Approximations of self-adjoint C 0-semigroups in the operator-norm topology

The paper improves approximation theory based on the Trotter– Kato product formulae. For self-adjoint C0-semigroups we develop a lifting of the strongly convergent Chernoff approximation (or product) formula to convergence in the operator-norm topology. This allows to obtain optimal estimate for the rate of operator-norm convergence of Trotter–Kato product formulae for Kato functions from the class K2.


Introduction
The aim of the paper is to present a new generalised proof of approximation theory developed in [6,7]. For self-adjoint Trotter-Kato product formulae it allows to obtain optimal estimate for the rate of convergence in operator norm for Kato functions of class K β , where β = 2 (see [7]).
Instead of a double-iteration procedure of [7] we extend in this paper the Chernoff approximation formula [5] and the Trotter-Neveu-Kato approximation theorem [8], Theorem IX.2.16, to the operator-norm topology. Essentially we follow here the idea of lifting the strongly convergent Chernoff approximation formula to operator-norm convergence [9,11], whereas majority of results concerning this formula are about the strong operator topology, see, for example, review [2]. In the same vein we quote a recent book [1], where different aspects of semigroup convergence in the strong operator topology are presented in great details.
To proceed, we first recall definition of the Kato functions that belong to the class K β . is such that for any ε > 0 there exists a positive constant δ ε < 1 implying and that for some β, where 1 < β ≤ 2, Note that the Kato functions of class K β are not necessarily monotonously decreasing, but it is true in a vicinity of s = +0. For more details about different types of Kato functions see Appendix C in [12]. By Definition 1.1 and by the spectral theorem one gets that for any non-negative self-adjoint operator A the bounded operator-valued function t → f (tA) ∈ L(H) is strongly continuous in R + and right-continuous on R + 0 = R + ∪ {0}, that is, s-lim t→+0 f (tA) = 1.
One of the main corollaries of the semigroup approximation results established in the present paper (Theorem 4.5) is the statement about operator norm convergence of the Trotter-Kato product formulae, see Section 5. Proposition 1.2. Let f, g ∈ K 2 . If A and B are non-negative self-adjoint operators in a separable Hilbert space H with domains dom A and dom B such that the operator sum C := A + B is self-adjoint on dom C = dom A ∩ dom B, then g(tB/n) 1/2 f (tA/n)g(tB/n) 1/2 n − e −tC = O(n −1 ), (1.5) f (tA/n)g(tB/n) n − e −tC = O(n −1 ), (1. 6) for n → ∞, hold in the operator norm topology. The convergence is locally uniform on R + 0 , but if operator C is strictly positive, it is uniform on R + 0 . Note that the rates of convergence in (1.5) and in (1.6) are optimal, i.e., they can not be improved in the general setup [7].

Chernoff approximation formula: strong operator topology
In this section we give a proof of the Chernoff approximation formula in the strong operator topology, which is alternative to the original one based on the √ n-Lemma [5]. In conclusion we relax some conditions of the main Theorem 2.3.
Let F (·) : R + 0 −→ L(H) be a measurable family of non-negative selfadjoint contractions F (t) ≤ 1 such that F (0) = 1. We set Then for each τ > 0 bounded operator S(τ ) is self-adjoint and positive. Let H ≥ 0 be self-adjoint operator in H. Then by the Trotter-Neveu-Kato convergence theorem we obtain that holds if and only if uniformly in t ∈ I for any closed bounded interval I ⊂ R + 0 . In this case we say that this convergence holds locally uniformly in t ∈ R + 0 , whereas if I ⊂ R + , then convergence holds locally uniformly away from zero. For example, setting τ = t/η for η ≥ 1, we obtain locally uniformly away from zero. To proceed, we need the following elementary estimate: The next assertion serves to lift the weak convergence of vectors {u n } n≥1 in H to the strong convergence of this sequence.

Lemma 2.2.
Let {u n } n≥1 be a weakly convergent sequence of vectors, w-lim n→∞ u n = u, in a Hilbert space H. If, in addition, lim n→∞ u n = u , then s-lim n→∞ u n = u.

Proof. Note that
Then by conditions of the lemma this yields s-lim n→∞ u n = u.
Now we are in position to prove the Chernoff approximation formula for self-adjoint semigroups in the strong operator topology.
be a measurable family of nonnegative self-adjoint contractions such that F (0) = 1 and let H ≥ 0 be a self-adjoint operator in H. The convergence holds locally uniformly in t away from zero if and only if the condition (2.2) is satisfied.
Proof. First we assume that condition (2.2) is satisfied. Let us show that locally uniformly in t > 0. To this end we use the spectral functional calculus for self-adjoint operators to obtain the representation Then inequality (2.5) yields the estimate Setting τ = t/η, η ≥ 1, and r = t/τ ≥ 1, we obtain This proves the limit (2.7) locally uniformly in t away from zero. Since condition (2.2) is equivalent to (2.4), the representation and (2.7) yield (2.6) . Conversely, assume that (2.6) is satisfied. Using representation we get from (2.6) and (2.7) that (2.4) holds locally uniformly in t away from zero. For τ = t/η, we verify that convergence in (2.3) holds locally uniformly in t ∈ R + 0 . Then by the Trotter-Neveu-Kato convergence theorem the limit (2.3) implies (2.2).
The limit (2.6) yields in particular that the semigroup approximation formula also holds locally uniformly away from zero for the sequence when η = n ∈ N.
Taking into account Lemma 2.4, the conditions of Theorem 2.3 can be relaxed as follows.
is satisfied.
We skip the proof since the line of reasoning is straightforward.

Lifting the Chernoff approximation formula to operator-norm topology
A natural question arises: can the limit (2.6) in Theorem 2.3 (or in Theorem 2.5) be lifted to convergence in the operator-norm topology? First we note that in contrast to quasi-sectorial contractions [10], the estimates for self-adjoint contraction C in the Chernoff √ n-Lemma [5] and in its refinement due to the 1/ 3 √ n -Theorem (see Lemma 2.1 and Theorem 3.3 in [11]) can be significantly improved. Namely, the spectral functional calculus of self-adjoint contraction C and Lemma 2.1 yield Similarly to the case of the strong operator topology, the next step in the program of lifting the approximation formula to operator-norm topology involves the lifting of the Trotter-Neveu-Kato convergence theorem. Therefore, we proceed with the following lemma, which is well suited for selfadjoint lifting of this theorem.
Lemma 3.1. Let K and L be non-negative self-adjoint operators in a Hilbert space H. Then with a constant c > 0 independent of operators K and L.
Proof. By the Riesz-Dunford functional calculus, one obtains for the difference of exponentials the representation where the contour Γ is a union of two branches: (3.5) Setting where the constant c Γ depends only on Γ but not on the operator K. Similarly, from (3.6) one also gets Using these estimates, we find from (3.5) that Since for z ∈ Γ ∞ the value of e z > 0, the integral is convergent and c depends only on the contour Γ.
The first step towards the proof the operator-norm convergence of the Chernoff approximation formula (2.6) would be lifting of the strong convergence in (2.2) to the operator-norm convergence. To study the consequence of this lifting we prove the following assertion.
is satisfied if and only if for any closed interval I ⊂ R + , i.e., locally uniformly away from zero.
Proof. A straightforward computation shows that Here we used the fact that if t > 0 and τ > 0, then for self-adjoint operator S(τ ) the closure For these values of arguments t and τ we get If I is a closed interval of R + , for example, for t ∈ I and τ > 0. Setting τ = t/η we find Since by (3.7) we obtain for the last factor in the right-hand side of (3.9) the estimate (3.9) yields (3.8). The converse is obvious.
Lemma 3.2 allows to advance in generalisation of self-adjoint Chernoff approximation formula for operator-norm convergence.
holds for some t 0 > 0, then (3.10) holds for any closed interval I ⊂ R + .
Proof. We use the representation Let X := e −t 0 H and X ν := F (t 0 /ν) ν . Then by assumption (3.19),  for any ε > 0. This completes the proof of (3.22). Now we are in position to prove another version of Theorem 3.3 for the operator-norm Chernoff approximation formula. We relax the restriction I ⊂ R + to condition I ⊂ R + 0 , but for (3.8) instead of (3.7).  Proof. By (3.14) and by assumption (3.24), we obtain the limit (3.23). Conversely, using (3.15) and assumption (3.23), one gets (3.21) for τ = t/η and for any bounded interval I ⊂ R + 0 . Then application of Theorem 3.6 yields (3.24).

Operator-norm approximation and estimates of the rate of convergence
Theorem 3.7 admits further modifications. In particular, it allows establishing estimates for the rate of operator-norm convergence.
In Theorem 4.1(i) it is shown that for ρ = 1 the condition (4.1) implies (4.2). Since integral in (4.13) diverges for ρ = 1, it is unclear whether the converse is also true. Hence, Theorem 4.1(ii) does not cover this case.
Proof. The line of reasoning that leads from (4.3) to the estimate (4.6) is obviously still valid if we assume 0 < τ ≤ t < ∞. Then setting τ := t/η, we deduce from (4.6) where c R + ρ := 1 + c M ρ and t ∈ R + 0 . For the case ρ = 1 the assumption (4.1) can be simplified and reduced to t-independent canonical form (3.7). To use Theorem 4.1 and Theorem 4.2, we return to local conditions: 0 < τ ≤ t ≤ 1. Proof. For t > 0 the identity where we used condition (4.17).
3. The proof of optimality of the asymptotic (1.5) and (1.6) is a subtle matter, see [7]. To this aim, one has to establish for convergence an estimate from below and also an example, where the operator-norm convergence is broken if operator A + B is not self-adjoint, but only essentially self-adjoint.
In the present paper we developed the lifting topology of convergence for self-adjoint Chernoff approximation. It yields optimal estimate for the rate of convergence for Trotter-Kato product formulae. For non-self-adjoint case one uses other schemes essentially based on analyticity of semigroups, see [3,4]. The results for quasi-sectorial contractions [10] improved by the 1/ 3 √ n -Theorem [11], are still not sufficiently refined to yield optimality for estimates of the rate of convergence.