.4 of[16].The second main result of this section upgrades Theorem 3.3.1 when the symmetric space in question is a Köthe dual.(It also generalizes Theorem 2.1.11.)Theorem 3.3.3(Köthe duals and property (M)).Let (E, • E ) be a strongly symmetric space with c E = 1.

Definition 4 . 4 . 5 (
I-differentiability). Fix a η M sa .A Borel measurable function f : R → C is called k-times (Fréchet) I-differentiable at a if there is an open set U ⊆ I sa with 0 ∈ U such that (a) f (a + b) − f (a) ∈ I for all b ∈ U (i.e., when b ∈ U , f (a + b) − f (a) is densely defined and bounded, and its unique bounded linear extension belongs to I), and (b) the map U ∋ b → f a,I (b) := f (a + b) − f (a) ∈ I is k-times Fréchet differentiable (with respect to • I ) at 0 ∈ U ⊆ I sa .