Abstract
Abstract: For any nonnegative self-adjoint operators A and B in a separable Hilbert space, we show that the Trotter-type formula [(e i2tA/n + e i2tB/n )/2] n converges strongly in dom(A 1/2 ) ∩ dom(B 1/2 ) for some subsequence and for almost every t ∈ R. This result extends to the degenerate case and to Katofunctions following the method of Kato [6] .
Abstract
In a famous paper [6] , T. Kato proved that for any nonnegative self-adjoint operators A and B in a Hilbert space H, the Trotter product formula (e −tA/n e −tB/n ) n converges strongly to the (degenerate) semigroup generated by the form-sum A+B for any t with Re t > 0. He also extended the result to a class of so-called Katofunctions, and to degenerate semigroups. However the convergence on the boundary iR remains an unclear problem in this generality [1, 3] . For Kato-functions f such that Im f ≤ 0 (for example f (s) = (1 + is) −1 ), Lapidus found such an extension [7] . For the case of the Trotter product formula with projector (e itA/n P ) n , Exner and Ichinose obtained recently a interesting result [4] .
Abstract
Since this note is closely related to Kato's paper [6] , it is convenient to use similar notations. A and B denote nonnegative self-adjoint operators defined in closed subspaces M A and M B of a separable Hilbert space H, and P A , P B denote the orthogonal projections on M A and M B . Let
Abstract
be the closure of D ′ , and let P ′ be the orthogonal projector on H ′ . The formsum C = A+B is defined as the self-adjoint operator in H ′ associated with the nonnegative, closed quadratic form u → A 1/2 u 2 + B 1/2 u 2 , u ∈ D ′ . We consider Trotter-type product formulae F (t/n) n based on the arithmetic mean
Abstract
The Kato-functions f and g are assumed here to be bounded, holomorphic functions in {t ∈ C : Re t > 0} with:
Abstract
and 0 ≤ f (s) ≤ 1 if s > 0, and the same conditions for g. By the functional calculus for normal operators, F (t) is well defined for Re t ≥ 0 and bounded by 1.

Note on a product formula for unitary groups by Vincent CACHIA 1 Abstract: For any nonnegative self-adjoint operators A and B in a separable Hilbert space, we show that the Trotter-type formula [(e i2tA/n + e i2tB/n )/2] n converges strongly in dom(A 1/2 ) ∩ dom(B 1/2 ) for some subsequence and for almost every t ∈ R. This result extends to the degenerate case and to Katofunctions following the method of Kato [6] .

In a famous paper [6] , T. Kato proved that for any nonnegative self-adjoint operators A and B in a Hilbert space H, the Trotter product formula (e −tA/n e −tB/n ) n converges strongly to the (degenerate) semigroup generated by the form-sum A+B for any t with Re t > 0. He also extended the result to a class of so-called Katofunctions, and to degenerate semigroups. However the convergence on the boundary iR remains an unclear problem in this generality [1, 3] . For Kato-functions f such that Im f ≤ 0 (for example f (s) = (1 + is) −1 ), Lapidus found such an extension [7] . For the case of the Trotter product formula with projector (e itA/n P ) n , Exner and Ichinose obtained recently a interesting result [4] .
Statement of the result
Since this note is closely related to Kato's paper [6] , it is convenient to use similar notations. A and B denote nonnegative self-adjoint operators defined in closed subspaces M A and M B of a separable Hilbert space H, and P A , P B denote the orthogonal projections on M A and M B . Let
Statement of the result
be the closure of D ′ , and let P ′ be the orthogonal projector on H ′ . The formsum C = A+B is defined as the self-adjoint operator in H ′ associated with the nonnegative, closed quadratic form
Statement of the result
We consider Trotter-type product formulae F (t/n) n based on the arithmetic mean
Statement of the result
The Kato-functions f and g are assumed here to be bounded, holomorphic functions in {t ∈ C : Re t > 0} with:
Statement of the result
and 0 ≤ f (s) ≤ 1 if s > 0, and the same conditions for g. By the functional calculus for normal operators, F (t) is well defined for Re t ≥ 0 and bounded by 1. 
Statement of the result
Moreover there exists a set L ⊂ R with zero Lebesgue measure and an increasing function ϕ : N → N, such that:
Statement of the result
One sees that the strong convergence, valid in the open right half-plane, cannot extend exactly to the boundary iR, as already remarked in [3, 4] : the strong convergence on the boundary is restricted to the subspace
Statement of the result
However the weaker convergence (3) was already observed [1, 5] .
Proof
Let us consider for Re t ≥ 0 and τ > 0:
Proof
which is a holomorphic operator-valued function of t in the open right half-plane. The main step of the proof is to show that the strong convergence:
Proof
holds for Re t > 0, and remains true for almost all t ∈ iR and on some subsequence. This will give the announced result (4) for u ∈ H ′ by Chernoff's lemma (see below). The convergence on the boundary is obtained by a useful result of Feldman [5, Th 5 .1], which we state here in a slightly more general form:
uniformly bounded family of bounded holomorphic H-valued function defined in the open right half-plane. Suppose that
Moreover for each (numerical) φ ∈ L 1 (R):
uniformly bounded family of bounded holomorphic H-valued function defined in the open right half-plane. Suppose that
Proof: The two results (7) and (8) follow from very similar arguments, so we present only the first one. The bounded holomorphic functions Ψ τ have boundary values for almost every is ∈ iR, and for any t > 0 and s ∈ R,
uniformly bounded family of bounded holomorphic H-valued function defined in the open right half-plane. Suppose that
The kernel P t is in fact an approximate identity:
uniformly bounded family of bounded holomorphic H-valued function defined in the open right half-plane. Suppose that
where we have used:
This leads to
The last term (10) can be made arbitrary small by choosing t sufficiently small, and for any t the integral in the right hand side of (9) tends to 0 as τ → 0 by Lebesgue's theorem. 2
This leads to
Remark: It can also be shown that (8) implies (7).
This leads to
It is convenient to introduce the following bounded accretive operators, for Re t ≥ 0 and τ > 0:
This leads to
Lemma 3 For any t with Re t > 0, s−lim τ →0 (I +S t,τ ) −1 = (I +tC) −1 P ′ . Moreover for any v ∈ L 1 (R, H), u ∈ H and t ∈ R, one has
This leads to
Proof: Since S t,τ = A t,2τ + B t,2τ , the strong convergence of (I + S t,τ ) −1 for t > 0 follows from [6, Lem. 2.2 and 2.3]. Then it extends to the open right half-plane by the theorem of Vitali: for any τ > 0, (I + S t,τ ) −1 is a holomorphic function of t, and is bounded by 1.
This leads to
The convergence on the boundary (12) follows from Lemma 2. 2 For any fixed u ∈ H and t ∈ R we set w t,τ = (I + S it,τ ) −1 u, τ > 0. Then one finds (u, w t,τ ) = w t,τ 2 + (A it,2τ w t,τ , w t,τ ) + (B it,2τ w t,τ , w t,τ )
This leads to
with Re (A it,2τ w t,τ , w t,τ ) ≥ 0 and Re (B it,2τ w t,τ , w t,τ ) ≥ 0. Therefore
This leads to
and thus w t,τ ≤ u , τ > 0.
This leads to
Lemma 4 Let α n be any sequence of positive numbers with limit zero. There exists a set L ⊂ R of zero Lebesgue measure and a subsequence τ n of α n , such that for each t ∈ R \ L, s − lim n→∞ (I + S it,τn )
This leads to
Proof: It follows from Lemma 3 that:
This leads to
Thus the same is true for the real part, and we have by (13):
This leads to
We observe that Re (u, w t ) = Re ((I + itC)w t , w t ) = w t 2 , and that
This leads to
Then one finds
This leads to
−→ 0. This means that the functions t → w t,τ − w t converge to 0 in L 2 (R, µ) as τ → 0, with the finite measure dµ = (1 + t 2 ) −1 dt. Let (e m ) m∈N be a basis of the separable Hilbert space H. For u = e 1 the above L 2 -convergence implies that there exists L 1 ⊂ R with µ(L 1 ) = 0 and some increasing function ϕ 1 : N → N such that (I + S it,α ϕ 1 (n) ) −1 e 1 → (I + itC) −1 P ′ e 1 as n → ∞, for any t ∈ R \ L 1 . Then for u = e 2 there exists L 2 ⊂ R with µ(L 2 ) = 0 and an increasing function ϕ 2 : N → N, such that (I +S it,α ϕ 1 •ϕ 2 (n) ) −1 e 2 → (I +itC) −1 P ′ e 2 as n → ∞, for any t ∈ R \ L 2 , and so on for each m ∈ N. Finally by the diagonal procedure we consider the sequence τ n = α ϕ 1 •···•ϕn(n) and find that convergence holds for each vector e m of the basis and for each t ∈ R \ L, where L = ∪ m∈N L m . We have µ(L) = 0 and thus L has also zero Lebesgue measure. Since the operators (I + S it,τ ) −1 are uniformly bounded, this implies the strong convergence for any vector u ∈ H.
This leads to
2
Proof of the theorem:
We consider Z t,n = (n/t)[F (it/n) − I] = −t −1 S it,1/n and α n = 1/n. Let L be as in Lemma 4 and let t ∈ R \ L, t = 0. By [2, Th. 3 .17] and Lemma 4 one obtains for some increasing function ϕ : N → N: 
Proof of the theorem:
Thus we obtain the convergence (4) in
Proof of the theorem:
follows by using Kato's result for Re t > 0 and Lemma 2. 2
Proof of the theorem:
Remark: the subsequence appearing in the theorem makes the result somewhat unsatisfactory. In fact this restriction is not necessary if we assume that the functions t → (I + S it,τ ) −1 u are equicontinuous with respect to τ , at some point t 0 = 0. In this case Lemma 4 can be improved in the following way:
Proof of the theorem:
For the proof, let us consider an approximate identity ρ n : R → R + . By Lemma 3 one has lim τ →0 [ρ n * (u, w ·,τ )](t 0 ) = [ρ n * (u, w · )](t 0 ) for each n = 1, 2, . . . , and by the equicontinuity of the functions t → (u, w t,τ ) at t 0 , lim n→∞ [ρ n * (u, w ·,τ )](t 0 ) = (u, w t 0 ,τ ) uniformly in τ . Then in the proof of the theorem we consider Z t,n = −t −1 0 S it 0 ,t/nt 0 for any t ∈ R. By (17) one has s − lim n→∞ (t
Proof of the theorem:
0 + iC) −1 P ′ , which leads to the result of the theorem without subsequence (the exceptional set L has also disappeared). Concerning the equicontinuity condition, we recall that in our first result the function t → F (it) (t ∈ R) is not necessarily continuous (whereas it is continuous for example if f and g are the exponential function).

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