# Note on a Product Formula Related
to Quantum Zeno Dynamics

Pavel Exner  Doppler Institute for Mathematical Physics and Applied
Mathematics
Czech Technical University
Břehová 7
11519 Prague
Czechia
and Department of Theoretical Physics
NPI
Academy of Sciences
25068 Řež near Prague, Czechia<EMAIL_ADDRESS>http://gemma.ujf.cas.cz/
exner/ and  Takashi Ichinose  Department of Mathematics
Faculty of Science
Kanazawa University
Kanazawa 920-1192
Japan<EMAIL_ADDRESS>
###### Abstract.

Given a nonnegative self-adjoint operator $H$ acting on a separable Hilbert
space and an orthogonal projection $P$ such that
$H_{P}:=(H^{1/2}P)^{*}(H^{1/2}P)$ is densely defined, we prove that
$\lim_{n\rightarrow\infty}(P\,\mathrm{e}^{-itH/n}P)^{n}=\mathrm{e}^{-itH_{P}}P$
holds in the strong operator topology. We also derive modifications of this
product formula and its extension to the situation when $P$ is replaced by a
strongly continuous projection-valued function satisfying $P(0)=P$.

_To the memory of our friend Hagen Neidhardt (1950–2019)_

##  1\. Introduction and the main result

The main aim of this paper is to prove the following product formulæ for
short-time Schrödinger unitary groups and orthogonal projections:

######  Theorem 1.1.

Let $H$ be a nonnegative self-adjoint operator acting on a separable Hilbert
space $\mathcal{H}$ and $P$ an orthogonal projection onto a closed subspace of
$\mathcal{H}$. Suppose that $H^{1/2}P$ is densely defined, so that
$H_{P}:=(H^{1/2}P)^{*}(H^{1/2}P)$ is a self-adjoint operator. Then for any
$f\in\mathcal{H}$ and $\varepsilon=\pm 1$ the following relations hold,

$\displaystyle\lim_{n\rightarrow\infty}(P\,\mathrm{e}^{-\varepsilon
itH/n}P)^{n}f=\mathrm{e}^{-\varepsilon itH_{P}}Pf\,,$ (1.1)
$\displaystyle\lim_{n\rightarrow\infty}(\mathrm{e}^{-\varepsilon
itH/n}P)^{n}f=\mathrm{e}^{-\varepsilon itH_{P}}Pf\,,$ (1.2)
$\displaystyle\lim_{n\rightarrow\infty}(P\,\mathrm{e}^{-\varepsilon
itH/n})^{n}f=\mathrm{e}^{-\varepsilon itH_{P}}Pf$ (1.3)

in the Hilbert space norm, and moreover, the convergence is uniform on every
bounded $t$-interval in $\mathbb{R}$.

Needless to say, the claim is nontrivial only if $H$ and $P$ do not commute.

The main motivation to study such product formulæ comes from the behavior of
quantum systems exposed to frequent measurements. Turing was the first to
notice [12] that if we ascertain repeatedly whether a quantum system is in a
given state, then in the limit of infinite measurement frequency it becomes
impossible to leave this state. The idea was rediscovered in the context of
unstable system decays – see, e.g., [1, 9] – but it attracted a wide attention
only after Misra and Sudarshan [20] invented a catchy name calling such a
behavior _quantum Zeno effect_ in an allusion to the classical Zeno aporia
about a flying arrow. More about the early history can be found in [4].

The second breaking moment came in 1990 when Itano et al. [14] demonstrated
the existence of the effect experimentally. Since then it became object of an
extensive examination, both from the theoretical and experimental points of
view, and it led even to various practical applications; a partial summary can
be found in the review paper [7].

If the projection $P$ describing the measurement has a dimension larger than
one, the question about the time evolution in the subspace to which the
permanent observation confines the state of the system becomes nontrivial. It
is natural to expect that such a ‘Zeno dynamics’ will be governed by the part
of the original Hamiltonian $H$ acting in the subspace $P\mathcal{H}$, and it
was shown in [4, Sect. 2.4] that the generator is indeed associated with
appropriate quadratic form constructed from the operator $H$ and the
projection $P$. It was not easy, however, to establish the existence of the
Zeno dynamics beyond the situations when the dimension of $P$ is finite or the
operator $H$ is bounded; needless to say that this is often not the case with
actual physical systems. A prime example is a permanent ascertaining whether a
free quantum particle dwells within a prescribed region $\Omega$ of the
configuration space discussed in [8], see also [7], with the conclusion that
the Zeno generator is (the multiple of) the corresponding Dirichlet Laplacian.
The argument made use of the stationary phase method but the existence of the
limit was not actually established by the authors.

Motivated by the said paper we addressed the question of the Zeno dynamics
existence in [5], where we have managed to establish the existence of the
limits of the expressions appearing on the left-hand sides of (1.1)–(1.3) in
the topology of a larger space, namely, the Fréchet space $L^{2}_{\text{\rm
loc}}({\mathbb{R}};{\mathcal{H}})=L^{2}_{\text{\rm
loc}}({\mathbb{R}})\otimes{\mathcal{H}}$, provided that $H$ is semibounded and
the operator $H_{P}$ is densely defined; the validity of the formulæ is
preserved if the exponential in (1.1)–(1.3) is replaced by functions of a
wider class, in particular, by the resolvent $(I+itH)^{-1}\,$ [6, 13]. We
argued in [5] that such a result can be regarded as sufficient from the
viewpoint of physics due to the fact that every measurement, in particular,
that of time is burdened with errors, and any actual experiment typically
involves averaging over a large number of system copies.

It is desirable, though, to answer the question without such a underpinning by
demonstrating the result with the convergence in a stronger sense, namely that
of the strong operator topology. This is the aim of the present paper. In
addition to the described physical motivation, the obtained relation are of
independent mathematical interest belonging to the genre of the product
formulæ of Trotter and Trotter-Kato [16, 24], see also [21]. In fact, we are
going to prove a slightly more general claim with a fixed $P$ replaced by a
projection-valued function of $t$ satisfying certain regularity assumptions.

######  Theorem 1.2.

Let $\mathcal{H}$ be a separable Hilbert space, and $H,\,P$, and thus also
$H_{P}$, be the same as in Theorem 1.1. Let further $P(\cdot)$ be a strongly
continuous function the values of which are orthogonal projections in
$\mathcal{H}$ defined in a right neighbourhood of zero and satisfying
$P(0)=P$. Moreover, suppose that

$\lim_{\tau\to 0+}[\tau^{-1}(I-\mathrm{e}^{-it\tau
H})]^{1/2}P(\tau)v=\mathrm{e}^{\pi i/4}(tH)^{1/2}Pv\,,$ (1.4)

for every $v\in D[H^{1/2}P]$. Then for any $f\in\mathcal{H}$ and
$\varepsilon=\pm 1$ we have

$\displaystyle\lim_{n\rightarrow\infty}(P(1/n)\,\mathrm{e}^{-\varepsilon
itH/n}P(1/n))^{n}f=\mathrm{e}^{-\varepsilon itH_{P}}Pf\,,$ (1.5)
$\displaystyle\lim_{n\rightarrow\infty}(\mathrm{e}^{-\varepsilon
itH/n}P(1/n))^{n}f=\mathrm{e}^{-\varepsilon itH_{P}}Pf\,,$ (1.6)
$\displaystyle\lim_{n\rightarrow\infty}(P(1/n)\,\mathrm{e}^{-\varepsilon
itH/n})^{n}f=\mathrm{e}^{-\varepsilon itH_{P}}Pf\,,$ (1.7)

in the Hilbert space norm, where the convergence is uniform on every bounded
$t$-interval in $\mathbb{R}$.

######  Remark 1.3.

Note that the hypothesis made in Theorem 1.2 about the convergence of
$[\tau^{-1}(I-\mathrm{e}^{-\tau H})]^{1/2}P(\tau)v$ is slightly weaker in
comparison with [5, Theorem 2.1] where we assumed that
$D[H^{1/2}P(\tau)]\supset D[H^{1/2}P]$ and $\lim_{\tau\rightarrow
0+}\|H^{1/2}P(\tau)v\|=\|H^{1/2}Pv\|$ holds for every $v\in D[H^{1/2}P]$.
This, in fact, was not fully necessary there as a footnote in [5, p. 206]
briefly mentioned.

Note that the assumption of positivity of $H$ is made for convenience only, it
is obvious that the result remains to be valid if $H$ is replaced by $H+cI$
with a fixed $c\in\mathbb{R}$, i.e. for any self-adjoint operator bounded from
below. On the other hand, the density hypothesis is crucial; in [5, Rem. 2.7]
we cited an example showing that in its absence the expressions
$(\mathrm{e}^{-itH/n}P)^{n}$ may not converge in any sense. It may also happen
that they converge but not strongly. Examples were found by Matolcsi and
Svidkoy [19], however, they do not contradict Theorem 1.2 because in one of
them the analogue of $H_{P}$ is not densely defined and in the other the
operator $H$ is not semibounded.

Let us now describe briefly our strategy to prove Theorem 1.2. The main tool
is Chernoff’s theorem [3], see also [2], which for reader’s convenience we
reproduce in Sect. 3 below. It will yield the sought result if we show that
the $\tau$-family $\\{[I+\tau^{-1}(I-P(\tau)\,\mathrm{e}^{-\varepsilon it\tau
H}P(\tau))]^{-1}\\}_{\tau>0}$ converges to $(I+\varepsilon itH_{P})^{-1}$ as
$\tau\to 0+$ in the strong operator topology. The proof might basically follow
the argument used by Kato in [16] to establish his celebrated self-adjoint
Trotter-Kato product formula for the form sum of two nonnegative self-adjoint
operators, since these two problems appear to have notable similarities.
However, a straightforward analogy of Kato’s argument is not sufficient due to
a difficulty one encounters, to be specified in Sect. 3, Step IV of the proof.
The point is that the argument _can_ be applied to a certain class of
admissible functions $\phi(x)$ which contains beside real exponentials, e.g.,
$(1+\varepsilon ix)^{-1}$, as shown in the mentioned papers [6, 13], but
unfortunately _this class fails to include_ $\mathrm{e}^{-\varepsilon ix}$
corresponding to our unitary group $\mathrm{e}^{-itH}$.

Our way to overcome this obstacle is to start from the weaker result mentioned
above. What we did in [5] was to complement the modified Kato’s argument by
the Vitali theorem from complex function theory; in that way we proved that
for each $f\in{\mathcal{H}}$ the $\tau$-family
$\\{[I+\tau^{-1}(I-P(\tau)\mathrm{e}^{-\varepsilon it\tau
H}P(\tau))]^{-1}f\\}_{\tau>0}$ converges in the the Fréchet space
$L^{2}_{\text{\rm loc}}({\mathbb{R}};{\mathcal{H}})$. This conclusion serves
as a departing point here, although it only implies that for some set
$M_{f}\subset[0,\infty)$ of Lebesgue measure zero, the $\tau$-family
$\\{[I+\tau^{-1}(I-P(\tau)\mathrm{e}^{-\varepsilon it\tau
H}P(\tau))]^{-1}f\\}_{\tau>0}$ converges in the Hilbert space norm for every
$t\in[0,\infty)\setminus M_{f}$, not excluding the possibility that the
convergence _does_ hold at some points of $M_{f}$. Furthermore, using the
separability hypothesis made about the Hilbert space ${\mathcal{H}}$, we can
choose a countable dense subset ${\mathcal{D}}:=\\{f_{l}\\}_{l=1}^{\infty}$ in
${\mathcal{H}}$. Putting then
$M=M_{\mathcal{D}}:=\cup_{l=1}^{\infty}M_{f_{l}}$, which is also a set of
Lebesgue measure zero, we may say that the above indicated $\tau$-family
converges for all $t\in{\mathbb{R}}\setminus M$ and for every
$f\in{\mathcal{D}}$, and therefore, in view of the density, also for every
$f\in{\mathcal{H}}$.

To pass from the ‘almost all $t$’ to the ‘all $t$’ stage, one has to
demonstrate that the exceptional set $M$ is in fact empty. This task may seem
a small step, but in reality it proved to be a deep and highly nontrivial
question. Our way to deal with it is to establish the equicontinuity of the
above $\tau$-family, cf. Lemma 3.6, which would allow us to achieve our goal
by means of the Ascoli-Arzelà theorem.

A detailed description of the argument we have sketched here, given in Sect. 3
and Sect. 4, is a core part of the paper. As a preliminary, we characterize in
the next section the limit self-adjoint operator $H_{P}$ appearing in the
theorems. Finally, the paper will be concluded with a short section in which
we return briefly to example of the permanent position measurement considered
in [8].

##  2\. Concerning the limit self-adjoint operator $H_{P}$

Throughout the paper $H$ will be a nonnegative self-adjoint operator in a
separable Hilbert space ${\mathcal{H}}$, and $P$ will be an orthogonal
projection. As we have indicated above, the nonnegativity assumption is made
for convenience; our main result extends easily to any self-adjoint operator
$H$ bounded from below as well as, by sign change, to one bounded from above,
i.e. to each semi-bounded self-adjoint operator in ${\mathcal{H}}$.

Our aim here is to elucidate what is the Zeno generator $H_{P}$ appearing in
Theorems 1.1 and 1.2 and to indicate some of its properties which we will need
in the sequel. Consider the quadratic form $u\mapsto\|H^{1/2}Pu\|^{2}$ with
_form domain_ $D[H^{1/2}P]$, being the domain of the _operator_ $H^{1/2}P$.
While the domains of $HP$ and $H^{1/2}P$ are nontrivial only as subspaces of
$P{\mathcal{H}}$, we consider these operators always as acting in the whole
Hilbert space ${\mathcal{H}}$ writing, if needed,
$HP=HP\\!\\!\restriction\\!_{P{\mathcal{H}}}\oplus\,0\\!\restriction\\!_{(I-P){\mathcal{H}}}$
with the domain

$D[HP]=D[HP\\!\\!\restriction\\!_{P{\mathcal{H}}}]\oplus(I-P){\mathcal{H}},\quad
D[HP\\!\\!\restriction\\!_{P{\mathcal{H}}}]=D[H]\cap P{\mathcal{H}},$ (2.1)

and similarly for $H^{1/2}P$. Note that the latter is a _closed_ linear
operator, because $H^{1/2}$ is closed; from the same reason the operator $HP$
is also closed. We have $(H^{1/2}P)^{*}\supset PH^{1/2}$ in general. On the
other hand, the operator $PH^{1/2}$ may not be closed. This would be the case
if and only if there were $c_{0},\,c_{1}>0$ such that $\|H^{1/2}u\|\leq
c_{0}\|PH^{1/2}u\|+c_{1}\|u\|$ holds for all $u$ in the domain $D[H^{1/2}]$ of
$PH^{1/2}$, however, the necessary part of this condition may not be
satisfied. The same is true for $PH$.

By $H_{P}$ we denote the unique self-adjoint operator associated with the
above mentioned quadratic form. Its (operator) domain $D[H_{P}]$ is a subspace
of $D[H^{1/2}P]$ consisting of all $u\in D[H^{1/2}P]$ which satisfy $|\langle
H^{1/2}Pu,H^{1/2}Pv\rangle|\leq C\|v\|$ for all $v\in D[H^{1/2}P]$ with a
constant $C\geq 0$, cf. [15, Sect. VI.2.1], so that

$H_{P}=(H^{1/2}P)^{*}(H^{1/2}P).$ (2.2)

Needless to say, the form domain of $H_{P}$ is the (operator) domain of
$H_{P}^{1/2}$, for which we have by polar decomposition, cf. [15, Sect.
VI.2.7],

$\displaystyle H_{P}^{1/2}=|H^{1/2}P|=[(H^{1/2}P)^{*}(H^{1/2}P)]^{1/2}\,,$
$\displaystyle D[H_{P}^{1/2}]=D[H^{1/2}P]=(D[H^{1/2}]\cap
P{\mathcal{H}})\oplus(I-P){\mathcal{H}}.$ (2.3)

Then it is not difficult to check the following claim:

######  Proposition 2.1.

The operator $H_{P}$ in (2.2) decomposes as

$H_{P}=H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}\oplus\,0\\!\restriction\\!_{(I-P){\mathcal{H}}}$
(2.4)

having the dense domain

$D[H_{P}]:=D[H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}]\oplus(I-P){\mathcal{H}}\,\subset\,P{\mathcal{H}}\oplus(I-P){\mathcal{H}}={\mathcal{H}}\,.$

Here $H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}$ is the
$P{\mathcal{H}}$-component of $H_{P}$, which is the self-adjoint operator in
the subspace $P{\mathcal{H}}\subset\mathcal{H}$ associated with the quadratic
form in $P{\mathcal{H}}$ with the form domain $D[{H_{P}}^{1/2}]\cap
P{\mathcal{H}}$,

$D[{H_{P}}^{1/2}]\cap P{\mathcal{H}}=D[H^{1/2}P]\cap P{\mathcal{H}}\ni
w\mapsto\|H^{1/2}Pw\|^{2}=\|H_{P}^{1/2}w\|^{2}\,,$ (2.5)

and the zero operator $0\\!\\!\restriction\\!\\!_{(I-P){\mathcal{H}}}$ is its
$(I\\!-\\!P){\mathcal{H}}$-component, trivially bounded and self-adjoint on
the subspace $(I\\!-\\!P){\mathcal{H}}$ orthogonal to $P\mathcal{H}$.

From Proposition 2.1, we can see that $H_{P}$ is in general not a restriction
of $H$, and furthermore, the inclusion $D[H_{P}]\subset D[H]$ does not hold
either. This is obvious, since $H$ is the unique self-adjoint operator in
${\mathcal{H}}$ associated with the quadratic form

$D[H^{1/2}]\ni w\mapsto\|H^{1/2}w\|_{\mathcal{H}}^{2}\,,$ (2.6)

so that $(I-P){\mathcal{H}}$ is not a subset of $D[H]$ as long as $H$ is
_unbounded_.

We can make, however, a weaker claim described in the following proposition;
we note in passing that it is well illustrated by the inclusion
$D[(-\Delta)_{\Omega}]\subset D[-\Delta]$ from the example treated in Sect. 5
below.

######  Proposition 2.2.

Let $H$ be a nonnegative unbounded self-adjoint operator acting on a Hilbert
space ${\mathcal{H}}$ and $P$ an orthogonal projection which may not commute
with $H$. Assume that $H^{1/2}P$ is densely defined, then the
$P{\mathcal{H}}$-component $H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}$ of
$H_{P}$ in (2.4) satisfies
$D[H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}]\subset D[H]$.

Proof: The form (2.6) restricted to $D[H^{1/2}P]$ becomes (2.5), so that
$\|H^{1/2}w\|^{2}=\|H^{1/2}Pw\|^{2}=\|H_{P}^{1/2}w\|^{2}$ holds if $w\in
D[H_{P}^{1/2}]$ proving thus the claim. ∎

######  Remark 2.3.

One can modify $H_{P}$ restricting it to the self-adjoint operator
$H_{P}^{\mathrm{mod}}$ such that
$H_{P}^{\mathrm{mod}}\\!\\!\restriction\\!_{P{\mathcal{H}}}=H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}$
and, at the same time, $\,D[H_{P}^{\mathrm{mod}}]\subset D[H]$. To this end,
it is enough to replace the zero operator in (2.4) on $(I-P){\mathcal{H}}$ by
its restriction to $(I-P)D[H]$, so that

$\displaystyle
H_{P}^{\mathrm{mod}}:=H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}\oplus
0\\!\\!\restriction\\!_{(I-P)D[H]}\,,$ (2.7) $\displaystyle
D[H_{P}^{\mathrm{mod}}]:=D[H_{P}\\!\\!\restriction\\!_{P{\mathcal{H}}}]\oplus(I-P)D[H]\,.$

Let us stress that in general $H$ is not an extension of
$H_{P}^{\mathrm{mod}}$ either.

Unless $H$ is bounded, the operator $H_{P}$ is generally different from $PHP$.
When $HP$ is densely defined, the symmetric operator $PHP$ is not necessarily
essentially self-adjoint on $D[HP]$, and consequently, $H_{P}$ may not be the
closure of $PHP$ either.

On the other hand, the quadratic form $u\mapsto\|H^{1/2}Pu\|^{2}$ defined on
$D[H^{1/2}P]$ is a closed extension of the form $u\mapsto\langle
Pu,HPu\rangle$ defined on $D[HP]$, but in general the former is not the
closure of the latter, because $D[HP]$ is not necessarily dense in
$D[H^{1/2}P]$. Indeed, if $H$ is unbounded, $D[H]$ is a proper subspace of
$D[H^{1/2}]$. Take $u_{0}\in D[H^{1/2}]\backslash D[H]$ such that the vector
$H^{1/2}u_{0}$ is nonzero, and set $P$ to be the orthogonal projection onto
the one-dimensional subspace spanned by $u_{0}$. Taking into account that
$D[HP]=\\{u\in{\mathcal{H}};\,\,Pu\in D[H]\\}$ which $u_{0}=Pu_{0}$ does not
belong to, we find $HPu=0$ for $u\in D[HP]$, while
$H^{1/2}Pu_{0}=H^{1/2}u_{0}\not=0$ by assumption.

##  3\. Proof of Theorem 1.2

Most parts of the argument work for a general Hilbert space, however, there is
a place where we have to assume ${\mathcal{H}}$ to be separable. It is clearly
sufficient to prove formula (1.5) in Theorem 1.2 because (1.6) and (1.7)
easily follow from it, and furthermore, it is enough to consider
$\varepsilon=1$ and $t\geq 0$. For the sake of definiteness we use here and in
the following the physicist convention about the inner product
$\langle\cdot,\cdot\rangle$ on ${\mathcal{H}}$ supposing that it is antilinear
in the _first_ argument.

The nonnegative self-adjoint operator $H$ we deal with can be conventionally
represented through its spectral family $\\{E(\lambda)\\}_{\lambda\geq 0}$,

$H=\int_{0-}^{\infty}\lambda E(\mathrm{d}\lambda)$ (3.1)

Given $\kappa>0$, we introduce

$\displaystyle K(\kappa):$
$\displaystyle=\frac{1}{\kappa}[I-\mathrm{e}^{-i\kappa H}]=\frac{I-\cos\kappa
H}{\kappa}+i\frac{\sin\kappa H}{\kappa}$
$\displaystyle=:G(\kappa)+iH(\kappa)\,,$ (3.2)

where $G(\kappa)$ and $H(\kappa)$ are bounded self-adjoint operators, and
$G(\kappa)$ is nonnegative. Let $H^{\pm}(\kappa)$ be nonnegative bounded self-
adjoint operators which are the nonnegative/negative parts of $H(\kappa)$,
respectively. Then we have

$\displaystyle|K(\kappa)|$
$\displaystyle=[G(\kappa)^{2}+H(\kappa)^{2}]^{1/2}=\tfrac{|\sin\frac{1}{2}\kappa
H|}{\frac{1}{2}\kappa}\,,$ $\displaystyle|I+K(\kappa)|$
$\displaystyle=\Big{(}I+(1+\kappa)\big{(}\tfrac{\sin\frac{1}{2}\kappa
H}{\frac{1}{2}\kappa}\big{)}^{2}\Big{)}^{1/2}\,,$ (3.3) $\displaystyle
H(\kappa)$
$\displaystyle=H^{+}(\kappa)-H^{-}(\kappa)\,,\quad|H(\kappa)|=H^{+}(\kappa)+H^{-}(\kappa)\,.$

Using the spectral family $\\{E(\lambda)\\}_{\lambda\geq 0}$ from (3.1), we
denote by $E_{H}^{+}(\kappa)$ the orthogonal projection

$E_{H}^{+}(\kappa):\mathcal{H}\rightarrow\bigoplus_{m=0}^{\infty}E\big{(}[\tfrac{2m\pi}{\kappa},\tfrac{(2m+1)\pi}{\kappa})\big{)}\mathcal{H}\equiv
E\Big{(}\bigcup_{m=0}^{\infty}[\tfrac{2m\pi}{\kappa},\tfrac{(2m+1)\pi}{\kappa})\Big{)}\mathcal{H}\,,$

and put $E_{H}^{-}(\kappa):=I-E_{H}^{+}(\kappa)$. Note that
$E_{H}^{+}(\kappa)\overset{s}{\to}I$ and $E_{H}^{-}(\kappa)\overset{s}{\to}0$
holds as $\kappa\rightarrow 0+$. Of course, both the $E_{H}^{\pm}(\kappa)$
commute with $H(\kappa)$. As $E_{H}^{+}(\kappa)$ and $E_{H}^{-}(\kappa)$ are
nothing but the projections onto the closed subspaces of $\mathcal{H}$ where
$H(\kappa)$ becomes respectively a nonnegative and negative self-adjoint
operator, we have

$H^{+}(\kappa)=H(\kappa)E_{H}^{+}(\kappa)\,,\quad
H^{-}(\kappa)=-H(\kappa)E_{H}^{-}(\kappa)\,.$ (3.4)

We put $\kappa=|t|\tau$ with $\tau>0$, so that $\kappa=\pm\,t\tau$ holds for
$\pm\,t\geq 0$.

We will use the same notation as in [5],
$F(\zeta;\tau):=P(\tau)\,\mathrm{e}^{-\zeta\tau H}P(\tau)$ and
$S(\zeta;\tau):=\tau^{-1}[I-F(\zeta;\tau)]$ with $\text{\rm Im}\,\zeta\geq 0$.
These operator families are uniformly bounded and holomorphic for $\text{\rm
Im}\,\zeta>0$ which are the properties we used there. In this paper, however,
we need only $\zeta=it$, in other words

$\displaystyle F(it;\tau)$ $\displaystyle=P(\tau)\,\mathrm{e}^{-it\tau
H}P(\tau)\,,$ (3.5) $\displaystyle S(it;\tau)$
$\displaystyle=\tau^{-1}[I-F(it;\tau)]=\tau^{-1}[I-P(\tau)\,\mathrm{e}^{-it\tau
H}P(\tau)]\,.$ (3.6)

It is easy to see that $F(it;\tau)$ in (3.5) is a contraction and $S(it;\tau)$
in (3.6) satisfies

$\displaystyle\mathrm{Re}\,(f,S(it;\tau)f)$
$\displaystyle=\tau^{-1}\big{[}(f,f)-(f,P(\tau)\,\mathrm{e}^{-it\tau
H}P(\tau)f)\big{]}$
$\displaystyle\geq\tau^{-1}\big{[}\|f\|^{2}-\|f\|\,\|P(\tau)\,\mathrm{e}^{-it\tau
H}P(\tau)f\|\big{]}$
$\displaystyle\geq\tau^{-1}\big{[}\|f\|^{2}-\|f\|^{2}\big{]}=0$

for all $f\in\mathcal{H}$, and consequently, $S(it;\tau)$ is an $m$-accretive
operator [15]. This means that $I+S(it;\tau)$ has a bounded inverse and that
$(I+S(it;\tau))^{-1}$ is also a contraction. The crucial observation is that
to prove Theorem 1.2 it is sufficient to refer to Chernoff’s result cited
below, and to verify that

$(I+S(it;\tau))^{-1}\overset{s}{\longrightarrow}(I+itH_{P})^{-1}P\quad\mathrm{as}\;\;\tau\rightarrow
0+$ (3.7)

holds for $t\in{\mathbb{R}}$.

Chernoff’s Theorem (cf. [3, Theorem 1.1, pp. 4–6], see also [2]): For a
$t$-family $\\{F(t)\\}_{t\geq 0}$ of linear contractions on a Banach space and
the generator $A$ of a strongly continuous contraction semigroup, the
following two conditions are equivalent:

(a) For some $\lambda_{0}>0$, the family
$\\{\lambda_{0}I+\tfrac{I-F(\varepsilon)}{\varepsilon}\\}_{\varepsilon>0}$
converges strongly to $(\lambda_{0}I+A)^{-1}$ as $\varepsilon\to 0+$.

(b) As $n\to\infty$, $\\{F(\tfrac{t}{n})\\}_{n=1}^{\infty}$ converges strongly
to $\mathrm{e}^{tA}$, uniformly on bounded $t$-intervals.

Let us stress that our proof requires to demonstrate the convergence in (3.7)
_pointwise for any fixed $t$_; this will be sufficient to establish the
convergence in the product formulæ (1.5)–(1.7) as _locally uniform_ in
$t\in{\mathbb{R}}$; in fact, we have only to deal with (1.5) as mentioned at
the beginning of this section.

######  Remark 3.1.

Since the function $F(it;\tau)$ in (3.5) differs slightly from $F(t)$
appearing in condition (a) of Chernoff’s Theorem, let us explain in detail how
the product formula (1.5) follows from (3.7), modifying to that purpose
Chernoff’s proof of the implication (a)$\,\Rightarrow\,$(b).

Consider the first the nontrivial part referring to the subspace
$P{\mathcal{H}}$. Note that $S(it;\tau)$ generates a strongly continuous
contraction semigroup $\\{\mathrm{e}^{-\theta S(it;\tau)}\\}_{\theta\geq 0}$
on ${\mathcal{H}}$, and the resolvent convergence (3.7) is equivalent to the
convergence of the corresponding semigroups [15, Theorem IX.2.16], hence for
any $f\in P{\mathcal{H}}$ we have $\mathrm{e}^{-\theta
S(it;\tau)}f\overset{s}{\longrightarrow}\mathrm{e}^{-i\theta tH_{P}}f$ as
$\tau\to 0+$, uniformly on bounded intervals of the variable $\theta\geq 0$.
In particular, choosing $\theta=1$ we get

$\mathrm{e}^{-S(it;\tau)}f\longrightarrow\mathrm{e}^{-itH_{P}}f\quad\text{\rm
as}\;\;\tau\to 0+\,$ (3.8)

for a fixed $t\geq 0$, and using the same equivalence in the opposite
direction we infer that

$(I+\lambda S(it;\tau))^{-1}f\longrightarrow(I+i\lambda
tH_{P})^{-1}Pf\quad\text{\rm as}\;\;\tau\to 0+$

holds for any $\lambda\geq 0$ and $t\geq 0$. In particular, using the diagonal
trick in the last relation with $\tau=1/n$ and $\lambda=1/\sqrt{n}$, we obtain

$(I+\tfrac{1}{\sqrt{n}}S(it;1/n))^{-1}f\longrightarrow Pf\quad\text{\rm
as}\;\;n\to\infty$ (3.9)

for every $t\geq 0$. Next we refer to [2, Lemma 2] by which we have

$\|[F(it;1/n)^{n}g-\mathrm{e}^{-n(I-F(it;1/n))}g\|\leq\sqrt{n}\,\|(I-F(it;1/n))g\|$

for any $g\in{\mathcal{H}}$. Choosing
$g=(I+\tfrac{1}{\sqrt{n}}S(it;1/n))^{-1}f$ and having in mind that $f=Pf$, we
conclude from here that

$\displaystyle\big{\|}[F(it;1/n)^{n}-\mathrm{e}^{-S(it;1/n)}](I+\tfrac{1}{\sqrt{n}}S(it;1/n))^{-1}f\big{\|}$
$\displaystyle\qquad\leq\big{\|}(I+\tfrac{1}{\sqrt{n}}S(it;1/n))^{-1}f-Pf\|\,,$

where by (3.9) the right-hand side tends to zero uniformly on bounded
$t$-intervals as $n\to\infty$. Using the diagonal trick once again we get

$\lim_{n\to\infty}\|F(it;1/n)^{n}f-\mathrm{e}^{-S(it;1/n)}f\|=0\,$

uniformly on bounded $t$-intervals. Then the sought conclusion (1.5) follows
immediately from relations (3.8) and (3.9), since by (3.5) we have
$F(it;1/n)^{n}=[P(1/n)\mathrm{e}^{-itH/n}P(1/n)]^{n}$. Having dealt with the
subspace $P\mathcal{H}$, the remaining case, $f\in(P{\mathcal{H}})^{\perp}$,
is trivial: we have

$[P(1/n)\mathrm{e}^{-itH/n}P(1/n)]^{n}f\longrightarrow 0\quad\text{\rm
as}\;\;n\to\infty$

for each $t\geq 0$, since $P(1/n)f=P(1/n)(I-P)f$ tends by assumption to
$P(I-P)f=0$ and $\mathrm{e}^{-itH_{P}}Pf=0$ holds at the same time.

Let us add a remark on the conventions: here and in the following the
convergence of operator families in the strong operator topology is denoted by
$\overset{s}{\longrightarrow}$, and in the weak operator topology by
$\overset{w}{\longrightarrow}$. A simple arrow is reserved for the convergence
with respect to the norm of the Hilbert space $\mathcal{H}$, sometimes also
dubbed ‘strong’ – we will occasionally use this term too – while for the weak
convergence in $\mathcal{H}$ we will again employ the symbol
$\overset{w}{\longrightarrow}$. Later, in Proposition 3.5, we will use still
another topology for convergence of families of Hilbert space vectors.

To proceed with the argument, we rewrite relation (3.6) as follows

$\displaystyle S(it;\tau)$ $\displaystyle=\tau^{-1}\big{[}I-P(\tau)(\cos t\tau
H-i\sin t\tau H)P(\tau)\big{]}$
$\displaystyle=\tfrac{I-P(\tau)}{\tau}+P(\tau)\,\tfrac{I-\cos t\tau
H}{\tau}\,P(\tau)+i\,P(\tau)\,\tfrac{\sin t\tau H}{\tau}\,P(\tau)$
$\displaystyle=\tfrac{I-P(\tau)}{\tau}+P(\tau)tG(t\tau)P(\tau)+i\,P(\tau)tH(t\tau)P(\tau)\,,$

and consequently,

$\displaystyle I$ $\displaystyle+S(it;\tau)$ (3.10)
$\displaystyle=I+\tau^{-1}(I-P(\tau))+P(\tau)tG(t\tau)P(\tau)+iP(\tau)tH(t\tau)P(\tau)$
$\displaystyle=(1+\tau^{-1})(I-P(\tau))\oplus
P(\tau)(I+tG(t\tau)+itH(t\tau))P(\tau)\,,$

where $\oplus$ denotes the direct sum corresponding to the decomposition of
the Hilbert space into $P(\tau)\mathcal{H}$ and its orthogonal complement.

To prove the sought relation (3.7) we need first a pair of lemmata.

######  Lemma 3.2.

The inverse of $I+S(it;\tau)$ in (3.10) is given by

$\displaystyle(I$ $\displaystyle+S(it;\tau))^{-1}$ (3.11)
$\displaystyle=(1+\tau^{-1})^{-1}(I-P(\tau))\oplus\big{[}P(\tau)(I+tG(t\tau)+itH(t\tau))P(\tau)\big{]}^{-1}.$

###### Proof.

The above expression and (3.10) clearly multiply to identity. ∎

######  Lemma 3.3.

Let $u\in D[H^{1/2}]$, then in the limit $\kappa\rightarrow 0+$ we have

$\displaystyle\mathrm{(i)}\,\,$ $\displaystyle G(\kappa)^{1/2}u\longrightarrow
0\,,$ $\displaystyle\mathrm{(ii)}\,\,$ $\displaystyle
H^{+}(\kappa)^{1/2}u\longrightarrow
H^{1/2}u\,,\;H^{-}(\kappa)^{1/2}u\longrightarrow
0\,,\;\mathrm{and}\;\;|H(\kappa)|^{1/2}u\longrightarrow H^{1/2}u$

###### Proof.

Using spectral theorem together with dominated convergence theorem we infer
that as $\kappa\rightarrow 0+$,

$\displaystyle G(\kappa)^{1/2}u$
$\displaystyle=\int_{0-}^{\infty}\Big{|}\tfrac{\sin\tfrac{\kappa\lambda}{2}}{\tfrac{\kappa\lambda}{2}}\Big{|}^{1/2}|\sin\tfrac{\kappa\lambda}{2}|^{1/2}\,\lambda^{1/2}E(\mathrm{d}\lambda)u\,\longrightarrow
0\,,$

$|H(\kappa)|^{1/2}u=\int_{0-}^{\infty}\Big{|}\tfrac{\sin\kappa\lambda}{\kappa}\Big{|}^{1/2}E(\mathrm{d}\lambda)u\longrightarrow\int_{0-}^{\infty}\lambda^{1/2}E(\mathrm{d}\lambda)u=H^{1/2}u$

and

$\displaystyle H^{+}(\kappa)^{1/2}u$
$\displaystyle=|H(\kappa)|^{1/2}E_{H}^{+}(\kappa)u$
$\displaystyle=\int_{0-}^{\infty}\Big{|}\tfrac{\sin\kappa\lambda}{\kappa}\Big{|}^{1/2}\chi_{\bigcup_{m=0}^{\infty}[\tfrac{2m\pi}{\kappa},\tfrac{(2m+1)\pi}{\kappa})}(\lambda)\,E(\mathrm{d}\lambda)u$
$\displaystyle\qquad\qquad\longrightarrow\int_{0-}^{\infty}\lambda^{1/2}E(\mathrm{d}\lambda)u=H^{1/2}u$

This implies at the same time $H^{-}(\kappa)^{1/2}u\longrightarrow 0\,$ by
(3.3). ∎

After these preliminaries, we are going to start the proof of (3.7). As we
said in the introduction, the core of our reasoning is the argument used by
Kato [16], see also [21, Supplements to Sect. VIII.8]. In the same vein, with
(3.6) in mind, we put for $\tau>0$ and $t\in{\mathbb{R}}$

$u_{\tau}(t):=(I+S(it;\tau))^{-1}f$ (3.12)

for an arbitrary but fixed $f\in\mathcal{H}$. We see from the expression
(3.11) of $(I+S(it;\tau))^{-1}$ that $u_{\tau}(t)$ is strongly continuous in
$t$. At the same time, the $\tau$-family
$\\{u_{\tau}(t)\\}\subset{\mathcal{H}}$ with $u_{\tau}(t)$ defined by (3.12)
and $t\in{\mathbb{R}}$ fixed is uniformly bounded by $\|f\|$, because
$(I+S(it;\tau))^{-1}$ is a contraction.

Our next aim is to show that for each fixed $t\in{\mathbb{R}}$, the family
$\\{u_{\tau}(t)\\}$ converges in the Hilbert space norm to some
$u(t)\in{\mathcal{H}}$ as $\tau\to 0+$ and that $u(t)=(I+itH_{P})^{-1}Pf$. To
achieve this goal we will have to analyze, in particular, a new uniform
property of the $\tau$-family $\\{u_{\tau}(t)\\}$ at the final stage of the
argument.

Using the identity (3.10) we can invert relation (3.12) explicitly as

$\displaystyle f$ $\displaystyle=(I+S(it;\tau))u_{\tau}(t)$
$\displaystyle=u_{\tau}(t)+\tau^{-1}(I-P(\tau))u_{\tau}(t)+P(\tau)tG(t\tau)P(\tau)u_{\tau}(t)$
$\displaystyle\qquad\qquad\qquad\qquad\qquad\qquad+iP(\tau)tH(t\tau)P(\tau)u_{\tau}(t)$
$\displaystyle=(1+\tau^{-1})(I-P(\tau))u_{\tau}(t)\oplus
P(\tau)[I+tG(t\tau)+itH(t\tau)]P(\tau)u_{\tau}(t)\,.$ (3.13)

Taking the inner product of $f$ with $(I-P(\tau))u_{\tau}(t)$ and
$P(\tau)u_{\tau}$, we get

$\langle(I-P(\tau))u_{\tau}(t),f\rangle=(1+\tau^{-1})\|(I-P(\tau))u_{\tau}(t)\|^{2}$
(3.14)

and

$\displaystyle\langle P(\tau)u_{\tau}(t),f\rangle$ $\displaystyle=\langle
P(\tau)u_{\tau}(t),u_{\tau}(t)\rangle+\langle
P(\tau)u_{\tau}(t),tG(t\tau)P(\tau)u_{\tau}(t)\rangle$
$\displaystyle\quad+i\langle
P(\tau)u_{\tau}(t),tH(t\tau)P(\tau)_{\tau}(t)\rangle$
$\displaystyle=\|P(\tau)u_{\tau}(t)\|^{2}+\|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}$
$\displaystyle\quad+i\big{(}\|(|t|H^{+}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}$
$\displaystyle\quad\quad-\|(|t|H^{-}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}\big{)}$
(3.15)

for $\tau>0\,$; recall that $H(\kappa)=H^{+}(\kappa)-H^{-}(\kappa)$.

From the real part of (3) with (3.14), we infer, using Schwarz inequality,
that the $\tau$-families $\\{P(\tau)u_{\tau}(t)\\}$ and
$\\{(I-P(\tau))u_{\tau}(t)\\}$, as well as
$\\{\tau^{-1}(I-P(\tau))u_{\tau}(t)\\}$, are uniformly bounded by $\|f\|$; the
last claim naturally extends to $\\{\tau^{-1/2}(I-P(\tau))u_{\tau}(t)\\}$ as
long as $\tau\leq 1$. Moreover, using the real part of (3) again we can
conclude that the same is true for
$\\{(|t|G(\tau))^{1/2}P(\tau)u_{\tau}(t)\\}$. Let $f\in{\mathcal{H}}$ and
$t\in{\mathbb{R}}$ be arbitrary but fixed. It follows from the obtained
uniform boundedness that for each $t\in{\mathbb{R}}$, there exist a
(sub)sequence $\\{\tau^{\prime}\\}_{0<\tau^{\prime}\leq 1}$ of the family
$\\{\tau\\}_{0<\tau\leq 1}$ with $\tau^{\prime}\rightarrow 0+$ and vectors
$u(t),\,u_{0}(t)$ and $g(t)$ in ${\mathcal{H}}$ such that the sequences
$\\{u_{\tau^{\prime}}(t)\\}$,
$\\{(\tau^{\prime})^{-1/2}(I-P(\tau^{\prime}))u_{\tau^{\prime}}(t)\\}$ and
$\\{t^{1/2}G(|t|\tau^{\prime})^{1/2}u_{\tau^{\prime}}(t)\\}$ converge weakly
to $u(t)$, $u_{0}(t)$ and $g(t)$, respectively, in ${\mathcal{H}}$ as
$\tau^{\prime}\to 0+$; thus $\\{P(\tau^{\prime})u_{\tau^{\prime}}(t)\\}$
converges weakly to $Pu(t)$. We keep in mind that, with the knowledge we have
at the present stage, the limit $u(t)$ may depend on the chosen sequence
$\\{u_{\tau^{\prime}}(t)\\}_{0<\tau^{\prime}\leq 1}$.

######  Lemma 3.4.

One has

$u(t)=Pu(t),\quad u_{0}(t)=0,\quad g(t)=0,$ (3.16)

and therefore, as $\tau^{\prime}\to 0+$,

$\displaystyle
u_{\tau^{\prime}}(t)\overset{w}{\longrightarrow}u(t)\,,\;\;({\tau^{\prime}})^{-1/2}(I-P(\tau^{\prime}))u_{\tau^{\prime}}(t)\overset{w}{\longrightarrow}0\,,\;\;$
(3.17) $\displaystyle
P(\tau^{\prime})u_{\tau^{\prime}}(t)\overset{w}{\longrightarrow}Pu(t)\,,\;\;(|t|G(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\overset{w}{\longrightarrow}0\,.$

_Proof._ To begin with, the second limit in (3.17) implies, in particular,
that $(I-P(\tau^{\prime}))u_{\tau^{\prime}}(t)\to 0$, hence we have
$(I-P)u(t)=0$ or $u(t)=Pu(t)$. Indeed, the uniform boundedness of
$\|(\tau^{\prime})^{-1/2}(I-P(\tau^{\prime}))u_{\tau^{\prime}}(t)\|$ means
that $\|(I-P(\tau^{\prime}))u_{\tau^{\prime}}(t)\|\to 0$ which together with
the weak convergence implies $u(t)=Pu(t)$. Moreover, the same uniform
boundedness following from (3.14) means that $\tau^{-1}(I-P(\tau))u_{\tau}(t)$
also converges weakly along $\\{\tau^{\prime}\\}$ which implies $u_{0}(t)=0$.

Secondly, for any fixed $w\in D[H^{1/2}]$ and $t\in{\mathbb{R}}$ we have the
relation

$\displaystyle\langle w,g(t)\rangle$ $\displaystyle=\lim_{\tau^{\prime}\to
0+}\,\langle
w,(|t|G(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\rangle$
$\displaystyle=\lim_{\tau^{\prime}\to
0+}\,\langle(|t|G(t\tau^{\prime}))^{1/2}w,P(\tau^{\prime})u_{\tau^{\prime}}(t)\rangle$
$\displaystyle=\langle 0,Pu(t)\rangle=0\,,$

because by Lemma 3.3 we have $(|t|G(t\tau^{\prime}))^{1/2}w\rightarrow 0$ as
$\tau^{\prime}\rightarrow 0+$. This means that $g(t)=0$, because $D[H^{1/2}]$
is dense by assumption. ∎

Next we want to find out what one can deduce from the imaginary part of (3)
about the properties of the operators $H^{\pm}(t\tau)$ introduced in (3.3) and
(3.4). To this aim we have to employ a topology different from those used up
to now. Given a dense subspace $\mathcal{K}$ of $\mathcal{H}$, the symbol
$\sigma(\mathcal{H},\mathcal{K})$ denotes the _weak topology on $\mathcal{H}$
defined by the dual pairing $\langle\mathcal{H},\mathcal{K}\rangle$_ or the
_$\mathcal{K}$ -weak topology on $\mathcal{H}$_ – see, e.g., [18, Sect.
IV.20.2] or [21, Sect. IV.5]. In general, it is weaker (coarser) than the
usual weak topology on $\mathcal{H}$, the latter being nothing else than
$\sigma(\mathcal{H},\mathcal{H})$ in the just introduced notation

We see from the imaginary part of (3) that the difference between the
respective elements of the $\tau$-families
$\\{\|(|t|H^{+}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}\\}$ and
$\\{\|(|t|H^{-}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}\\}$ is uniformly
bounded, because by Schwarz inequality the modulus of the left-hand side in

$\displaystyle\mathrm{Im}\,\langle Pu(t),f\rangle$
$\displaystyle=\lim_{\tau\to
0+}\,\big{[}\|(|t|H^{+}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}-\|(|t|H^{-}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}\big{]}$

does not exceed $\|f\|^{2}$. This fact, unfortunately, does not tell us
whether the two $\tau$-families of vectors,
$\\{(|t|H^{\pm}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\\}$, are separately
(uniformly) bounded, that is, whether each of them is (uniformly) weakly
bounded. We have, however, at least the following result.

######  Proposition 3.5.

Let $\\{u_{\tau^{\prime}}(t)\\}$ be the subsequence appearing in Lemma 3.4,
weakly convergent to $u(t)$. Then the $\tau^{\prime}$-families
$\\{(|t|H^{\pm}(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\\}$
are _Cauchy_ sequences in the $\sigma(\mathcal{H},D[H^{1/2}])$-weak topology,
and as a result they are $\sigma(\mathcal{H},D[H^{1/2}])$-weakly bounded.
Furthermore, the family
$\\{(|t|H^{-}(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\\}$
converges to zero in this topology as $\tau^{\prime}\rightarrow 0+$.

###### Proof.

Take an arbitrary $\phi\in D[H^{1/2}]$. We use Lemma 3.3 which states, in
particular, that $(|t|H^{+}(t\tau))^{1/2}\phi\rightarrow(|t|H)^{1/2}\phi$
holds when $\tau\rightarrow 0+$. In combination with (3.17), this yields

$\displaystyle\langle\phi,(|t|H^{+}(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\rangle$
$\displaystyle=\langle(|t|H^{+}(t\tau^{\prime}))^{1/2}\phi,P(\tau^{\prime})u_{\tau^{\prime}}(t)\rangle$
$\displaystyle\longrightarrow\langle(|t|H)^{1/2}\phi,Pu(t)\rangle$

as $\tau^{\prime}\rightarrow 0+$. For the minus sign, on the other hand, Lemma
3.3 says that $(|t|H^{-}(t\tau))^{1/2}\phi\rightarrow 0$, and this implies

$\langle\phi,(|t|H^{-}(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\rangle\longrightarrow
0,\quad\tau^{\prime}\rightarrow 0+,$

because the self-adjointness of $(|t|H^{-}(t\tau^{\prime}))^{1/2}$ in
combination with Schwarz inequality gives

$\displaystyle|\langle(|t|H^{-}(t\tau^{\prime}))^{1/2}\phi,P(\tau^{\prime})u_{\tau^{\prime}}(t)\rangle|$
$\displaystyle\leq\|(|t|H^{-}(t\tau^{\prime}))^{1/2}\phi\|\,\|P(\tau^{\prime})u_{\tau^{\prime}}(t)\|$
$\displaystyle\leq\|(|t|H^{-}(t\tau^{\prime}))^{1/2}\phi\|\,\|f\|\
\longrightarrow 0\,.$

This yields the stated assertion, including the fact that the family
$\\{(|t|H^{-}(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\\}$
is $\sigma(\mathcal{H},D[H^{1/2}])$-weakly convergent to zero. ∎

On the other hand, it is not clear whether $Pu(t)$ belongs to $D[H^{1/2}]$,
that is, whether the plus-sign family
$\\{(|t|H^{+}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\\}$ converges to $H^{1/2}Pu(t)$
in the $\sigma(\mathcal{H},D[H^{1/2}])$ topology. What is important, the
information we were able to deduce in this way about the convergence of the
families
$\\{(|t|H^{\pm}(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\\}$
is too limited; it does not seem possible to apply the same procedure as we
used in Lemma 3.4 for
$\\{(|t|G(t\tau^{\prime}))^{1/2}P(\tau^{\prime})u_{\tau^{\prime}}(t)\\}$.

This forces us to seek a different strategy for the proof of Theorem 1.2 that
would allow us to identify the vector $u(t)$ with $(I+itH_{P})^{-1}Pf$, in
other words, to demonstrate relation (3.7) claiming that for every fixed
$t\in{\mathbb{R}}$, the family $\\{(I+S(it;\tau))^{-1}f\\}$, or otherwise
$\\{u_{\tau}(t)\\}$ in accordance with (3.12), converges to
$(I+itH_{P})^{-1}Pf$ in the Hilbert space norm as $\tau\to 0+$.

The argument is somewhat subtle and relies on our previous work [5], see also
[6], about the Zeno product formulæ related to Theorems 1.1 and 1.2. In those
papers we demonstrated that the family $\\{u_{\tau}(t)\\}$ has a unique limit,
namely $(I+itH_{P})^{-1}Pf$, as $\tau\to 0+$. As we mentioned in the
introduction, however, the obtained convergence referred neither to the norm
of ${\mathcal{H}}$ nor even to the weak topology. Precisely speaking, it is
shown in [5] that (1.1)–(1.3) and (1.5)–(1.7) hold in the topology of the
Fréchet space $L^{2}_{\text{\rm
loc}}({\mathbb{R}};{\mathcal{H}})=L^{2}_{\text{\rm
loc}}({\mathbb{R}})\otimes{\mathcal{H}}$ of the ${\mathcal{H}}$-valued
strongly measurable functions $v(\cdot)$ on ${\mathbb{R}}$ such that the
$\|v(\cdot)\|$ are locally square integrable there, equipped with the topology
induced by the family of semi-norms
$v\mapsto\big{(}\int_{a}^{b}\|v(t)\|^{2}\,\mathrm{d}t\big{)}^{1/2}$ for any
bounded interval $(a,b)$ with $a<b$. This follows from [5, Lemma 3.1, p. 200]
which says that for every bounded closed interval $[a,b]\subset{\mathbb{R}}$
one has

$\int_{a}^{b}\|u_{\tau}(t)-(I+itH_{P})^{-1}Pf\|^{2}\,\mathrm{d}t\;\longrightarrow\,0\quad\mathrm{as}\;\>\tau\to
0+\,;$ (3.18)

the reader may compare this result with (3.7).

Note also that (3.18) implies that for every $f\in{\mathcal{H}}$, there exist
a set $M_{f}\subset{\mathbb{R}}$ of Lebesgue measure zero, possibly dependent
on $f$, and a (sub)sequence $\\{\tau^{\prime}\\}_{0<\tau^{\prime}\leq 1}$ of
$\\{\tau\\}_{0<\tau\leq 1}$ along which it holds that for all
$s\in{\mathbb{R}}\setminus M_{f}$,

$u_{\tau}(s)\longrightarrow(I+isH_{P})^{-1}Pf\quad\text{\rm
in\;the\;norm\;of}\;\>{\mathcal{H}},$ (3.19)

in other words, $u_{\tau^{\prime}}(s)\longrightarrow(I+isH_{P})^{-1}Pf$ as
$\tau^{\prime}\to 0$. With the coming argument in mind, it is useful to note
that the set $\mathbb{R}\setminus M_{f}$ at which the convergence takes place
is dense in ${\mathbb{R}}$. Furthermore, since ${\mathcal{H}}$ is separable by
assumption, we can choose a countable dense subset
${\mathcal{D}}:=\\{f_{l}\\}_{l=1}^{\infty}$ in ${\mathcal{H}}$. Putting
$M=M_{\mathcal{D}}:=\cup_{l=1}^{\infty}M_{f_{l}}$, which is also a set of
Lebesgue measure zero, we may then say that (3.19) holds for all
$s\in{\mathbb{R}}\setminus M$ and for every $f\in{\mathcal{D}}$, and hence, in
view of the density, also for every $f\in{\mathcal{H}}$.

Moreover, we note that $s=0$ does not belong to $M_{f}$ for any
$f\in{\mathcal{H}}$, and therefore it neither belongs to $M$. Indeed, using
Lemma 3.2 and (3.11) in combination with the continuity of $\tau\mapsto
P(\tau)$ it is easy to see that

$u_{\tau}(0)=(I+S(0;\tau))^{-1}f=(I+\tau^{-1})^{-1}(I-P(\tau))\oplus
P(\tau)f\longrightarrow Pf,$ (3.20)

which means $0\notin M_{f}$.

Now let us first briefly outline our plan of _how to complete the proof of
Theorem 1.2_; we recall that one has to verify relation (3.7) showing that for
every $t\in{\mathbb{R}}$, $\\{u_{\tau}(t)=(I+S(it;\tau))^{-1}f\\}$ converges
to $(I+itH_{P})^{-1}Pf$ in the Hilbert space norm for all $t$ as $\tau\to 0+$.
Let us recall here that, as already mentioned in the text following (3.7), the
use of Chernoff’s theorem only requires to establish the convergence in (3.7)
pointwise for all $t\in{\mathbb{R}}$. Here and in the following we keep in
mind that $\\{u_{\tau}(t)\\}_{0<\tau\leq 1}$ is uniformly bounded in both $t$
and $0<\tau\leq 1$, i.e.

$\sup_{0<\tau\leq 1}\sup_{t\in{\mathbb{R}}}\|u_{\tau}(t)\|\leq\|f\|,$ (3.21)

since $(I+S(it;\tau))^{-1}$ is a contraction. We need not strive to show its
local uniformity, however, the following reasoning will establish this
property with respect to $t\in{\mathbb{R}}\setminus\\{0\\}$. We will proceed
in four steps demonstrating validity of the following claims:

I. The $\tau$-family $\\{u_{\tau}(t)\\}_{0<\tau\leq 1}$ of vectors
$u_{\tau}:{\mathbb{R}}\ni t\mapsto u_{\tau}(t)\in{\mathcal{H}}$ is
equicontinuous in $t\in{\mathbb{R}}\setminus\\{0\\}$ with respect to the
strong topology (i.e., Hilbert space norm) of ${\mathcal{H}}$.

II. The family $\\{u_{\tau}(t)\\}_{0<\tau\leq 1}$ converges as $\tau\to 0+$
for each fixed $t\in{\mathbb{R}}$ to some $u(t)\in{\mathcal{H}}$ in the weak
topology of ${\mathcal{H}}$, and furthermore, the convergence is _even_
locally uniform with respect to $t\in{\mathbb{R}}\setminus\\{0\\}$. The limit
function $u(t)$ turns out to be continuous in $t\in{\mathbb{R}}$ in the weak
topology of ${\mathcal{H}}$.

III. The limit satisfies

$u(t)=(I+itH_{P})^{-1}Pf\quad\text{for\,\, all}\,\,t.$ (3.22)

IV. Finally, the family $\\{u_{\tau}(t)\\}_{0<\tau\leq 1}$ converges as
$\tau\to 0+$ for any fixed $t\in{\mathbb{R}}$ to
$u(t)\equiv(I+itH_{P})^{-1}Pf$ in the strong topology of ${\mathcal{H}}$, and
furthermore, the convergence is _even_ locally uniform with respect to
$t\in{\mathbb{R}}\setminus\\{0\\}$.

_Step I._ The claim is expressed in the following lemma about the local
equicontinuity of the $\tau$-family $\\{u_{\tau}(t)\\}$ in a neighbourhood of
$t=s$ in ${\mathbb{R}}\setminus\\{0\\}$. It plays a crucial role in concluding
the proof of Theorem 1.2.

######  Lemma 3.6.

Let $f\in{\mathcal{H}}$. Then the $\tau$-family $\\{u_{\tau}(t)\\}_{0<\tau\leq
1}$ of vectors $u_{\tau}:{\mathbb{R}}\setminus\\{0\\}\ni t\mapsto
u_{\tau}(t)\in{\mathcal{H}}$ is equicontinuous locally in $t$ with respect to
the strong topology on ${\mathcal{H}}$. More explicitly, for every
$\varepsilon>0$ and for every $s\in{\mathbb{R}}\setminus\\{0\\}$ there exists
an $s$-dependent constant $\delta=\delta(f;\varepsilon;s)>0$ such that if
$t,s>0$ or $\>t,s<0$ with $|t-s|<\delta$, then
$\|u_{\tau}(t)-u_{\tau}(s)\|<\varepsilon$ holds for all $0<\tau\leq 1$.

We postpone for the moment the proof of Lemma 3.6, returning to it in Sect. 4,
and accept its claim, to finish first the proof of Theorem 1.2.

_Step II._ Without loss of generality we may suppose $f\not=0$. If $t=0$, the
family $\\{u_{\tau}(0)\\}_{0<\tau\leq 1}$ converges strongly to $Pf$ as
$\tau\to 0+\,$ as mentioned above, cf. (3.20). Consider thus a nonzero
$t\in{\mathbb{R}}\setminus\\{0\\}$ and take _arbitrary_ (sub)sequence
$\\{\tau^{\prime}\\}_{0<\tau^{\prime}\leq 1}$ of $\\{\tau\\}_{0<\tau\leq 1}$
with $\tau^{\prime}\to 0+$. By Lemma 3.6, to be yet proven, we see the
$\tau^{\prime}$-family $\\{u_{\tau^{\prime}}(t)\\}_{0<\tau^{\prime}\leq 1}$ is
equicontinuous in the strong topology and therefore in the weak topology,
because the ‘full’ $\tau$-family $\\{u_{\tau}(t)\\}_{0<\tau\leq 1}$ is. We
observed in (3.21) that the vectors $u_{\tau}(t)$ with any
$t\in{\mathbb{R}}\setminus\\{0\\}$ and $\tau\in(0,1]$ lie in the closed ball
$\bar{B}(0;\|f\|)\subset{\mathcal{H}}$ with the center at the origin and
radius $\|f\|$, which is weakly compact. To proceed, note that
$\bar{B}(0;\|f\|)$ is metrizable in the weak topology, since ${\mathcal{H}}$
is separable by assumption, see e.g. [11, Problem/Solution 18, p. 12 and 181].
Thus the equicontinuity holds with respect to the metric on the space
$\bar{B}(0;\|f\|)$ equivalent to the weak topology on it. Then, by virtue of
the Ascoli–Arzelà theorem, see e.g. [17, p. 81] or [23, Thm. 1.5.3], there
exists a (sub)sequence $\\{\tau^{\prime\prime}\\}_{0<\tau^{\prime\prime}\leq
1}$ of $\\{\tau^{\prime}\\}_{0<\tau^{\prime}\leq 1}$ with
$\tau^{\prime\prime}\to 0+$, along which $\\{u_{\tau}(t)\\}$ converges
_weakly_ to some limit $u(t)$, and moreover, the convergence is _(locally)
uniform_ in $t\in{\mathbb{R}}\setminus\\{0\\}$. This means that the numerical
family $\\{\langle\psi,u_{\tau^{\prime\prime}}(t)\rangle\\}$ converges for
every fixed $\psi\in{\mathcal{H}}$ as $\tau^{\prime\prime}\to 0+$, (locally)
uniformly in $t\in{\mathbb{R}}\setminus\\{0\\}$. Putting this together with
the case $t=0$ mentioned above, we see that the limit
$\langle\psi,u(t)\rangle$ is $t$-continuous everywhere in ${\mathbb{R}}$, in
other words, that $u(t)$ is $t$-continuous everywhere in ${\mathbb{R}}$ with
respect to the weak topology of ${\mathcal{H}}$.

_Step III._ We have established above two ways of convergence of the original
$\tau$-family $\\{u_{\tau}(t)\\}_{0<\tau\leq 1}$ of ${\mathcal{H}}$-valued
$t$-continuous functions as $\tau\to 0+$, or more specifically, convergence
with respect to two different topologies. One is the convergence to $u(t)$ in
the weak topology of ${\mathcal{H}}$, (locally) uniformly for
$t\in{\mathbb{R}}\setminus\\{0\\}$, and the other is the convergence to
$(I+itH_{P})^{-1}Pf$ in $L^{2}_{\text{\rm loc}}({\mathbb{R}};{\mathcal{H}})$
in the strong, and therefore also weak sense. This allows us to conclude that
these two limit vectors coincide for all $t$, in other words, to establish the
equality (3.22). Indeed, for any $\psi\in{\mathcal{H}}$ and an arbitrary
bounded interval $(a,b)$ with $a<b$ we have the following elementary estimate,

$\displaystyle\int_{a}^{b}|\langle\psi,u(t)-(I+itH_{P})^{-1}Pf\rangle|\,\mathrm{d}t$
$\displaystyle\leq\int_{a}^{b}|\langle\psi,u(t)-u_{\tau}(t)\rangle|\,\mathrm{d}t+\int_{a}^{b}|\langle\psi,u_{\tau}(t)-(I+itH_{P})^{-1}Pf\rangle|\,\mathrm{d}t\,,$

the last term of which tends to zero as $\tau\to 0+$ by (3.18). At the same
time, the first term on the right-hand side also tends to zero in view of the
weak convergence and the dominated convergence theorem. Consequently the
integrated expression on the left-hand side vanishes identically,

$\langle\psi,u(t)-(I+itH_{P})^{-1}Pf\rangle=0,$

for all $t$ and for all $\psi\in{\mathcal{H}}$, which yields the desired claim
(3.22). In this way, we have obtained the relations

$\displaystyle u(t)=Pu(t)\in D[H_{P}]=D[(H^{1/2}P)^{*}(H^{1/2}P)]\,,$
$\displaystyle Pf=P(I+itH_{P})u(t)=P[I+it(H^{1/2}P)^{*}(H^{1/2}P)]u(t)\,,$

which show that $\\{u_{\tau}(t)\\}$ converges to $u(t)=(I+itH_{P})^{-1}Pf$ in
the weak topology of ${\mathcal{H}}$ along $\tau\to 0+$, and therefore also
along $\tau\to 0+$.

_Step IV._ It remains to demonstrate that $\\{u_{\tau}(t)\\}$ converges to
$u(t)=(I+itH_{P})^{-1}Pf$ also _in the Hilbert space norm_ for all
$t\in{\mathbb{R}}$. For the sake of completeness we shall show that the same
claim can also be made about the $\tau$-family
$\\{(tG(\tau))^{1/2}P(\tau)u_{\tau}(t)\\}$ in (3); note that the analogous
question concerning $\\{(|t|H^{\pm}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\\}$ also
appearing in (3) is more complicated and we are able to provide only a partial
answer to it, cf. Proposition 3.5.

Since we have already established the weak convergence of the $\tau$-family
$\\{u_{\tau}(t)\\}$ for all $t$, we need only to show that the $\tau$-families
of the norms of these vectors converge. To this end, we observe again the real
part of (3), however, replacing now $f$ by $P(\tau)f$ in the inner product; we
write also the imaginary part for purpose of a further discussion. Writing the
left-hand side of (3) as $\langle u_{\tau}(t),P(\tau)f\rangle$, we obtain for
its real and imaginary part

$\displaystyle\mathrm{Re}\,\langle u_{\tau}(t),P(\tau)f\rangle$
$\displaystyle=\|P(\tau)u_{\tau}(t)\|^{2}+\|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2},$
(3.24) $\displaystyle\mathrm{Im}\,\langle u_{\tau}(t),P(\tau)f\rangle$
$\displaystyle=\|(|t|H^{+}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}$
$\displaystyle\qquad\qquad\qquad-\|(|t|H^{-}(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}.$
(3.25)

In view of (LABEL:limits2), the continuity of the projection family
$\\{P(\tau)\\}$, and the weak convergence of $\\{u_{\tau}(t)\\}$ as
$\tau\rightarrow 0+$, which we have already established, the left-hand sides
of the last two relations converge, leading to the following limits

$\displaystyle\text{\rm Re}\,\langle u_{\tau}(t),P(\tau)f\rangle$
$\displaystyle\longrightarrow\langle u(t),Pu(t)\rangle=\|Pu(t)\|^{2}.$ (3.26)
$\displaystyle\text{\rm Im}\,\langle u_{\tau}(t),P(\tau)f\rangle$
$\displaystyle\longrightarrow\langle
u(t),tH_{P}u(t)\rangle=\|(|t|H)^{1/2}Pu(t)\|^{2}$ (3.27)

valid for _all_ $t$. Comparing now the right-hand sides of relations (3.24)
and (3.26) we get

$\displaystyle\|Pu(t)\|^{2}$ $\displaystyle=\lim_{\tau\to
0+}\,\big{[}\|P(\tau)u_{\tau}(t)\|^{2}+\|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}\big{]}$
$\displaystyle=\liminf_{\tau\to
0+}\,\big{[}\|P(\tau)u_{\tau}(t)\|^{2}+\|(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}\big{]}$
$\displaystyle\geq\liminf_{\tau\to
0+}\,\|P(\tau)u_{\tau}(t)\|^{2}+\liminf_{\tau\to
0+}\,\|(|t|G(\tau))^{1/2}P(\tau)u_{\tau}(t)\|^{2}$
$\displaystyle\geq\|Pu(t)\|^{2}+\|0\|^{2}=\|Pu(t)\|^{2},$

which means that the two terms on the right-hand side of (3.24) converge to
$\|Pu(t)\|^{2}$ and to zero, respectively, yielding thus the sought
convergence in the Hilbert space norm along the sequence $\tau\to 0+$,

$P(\tau)u_{\tau}(t)\longrightarrow
Pu(t)\,,\quad(|t|G(t\tau))^{1/2}P(\tau)u_{\tau}(t)\longrightarrow 0\,.$ (3.28)

The first convergence in (3.28) also implies $u_{\tau}(t)\rightarrow
u(t)=Pu(t)$, i.e. convergence of $\tau$-family$\\{u_{\tau}(t)\\}$ to
$u(t)=Pu(t)$ in the Hilbert space norm, because we know already that
$(I-P(\tau^{\prime}))u_{\tau^{\prime}}(t)\to 0$.

This shows nothing but our sought statement of the convergence (3.7) for the
family $\\{u_{\tau}(t)\\}$ which, with the reference to Chernoff’s criterion
[2, 3], concludes the proof of Theorem 1.2. ∎

The proof of Lemma 3.6 will be given in the next section.

##  4\. Proof of Lemma 3.6

Recall first that the Hilbert space ${\mathcal{H}}$ we are handling is assumed
to be separable which is the property we needed to construct the exceptional
set $M$, cf. the text following (3.19). Recall also the notation we introduced
in (3.12) for the vector obtained by application of the operator
$(I+S(it;\tau))^{-1}$ to an arbitrary $f\in{\mathcal{H}}$,

$\displaystyle u_{\tau}$ $\displaystyle(t)=(I+S(it;\tau))^{-1}f$
$\displaystyle=\big{\\{}(1+\tau^{-1})^{-1}(I-P(\tau))\oplus[P(\tau)(I+tG(t\tau)+itH(t\tau))P(\tau)]^{-1}\big{\\}}f$
$\displaystyle=\big{\\{}(1+\tau^{-1})^{-1}(I-P(\tau))\oplus
T^{P}(t;\tau)\big{\\}}f\,,$ (4.1)

where the operator

$T^{P}(t;\tau):=[P(\tau)(I+tG(t\tau)+itH(t\tau))P(\tau)]^{-1}$ (4.2)

may be also considered on the whole Hilbert space ${\mathcal{H}}$, although in
the proper sense it is an operator with the domain and range included in the
closed subspace $P(\tau){\mathcal{H}}$; one may regard it as vanishing on the
orthogonal complement $(I-P(\tau)){\mathcal{H}}$.

Next, for fixed $t,s>0$ or $t,s<0$, we put

$\displaystyle D(t,s;\tau)f:$ $\displaystyle=u_{\tau}(t)-u_{\tau}(s)$
$\displaystyle=[(I+S(it;\tau))^{-1}-(I+S(is;\tau))^{-1}]f$
$\displaystyle=(I+\tau^{-1})^{-1}(I-P(\tau))\oplus\big{[}T^{P}(t;\tau)-T^{P}(s;\tau)\big{]}f\,.$
(4.3)

With the above direct sum decomposition in mind, one may for a fixed $\tau$
deal with the operator on the subspace $P(\tau){\mathcal{H}}$,

$T^{P}(t,s;\tau):=D(t,s;\tau)\restriction_{P(\tau){\mathcal{H}}}=T^{P}(t;\tau)-T^{P}(s;\tau)\,,$
(4.4)

however, since $\tau$ is varying, we have to consider it on the whole space
${\mathcal{H}}$. Note that the $\tau$-family $\\{D(t,s;\tau)\\}_{0<\tau\leq
1}$ is strongly continuous in $0<\tau\leq 1$, and uniformly bounded, i.e.
$\|D(t,s;\tau)\|\leq 2$, because both $(I+S(it;\tau))^{-1}$ and
$(I+S(is;\tau))^{-1}$ are contractions. Note also that we have
$D(t,s;\tau)=P(\tau)D(t,s;\tau)$.

To verify the assertion of Lemma 3.6, we need to show that for any fixed
$f\in{\mathcal{H}}$, the difference $D(t,s;\tau)f$ in (4) with
$t,s\in{\mathbb{R}}\setminus\\{0\\}$ converges to zero in the Hilbert space
norm as $|t-s|\to 0$, and that the convergence is uniform with respect to
$\tau\in(0,1]$.

We use a small trick showing first that it is sufficient to establish the
claim of Lemma 3.6 _under the additional assumption_ that one of the $t$ and
$s$, say the latter, belongs to ${\mathbb{R}}\setminus(M\cup\\{0\\})$. Indeed,
if this is the case, i.e. if for any fixed
$s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$ and arbitrary
$f\in{\mathcal{H}},\;\varepsilon>0$, there is a
$\delta=\delta(f;\varepsilon;s)>0$ such that

$\|u_{\tau}(t)-u_{\tau}(s)\|<\tfrac{\varepsilon}{2}\;\;\text{\rm holds for
all}\;\;t\in(s-\delta,s+\delta)\;\;\text{\rm and}\;\;\tau\in(0,1],$

the lemma is valid in the general case as well. To see that, we take any two
points $t_{1},\,t_{2}\in{\mathbb{R}}\setminus\\{0\\}$ in the vicinity of the
chosen $s$ satisfying $|t_{i}-s|<\tfrac{\delta}{2}$ for $i=1,2$, then

$\displaystyle|t_{1}-t_{2}|\leq|t_{1}-s|+|s-t_{2}|<\tfrac{\delta}{2}+\tfrac{\delta}{2}=\delta,$
$\displaystyle\|u_{\tau}(t_{1})-u_{\tau}(t_{2})\|\leq\|u_{\tau}(t_{1})-u_{\tau}(s)\|+\|u_{\tau}(s)-u_{\tau}(t_{2})\|<\tfrac{\varepsilon}{2}+\tfrac{\varepsilon}{2}=\varepsilon$

holds independently of $\tau$. This yields the ‘full’ claim of Lemma 3.6 in
view of the fact that the set ${\mathbb{R}}\setminus(M\cup\\{0\\})$ from which
the number $s$ is chosen is dense in ${\mathbb{R}}\setminus\\{0\\}$.

Let us thus turn to the nontrivial part of the proof which consists of
establishing the assertion of Lemma 3.6 for
$s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$ and
$t\in{\mathbb{R}}\setminus\\{0\\}$; the aim is to complete the proof of our
main result in the forthcoming Lemma 4.5(ii). We will work out the argument
assuming that $t,s>0$, the case with the opposite signs is completely
analogous.

To begin with, we use functional calculus with (3.2), (3.3) to rewrite the
differences containing $H(t\tau)$ and $G(t\tau)$ in (4) as

$\displaystyle tH(t\tau)-sH(s\tau)$ $\displaystyle=\tfrac{\sin t\tau H-\sin
s\tau H}{\tau}=\tfrac{2}{\tau}\cos(\tfrac{t+s}{2}\tau
H)\,\sin(\tfrac{t-s}{2}\tau H),$ (4.5) $\displaystyle tG(t\tau)-sG(s\tau)$
$\displaystyle=\tfrac{-\cos t\tau H+\cos s\tau
H}{\tau}=\tfrac{2}{\tau}\sin(\tfrac{t+s}{2}\tau H)\,\sin(\tfrac{t-s}{2}\tau
H).$ (4.6)

To simplify expressions in the discussion to follow, we recall the quantity
$K(\kappa)$ introduced in (3.2) which for $\kappa=t\tau$ allows us to write

$I+tK(t\tau)=I+tG(t\tau)+itH(s\tau)=I+\tfrac{I-\cos t\tau
H}{\tau}+i\tfrac{\sin t\tau H}{\tau}\,,$ (4.7)

and similarly for $k=s\tau$. Furthermore, for a given $\tau\in(0,1]$ we
introduce the self-adjoint operator

$H_{\tau}:=H(I+\tau H)^{-1}$ (4.8)

which is positive and bounded, and note that $I+|s|H_{\tau}$ has a bounded
inverse for any $s\in\mathbb{R}$. The difference $D(t,s;\tau)$ in (4) which
can be identified with its nontrivial part $T^{P}(t,s;\tau)$ in (4.4) can be
then rewritten as

$\displaystyle D(t,s;\tau)$ $\displaystyle=T^{P}(t,s;\tau)$
$\displaystyle=[P(\tau)(I+tK(t\tau))P(\tau)]^{-1}-[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}$
$\displaystyle=[P(\tau)(I+tK(t\tau))P(\tau)]^{-1}$
$\displaystyle\qquad\cdot\big{\\{}P(\tau)(I+sK(s\tau))P(\tau)-P(\tau)(I+tK(t\tau))P(\tau)\big{\\}}$
$\displaystyle\quad\qquad\cdot[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}$
$\displaystyle=[P(\tau)(I+tK(t\tau))P(\tau)]^{-1}[P(\tau)(sK(s\tau)-tK(t\tau))P(\tau)]$
$\displaystyle\qquad\cdot[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}$
$\displaystyle=[P(\tau)(I+tK(t\tau))P(\tau)]^{-1}$
$\displaystyle\qquad\cdot[(sK(s\tau)-tK(t\tau))(I+|s|H_{\tau})^{-1}]$
$\displaystyle\quad\qquad\cdot(I+|s|H_{\tau})P(\tau)[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}$
$\displaystyle=:T_{1}(t;\tau)\,T_{2}(t,s;\tau)\,T_{3}(s;\tau),$ (4.9)

where we have introduced

$\displaystyle
T_{1}(t;\tau)=T^{P}(t;\tau)=[P(\tau)(I+tK(t\tau))P(\tau)]^{-1},$ (4.10)
$\displaystyle T_{2}(t,s;\tau)=(sK(s\tau)-tK(t\tau))(I+|s|H_{\tau})^{-1},$
(4.11) $\displaystyle
T_{3}(s;\tau)=(I+|s|H_{\tau})P(\tau)[P(\tau)(I+sK(s,\tau))P(\tau)]^{-1},$
(4.12)

which are all bounded operators on $P(\tau){\mathcal{H}}$ as well as on
${\mathcal{H}}$; in (4.11) we are able to drop the projections $P(\tau)$
appearing in the last formula since the operator (4.2) maps the subspace
$P(\tau)\mathcal{H}$ onto itself.

Next we consider the following two operator-valued functions, which may be
thought of as the $\tau$-limits of the families
$\\{T_{1}(t;\tau)\\}_{0<\tau\leq 1}\,$ in (4.10), and
$\\{T_{3}(s;\tau)\\}_{0<\tau\leq 1}\,$ in (4.12), namely

$\displaystyle T_{1}(t):$ $\displaystyle=[(I+itH_{P})^{-1}P],$ (4.13)
$\displaystyle T_{3}(s):$
$\displaystyle=(I+|s|H)P[(I+isH_{P})^{-1}P]=(I+|s|H)[(I+isH_{P})^{-1}P],$
(4.14)

where we can remove again one $P$ from the second expression of (4.14) since
$(I+isH_{P})^{-1}$ maps $P{\mathcal{H}}$ to itself. We already know from
(3.28) that $T_{1}(t)$ is the strong limit as $\tau\to 0+$ of the family
$\\{T_{1}(t;\tau)\\}$ of contractions, for the moment at least as long as
$t\in{\mathbb{R}}\setminus(M\cup\\{0\\})$.

Next we are going to show that $T_{3}(s)$ can be extended to a bounded
operator on ${\mathcal{H}}$. We begin with a crucial observation.

######  Lemma 4.1.

Let $H$ be our nonnegative self-adjoint operator acting in ${\mathcal{H}}$ and
$H_{P}$ the self-adjoint operator introduced in Sect. 2 referring to the
orthogonal projection $P$. Consider the operator

$T_{0}:=(I+H)[(I+H_{P})^{-1}P]$ (4.15)

in ${\mathcal{H}}$ for which we have: (i) the domain and range of $T_{0}$ are

$\displaystyle D[T_{0}]$ $\displaystyle=P(I+H_{P})D[HP]=P(I+H_{P})(D[H]\cap
P{\mathcal{H}})\oplus(P{\mathcal{H}})^{\perp},$ $\displaystyle R[T_{0}]$
$\displaystyle=(I+H)D[HP]=(I+H)(D[H]\cap P{\mathcal{H}}),$ (4.16)

as $T_{0}$ is the direct sum,
$T_{0}=T_{0}\\!\restriction_{P{\mathcal{H}}}\oplus\,0\,$, in accordance with
(2.1),
(ii) and, in addition,

$T_{0}\,g=g\,,\quad g\in D[T_{0}\\!\restriction_{P{\mathcal{H}}}],$ (4.17)

thus $T_{0}$ can be extended to a bounded operator $\tilde{T}_{0}$ on
${\mathcal{H}}$ such that

$\displaystyle\text{\it the closure of
\,$T_{0}\\!\restriction_{P{\mathcal{H}}}$\, is the identity
operator\,$I_{P{\mathcal{H}}}$\,\,on\,}P{\mathcal{H}},$
$\displaystyle\text{\it and}\,\,\tilde{T_{0}}=I_{P{\mathcal{H}}}\oplus
0\,\,\,\text{\rm on
}\,\,{\mathcal{H}}=P{\mathcal{H}}\oplus(P{\mathcal{H}})^{\perp}.$ (4.18)

In particular, $\tilde{T_{0}}$ is a contraction.

###### Proof.

Product of linear operators $A$ and $B$ in ${\mathcal{H}}$ has the domain
$D[AB]=B^{(-1)}D[A]:=\\{\,g\in D[B]:\,Bg\in D[A]\,\\}$. Since $P$ commutes
with $H_{P}$, we can rewrite (4.15) as $T_{0}:=[(I+H)P][(I+H_{P})^{-1}P]$ with
the domain

$\displaystyle D[T_{0}]$ $\displaystyle=[P(I+H_{P})^{-1}]^{(-1)}D[(I+H)P]$
$\displaystyle=\\{\,g\in D[(I+H_{P})^{-1}P]:\,(I+H_{P})^{-1}Pg\in
D[(I+H)P]\\}$ $\displaystyle=\\{\,g\in D[H_{P}]:\,(I+H_{P})^{-1}Pg\in
D[HP]\\}$ $\displaystyle=P(I+H_{P})D[HP]=P(I+H_{P})(D[H]\cap
P{\mathcal{H}})\oplus(P{\mathcal{H}})^{\perp}$

and the range

$\displaystyle R[T_{0}]$ $\displaystyle=T_{0}D[T_{0}]$
$\displaystyle=[(I+H)P][(I+H_{P})^{-1}P]P(I+H_{P})D[HP]$
$\displaystyle=[(I+H)P]D[HP]=(I+H)(D[H]\cap P{\mathcal{H}}).$

Let us now turn to the claim (ii). Just as $H_{P}$ is the self-adjoint
operator in ${\mathcal{H}}$ associated with the quadratic form
$u\mapsto\|H^{1/2}Pu\|^{2}$ defined on $D[H^{1/2}P]=(D[H^{1/2}]\cap
P{\mathcal{H}})\oplus(P{\mathcal{H}})^{\perp}$, the self-adjoint operator
$P(I+H_{P})$ is associated with the form $u\mapsto\|(I+H)^{1/2}Pu\|^{2}$
defined in view of the inequalities

$\frac{1}{\sqrt{2}}(I+H^{1/2})\leq(I+H)^{1/2}\leq I+H^{1/2}.$

on the same domain $D[(I+H)^{1/2}P]=D[H^{1/2}P]$. Consequently,

$P(I+H_{P})=P(I+H)_{P}=((I+H)^{1/2}P)^{*}(I+H)^{1/2}P\,.$ (4.19)

We are going to use it to show (4.17). The adjoint $((I+H)^{1/2}P)^{*}$ to
$(I+H)^{1/2}P$ is a closed extension of the closable (in general, non-closed)
operator $P(I+H)^{1/2}$, in other words, $((I+H)^{1/2}P)^{*}\supset
P(I+H)^{1/2}$, which yields

$((I+H)^{1/2}P)^{*}(I+H)^{1/2}P\supset P(I+H)^{1/2}(I+H)^{1/2}P=P(I+H)P,$

i.e. the operator on the left-hand side is an extension of the operator on the
right. As both sides are invertible, the analogous inclusion holds for their
inverses,

$[(I+H_{P})^{-1}P]=[((I+H)^{1/2}P)^{*}(I+H)^{1/2}P]^{-1}\supset[P(I+H)P]^{-1}.$

It follows that for any $g\in D[T_{0}]$ specified in (4.1) we have

$T_{0}\,g=(I+H)[(I+H_{P})^{-1}P]g=(I+H)[P(I+H)P]^{-1}g=Pg,$

which yields the desired claim (4.17), since $T_{0}$ is reduced by the
projection $P$ and $D[T_{0}\\!\restriction_{P{\mathcal{H}}}]\subseteq
P{\mathcal{H}}$. Thus we see that the closure of
$T_{0}\\!\restriction_{P{\mathcal{H}}}$ is the identity operator
$I_{P{\mathcal{H}}}$ on $P{\mathcal{H}}$, and $\tilde{T_{0}}$ as the closed
extension of $T_{0}$ to the whole ${\mathcal{H}}$ has the norm not exceeding
one. ∎

######  Lemma 4.2.

The operator $T_{3}(s)$ in (4.14), acting in ${\mathcal{H}}$ with the domain
$P(I+isH_{P})D[HP]=(I+isH_{P})(D[H]\cap P{\mathcal{H}})$, can be extended to a
bounded operator on the whole space ${\mathcal{H}}$ with the norm satisfying
$\|T_{3}(s)\|\leq\sqrt{2}$.

###### Proof.

Note first that the claim of Lemma 4.1 remains valid when we replace
$H_{P},\,H$ by $|s|H_{P},\,|s|H$, respectively, for any $s\not=0$. We rewrite
the operator $T_{3}(s)$ in (4.14) as

$\displaystyle T_{3}(s)$
$\displaystyle=\big{\\{}(I+|s|H)[(I+|s|H_{P})]^{-1}P]\big{\\}}$
$\displaystyle\qquad\qquad\cdot\big{\\{}[P(I+|s|H_{P})][(I+isH_{P})^{-1}P]\big{\\}}\,.$

By spectral theorem the norms of the first and second factors on the right-
hand side are one and $\sqrt{2}$, respectively, independently of $s$, and
Lemma 4.1 (ii) in combination with the above observation implies that the
first factor is a contraction, which gives $\|T_{3}(s)\|\leq\sqrt{2}$. ∎

In the next step, we turn to investigation of the second and third factors,
(4.11) and (4.12), of the operator $T^{P}(t,s;\tau)$ in (4.9). We begin with
proving a crucial property of the $\tau$-family
$\\{T_{3}(s;\tau)\\}_{0<\tau\leq 1}$ with
$s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$, where the set
$M\subseteq{\mathbb{R}}$ of Lebesgue measure zero was introduced in the text
following (3.19); recall that we adopted the separability assumption. After
doing that, we will focus on the $\tau$-family
$\\{T_{2}(t,s;\tau)\\}_{0<\tau\leq 1}$ with
$t,\,s\in{\mathbb{R}}\setminus\\{0\\}$.

######  Lemma 4.3.

Let the Hilbert space ${\mathcal{H}}$ be separable and consider a number
$s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$, so that for every vector
$f\in{\mathcal{H}}$ the $\tau$-family
$\\{[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}f\\}$ converges in the Hilbert space
norm to $[(I+isH_{P})^{-1}P]f$ as $\tau\to 0+$. Then the following claims are
valid:

(i) For fixed $s$, the operator family $\\{T_{3}(s;\tau)\\}_{0<\tau\leq 1}$
defined by (4.12) is uniformly bounded on ${\mathcal{H}}$, and converges
strongly to $T_{3}(s)$ in (4.14) as $\tau\to 0+$.

(ii) To be specific, for a fixed $s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$
there is a constant $C_{T_{3}}(s)\geq\sqrt{2}$ such that

$\|T_{3}(s;\tau)\|\leq C_{T_{3}}(s)\quad\text{for all}\;\>\tau\in(0,1]\,.$
(4.20)

###### Proof.

(i) Our aim is to verify that $T_{3}(s;\tau)g$ converges for
$g\in{\mathcal{H}}$ to $T_{3}(s)g$ as $\tau\to 0+$ in the Hilbert space norm
and to use this fact to establish the uniform boundedness of
$\\{T_{3}(s;\tau)g\\}_{0<\tau\leq 1}$.

To this end, let us recall the notion of convergence in the strong graph
sense: given a sequence $\\{A_{n}\\}_{n=1}^{\infty}$ of operators in a Hilbert
space ${\mathcal{H}}$ we say that
$(\psi,\varphi)\in{\mathcal{H}}\times{\mathcal{H}}$ belongs to the strong
graph limit if one can find $\psi_{n}\in D[A_{n}],\,n=1,2,\dots,$ such that

$\psi_{n}\longrightarrow\psi,\quad
A_{n}\psi_{n}\longrightarrow\varphi\quad\;\text{\rm as}\;\;n\to\infty\,.$

We denote the set of such pairs $(\psi,\varphi)$ by $\Gamma_{\infty}^{s}$. If
$\Gamma_{\infty}^{s}$ is the graph of an operator $A$, i.e.
$\Gamma_{\infty}^{s}=\\{(\psi,A\psi)\in{\mathcal{H}}\times{\mathcal{H}};\,\psi\in
D[A]\\}$, $\>A$ is also said to be the _strong graph limit_ of $\\{A_{n}\\}$.
We have the following result [10, Thm 4], see also [21, Thm VIII.26]:

######  Proposition 4.4.

Let $\\{A_{n}\\}_{n=1}^{\infty}$ and $A$ be self-adjoint operators in
${\mathcal{H}}$. Then $\\{A_{n}\\}$ converges to $A$ in strong resolvent sense
if and only if $A$ is the strong graph limit of $\\{A_{n}\\}$.

To make use of this result, let us consider the family
$\\{A_{\tau}\\}_{0<\tau\leq 1}$ of the following bounded and self-adjoint
operators,

$A_{\tau}:=I+|s|H_{\tau}\,,\quad\;0<\tau\leq 1.$

By spectral theorem its elements converge in the strong operator topology to
the self-adjoint operator $I+|s|H$ as $\tau\to 0+$, and that by Proposition
4.4 means that $I+|s|H$ is the strong graph limit of $\\{A_{\tau}\\}$. Since
$s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$ holds by assumption, we have

$\displaystyle\quad\psi_{\tau}:=[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}g\longrightarrow\psi:=[(I+isH_{P})^{-1}P]g\,,$
as $\tau\to 0+$ for $g\in{\mathcal{H}}$, with
$\displaystyle\quad[(I+isH_{P})^{-1}P]g\in D[H_{P}]\subset D[(I+|s|H)P]\subset
D[I+|s|H]\,.$

In this way, the pair $(\psi,(I+|s|H)\psi)$ belongs to the set
$\Gamma_{\infty}^{s}$ resulting from the strong graph limit of the family
$\\{A_{\tau}\\}$, and consequently,
$A_{\tau}\psi_{\tau}\longrightarrow(I+|s|H)\psi$, i.e.

$\displaystyle
A_{\tau}[P(\tau)(I+sK(s\tau))P(\tau)]^{-1}g\longrightarrow(I+|s|H)[(I+isH_{P})^{-1}P]g\,,$

in other words, the sought convergence,
$T_{3}(s;\tau)g\longrightarrow\,T_{3}(s)g$. This implies, by the uniform
boundedness principle (or Banach-Steinhaus theorem), the family
$\\{T_{3}(s;\tau)\\}_{0\leq\tau<1}$ is uniformly bounded in the operator norm.

(ii) Consider again an arbitrary $g\in{\mathcal{H}}$. In view of the
convergence established above in combination with the bound on the norm of
$T_{3}(s)$ from Lemma 4.2(ii), we see that for a fixed
$s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$ there is a positive, $g$-dependent
number $\tau_{s}=\tau_{s}(g)\leq 1$ such that

$\displaystyle\|T_{3}(s;\tau)g\|\leq\|T_{3}(s)g\|+\tfrac{1}{2}\leq\|T_{3}(s)\|\|g\|+\tfrac{1}{2}\leq\sqrt{2}\|g\|+\tfrac{1}{2}\,,\;\>\tau\in(0,\tau_{s}]\,.$

This helps us to complement our knowledge of the uniform boundedness using the
estimate

$\sup_{0<\tau\leq
1}\|T_{3}(s;\tau)g\|\leq\max\big{\\{}\sqrt{2}\|g\|+\tfrac{1}{2},\,\sup_{\tau_{s}<\tau\leq
1}\|T_{3}(s;\tau)g\|\,\big{\\}}\,.$

We note that the function $\tau\mapsto\|T_{3}(s;\tau)g\|$ is continuous on the
compact set $[\tau_{s},1]\subset{\mathbb{R}}$, hence the right-hand side is
bounded for any $g$ and, applying again the uniform boundedness principle, we
conclude that the $\tau$-family $\\{T_{3}(s;\tau)\\}_{0<\tau\leq 1}$ is
uniformly bounded by a constant $C_{T_{3}}(s)\geq\sqrt{2}$ depending on $s$.
This completes the proof of Lemma 4.3. ∎

It remains to deal with the middle factor $T_{2}(t,s;\tau)$ in (4.11). We
rewrite it as

$\displaystyle T_{2}(t,s;\tau)$
$\displaystyle=i(sH(s\tau)-tH(t\tau))(I+|s|H_{\tau})^{-1}$
$\displaystyle\quad+(sG(s\tau)-tG(t\tau))I+|s|H_{\tau})^{-1}$
$\displaystyle=:iT_{2,H}(t,s;\tau)+T_{2,G}(t,s;\tau).$ (4.21)

For a fixed $\tau\in(0,1]$, the operators $T_{2,H}(t,s;\tau)$,
$T_{2,G}(t,s;\tau)$, as well as their linear combination, are bounded
operators which are in the functional calculus sense obtained from $H$ as its
appropriate bounded and continuous functions; in view of (4.5) and (4.6) we
have

$\displaystyle T_{2,H}(t,s;\tau)$
$\displaystyle=\phi_{H}(t,s;\tau,H):=(tH(t\tau)-sH(s\tau))(I+|s|H_{\tau})^{-1}$
$\displaystyle=\frac{\tfrac{2}{\tau}\sin(\tfrac{t-s}{2}\tau\lambda)\cos(\tfrac{s+t}{2}\tau\lambda)}{I+|s|\lambda(I+\tau\lambda)^{-1}}\,,$
$\displaystyle T_{2,G}(t,s;\tau)$
$\displaystyle=\phi_{G}(t,s;\tau,H):=(tG(t\tau)-sG(s\tau))(I+|s|H_{\tau})^{-1}$
$\displaystyle=\frac{\tfrac{2}{\tau}\sin(\tfrac{t-s}{2}\tau\lambda)\,\sin(\tfrac{s+t}{2}\tau\lambda)}{I+|s|\lambda(I+\tau\lambda)^{-1}}\,,$

and

$T_{2}(t,s;\tau)=\phi(t,s;\tau,H):=i\phi_{H}(t,s;\tau,H)+\phi_{G}(t,s;\tau,H)$

It is straightforward to check that

$|\phi(t,s;\tau,\lambda)|^{2}=\Big{|}\frac{\tfrac{2}{\tau}\sin(\tfrac{t-s}{2}\tau\lambda)}{I+|s|\lambda(I+\tau\lambda)^{-1}}\Big{|}^{2},$
(4.22)

which means that for any $g\in{\mathcal{H}}$ we have

$\displaystyle\|T_{2}(t,s;\tau)g\|^{2}$
$\displaystyle=\|T_{2,H}(t,s;\tau)g\|^{2}+\|T_{2,G}(t,s;\tau)g\|^{2}$
$\displaystyle=\int_{0-}^{\infty}\Big{|}\frac{\tfrac{2}{\tau}\sin(\tfrac{t-s}{2}\tau\lambda)}{I+|s|\lambda(I+\tau\lambda)^{-1}}\Big{|}^{2}\|E(\mathrm{d}\lambda)g\|^{2}\,.$
(4.23)

The following lemma represents the second crucial element in establishing
Lemma 3.6, and in this way, proving our main theorems.

######  Lemma 4.5.

Consider $t,s\in{\mathbb{R}}\setminus\\{0\\}$ such that either $t,s>0$ or
$t,s<0$. Then for $T_{2}(t,s;\tau)$ given by (4.11) and for $D(t,s;\tau)$
given by (4) or (4.9) the following holds:

(i) The $\tau$-family $\\{T_{2}(t,s;\tau\\}$ of bounded operators in (4.21) is
uniformly bounded locally uniformly for $t,s\in{\mathbb{R}}\setminus\\{0\\}$.
Further, for every $f\in{\mathcal{H}}$ and any $\varepsilon>0$, there is an
$s$-dependent number $\delta=\delta(f;\varepsilon;s)>0$ such that

$t\in{\mathbb{R}}\setminus\\{0\\}\;\;\text{\it
with}\;\,|t-s|<\delta\;\;\Longrightarrow\;\;\|T_{2}(t,s;\tau)f\|<\varepsilon,$
(4.24)

uniformly with respect to $\tau\in(0,1]$.

(ii) Let further $s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$, then for every
$f\in{\mathcal{H}}$ and any $\varepsilon>0$, there is a number
$\delta=\delta(f;\varepsilon;s)>0$ such that

$t\in{\mathbb{R}}\setminus\\{0\\}\;\;\text{\it
with}\;\;|t-s|<\delta\;\;\Longrightarrow\;\;\|D(t,s;\tau)f\|<\,\varepsilon,$
(4.25)

uniformly with respect to $\tau\in(0,1]$.

###### Proof.

(i) The $\tau$-families of bounded operators appearing in (4.21) converge in
the strong operator topology for any fixed pair
$t,s\in{\mathbb{R}}\setminus\\{0\\}$; by functional calculus we find easily
that

$\displaystyle
T_{2,H}(t,s;\tau)\overset{s}{\to}T_{2,H}(t,s):=\tfrac{(t-s)H}{I+|s|H},\quad
T_{2,G}(t,s;\tau)\overset{s}{\to}T_{2,G}(t,s):=0,$ $\displaystyle
T_{2}(t,s;\tau)\overset{s}{\longrightarrow}T_{2}(t,s):=iT_{2,H}(t,s)+T_{2,G}(t,s)$

holds as $\tau\to 0+$, and therefore

$|T_{2}(t,s;\tau)|^{2}\overset{s}{\longrightarrow}|T_{2}(t,s)|^{2}=T_{2,H}(t,s)^{2}+T_{2,G}(t,s)^{2}=\big{(}\tfrac{(t-s)H}{I+|s|H}\big{)}^{2}.$

Our way to prove the claim is to apply the uniform boundedness principle to
these operator families. From (4.23) we obtain

$\|T_{2}(t,s;\tau)\|=\sup_{\|g\|=1}\|T_{2}(t,s;\tau)g\|=\sup_{\lambda\geq
0}|\phi(t,s;\tau,\lambda)|\,,$

and for $T_{2}(t,s)=\phi(t,s;H)$ with
$\phi(t,s;\lambda):=\tfrac{(t-s)\lambda}{1+|s|\lambda}$ we have similarly

$\|T_{2}(t,s)\|=\sup_{\lambda\geq 0}|\phi(t,s;\lambda)|=\tfrac{|t-s|}{|s|}\,.$
(4.26)

We note that $\phi(t,s;\tau,\lambda)$ is continuous with respect to all the
variables, $\tau\in(0,1]$, $\lambda\in[0,\infty)$, and
$t,s\in{\mathbb{R}}\setminus\\{0\\}$. Our intention is now to regard the
family $\\{T_{2}(t,s;\tau)\\}$ as depending on a broader set of parameters: we
fix a nonzero $s$ and consider $\tau$ in $(0,1]$ and $t$ in a neighborhood of
$s$, more specifically,

$\tau\in(0,1],\quad t\in
D(s):=\\{t\in{\mathbb{R}}\setminus\\{0\\};\,|t-s|\leq\tfrac{|s|}{2}\\}.$
(4.27)

For each fixed $f\in{\mathcal{H}}$ and $s\in{\mathbb{R}}\setminus\\{0\\}$ we
put

$\Phi_{f,s}(t;\tau):=\left\\{\begin{array}[]{lcc}\|T_{2}(t,s;\tau)f\|&\quad\dots&0<\tau\leq
1,\\\\[3.00003pt] \|T_{2}(t,s)f\|&\quad\dots&\tau=0,\end{array}\right.$ (4.28)

which is a bounded and continuous function on the compact set
$D(s)\times[0,1]\subset{\mathbb{R}}^{2}$. Then $C_{f,s}:=\max_{t,\tau\in
D(s)\times[0,1]}\Phi_{f,s}(t;\tau)$ exists, depending on $f$ and $s$, but
being independent of $t\in D(s)$ and $\tau\in(0,1]$, and from the strong
convergence of $\\{T_{2}(t,s;\tau)\\}$ to $T_{2}(t,s)$ noted above we infer
that

$\|T_{2}(t,s;\tau)f\|\leq C_{f,s}$

holds for given $f\in{\mathcal{H}}$, $\>s\neq 0$, and all $(t,\tau)\in
D(s)\times(0,1]$. Applying now the uniform boundedness principle to this
operator family with the extended set of parameters (4.27), we establish the
existence of a constant $C_{T_{2}}(s)>0$, depending on $s$ but independent of
$t\in D(s)$ and $\tau\in(0,1]$, such that

$\|T_{2}(t,s;\tau)\|\leq C_{T_{2}}(s).$ (4.29)

In view of the strong convergence, it cannot be smaller than the norm of
$T_{2}(t,s)$ in (4.26), that is, $C_{T_{2}}(s)\geq\tfrac{|t-s|}{|s|}$.

Since $E([\lambda,\infty))\overset{s}{\to}0$ holds as $\lambda\to\infty$, one
can find for a fixed $s$, any $f\in{\mathcal{H}}$, and each $\varepsilon>0$ a
number $\lambda_{0}=\lambda_{0}(f;\varepsilon;s)>0$ such that

$\|E([\lambda_{0},\infty))f\|^{2}<\tfrac{1}{C_{T_{2}}(s)^{2}}\cdot\tfrac{\varepsilon^{2}}{2}\,.$
(4.30)

In this case using (4.22) and (4.30), we get

$\displaystyle\|T_{2}$
$\displaystyle(t,s;\tau)f\|^{2}=\|\phi(t,s;\tau,H)f\|^{2}$
$\displaystyle=\Big{(}\int_{0-}^{\lambda_{0}}\,\,+\,\,\int_{\lambda_{0}}^{\infty}\Big{)}|\phi(t,s;\tau,\lambda)|^{2}\|E(\mathrm{d}\lambda)f\|^{2}$
$\displaystyle\leq\int_{0-}^{\lambda_{0}}\bigg{|}\tfrac{\tfrac{2}{\tau}\sin(\tfrac{t-s}{2}\tau\lambda)}{1+(s\lambda)(I+\tau\lambda)^{-1}}\bigg{|}^{2}\|E(\mathrm{d}\lambda)f\|^{2}+C_{T_{2}}(s)^{2}\|E([\lambda_{0},\infty))f\|^{2}$
$\displaystyle\leq(|t-s|\lambda_{0})^{2}\|E([0,\lambda_{0}))f\|^{2}+\tfrac{\varepsilon^{2}}{2}\,,$

where the last inequality comes from the estimate of the numerator of the
squared modulus factor in the above integral,

$\big{|}\tfrac{2}{\tau}\sin(\tfrac{t-s}{2}\tau\lambda)\big{|}\leq|t-s|\lambda_{0},\qquad
0\leq\lambda\leq\lambda_{0}.$

Since the set $\\{(\tau,\lambda);\,0<\tau\leq
1,\,0\leq\lambda\leq\lambda_{0}\\}$ is bounded, and therefore its closure is
compact, we infer that for a fixed $s\in{\mathbb{R}}\setminus\\{0\\}$ there is
a number $\delta=\delta(f;\varepsilon;s)>0$ such that

$t\in{\mathbb{R}}\setminus\\{0\\}\,\,\text{\rm
with}\,\,|t-s|<\delta\,\,\Longrightarrow(|t-s|\lambda_{0})^{2}(1+\|f\|^{2})<\tfrac{\varepsilon^{2}}{2}$

uniformly for $0<\tau\leq 1$. In fact, we can find it explicitly; it is enough
to choose

$\delta(f;\varepsilon;s):=\min\big{\\{}\tfrac{1}{\sqrt{2}\lambda_{0}(1+\|f\|^{2})^{1/2}}\,\varepsilon,\tfrac{|s|}{2}\big{\\}}.$

Thus we have

$\|T_{2}(t,s;\tau)f\|^{2}\leq(1+\|f\|^{2})^{-1}\tfrac{\varepsilon^{2}}{2}\|E([0,\lambda_{0}))f\|^{2}+\tfrac{\varepsilon^{2}}{2}<\varepsilon^{2},$

in other words,

$\|T_{2}(t,s;\tau,H)f\|<\varepsilon,$

which yields the implication (4.24).

(ii) To prove the remaining claim of the lemma, we first note that
$T_{1}(t;\tau)\equiv T^{P}(t;\tau)$ is a contraction, $\|T_{1}(t;\tau)\|\leq
1$, and therefore

$\displaystyle\|D(t,s;\tau)f\|$
$\displaystyle=\|T^{P}(t,s;\tau)f\|=\|T_{1}(t;\tau)T_{2}(t,s;\tau)T_{3}(s;\tau)f\|$
$\displaystyle\leq\|T_{2}(t,s;\tau)T_{3}(s;\tau)f\|.$

To deal with the last norm we repeat the argument from the part (i) replacing
$f$ by $T_{3}(s;\tau)f$, obtaining for any $\lambda_{0}>0$ the estimate

$\displaystyle\|T_{2}(t,s;\tau)T_{3}(s;\tau)f\|^{2}$
$\displaystyle=\Big{(}\int_{0-}^{\lambda_{0}}+\int_{\lambda_{0}}^{\infty}\Big{)}|\phi(t,s;\tau,\lambda)|^{2}\|E(\mathrm{d}\lambda)T_{3}(s;\tau)f\|^{2}$
$\displaystyle\leq\Big{[}\int_{0-}^{\lambda_{0}}\Big{|}\tfrac{\tfrac{2}{\tau}\sin\tfrac{(t-s)}{2}\tau\lambda}{1+(s\lambda)(I+\tau\lambda)^{-1}}\Big{|}^{2}\|E(\mathrm{d}\lambda)T_{3}(s;\tau)f\|^{2}$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+C_{T_{2}}(s)^{2}\|E([\lambda_{0},\infty))T_{3}(s;\tau)f\|^{2}\Big{]}$
(4.31)
$\displaystyle\leq(|t-s|\lambda_{0})^{2}\|E([0,\lambda_{0}))T_{3}(s;\tau)f\|^{2}+C_{T_{2}}(s)^{2}\|E([\lambda_{0},\infty))T_{3}(s;\tau)f\|^{2}$

with the constant $C_{T_{2}}(s)$ from (4.29). The second term of the last
expression converges to zero as $\lambda_{0}\to\infty$ for any _fixed_
$\tau\in(0,1]$.

To make use of the estimate, however, we need to find a $\lambda_{0}$
independent of $\tau$ in order to obtain convergence _uniform_ with respect to
$0<\tau\leq 1$. This can be achieved, however, because for fixed
$f\in{\mathcal{H}}$ and $s\in{\mathbb{R}}\setminus(M\cup\\{0\\})$, the set

$V_{f}:=\\{T_{3}(s;\tau)f;\,\,0<\tau\leq 1\\}\subset{\mathcal{H}}$

is bounded by (4.20), and has a compact closure, as a consequence of the
strong continuity of $T_{3}(s;\tau)f$ with respect to the variable $\tau>0$.
Then for every $f\in{\mathcal{H}}$ and every $\varepsilon>0$ one can find an
$s$-dependent number $\lambda_{00}=\lambda_{00}(f;\varepsilon;s)>0$ such that

$\sup_{0<\tau\leq
1}\|E([\lambda_{00},\infty))T_{3}(s;\tau)f\|^{2}<\tfrac{1}{C_{T_{2}}(s)^{2}}\cdot\tfrac{\varepsilon^{2}}{2}$
(4.32)

with the constant $C_{T_{2}}(s)$ in (4.29). Let us be more explicit about the
last claim. Since $V_{f}$ is totally bounded, there is a finite family of
vectors $\\{y_{j}\\}_{j=1}^{N}\subset V_{f}$, $y_{j}=T_{3}(s;\tau_{j})f$ for
some $\tau_{j}\in(0,1]$, and an open ball
$B(0;\tfrac{\varepsilon}{\sqrt{8}C_{T_{2}}(s)})$ with the center at the origin
and radius $\tfrac{\varepsilon}{\sqrt{8}C_{T_{2}}(s)}$ for which we have
$V_{f}\,\,\subset\,\,\cup_{j=1}^{N}\big{(}y_{j}+B(0;\tfrac{\varepsilon}{\sqrt{8}C_{T_{2}}(s)})\big{)}.$
Using again the fact that $E([\lambda,\infty))\overset{s}{\to}0$ holds as
$\lambda\to\infty$ we infer that there is a family
$\\{\lambda_{j}\\}_{j=1}^{N}$ of large positive numbers such that
$C_{T_{2}}(s)\|E([\lambda_{j},\infty))y_{j}\|\leq\tfrac{\varepsilon}{\sqrt{8}}$
holds for $j=1,2,\dots,N$ and we put
$\lambda_{00}:=\max_{j=1,2,\dots,N}{\lambda_{j}}$. The finite union coverage
means that any $\phi\in V_{f}$ satisfies $\phi\in
y_{j_{\phi}}+B(0;\tfrac{\varepsilon}{\sqrt{8}C_{T_{2}}(s)})$ for some $1\leq
j_{\phi}\leq N$ with $0<\tau_{j_{\phi}}\leq 1$. Noting that
$\|E([\lambda_{00},\infty))\|\leq 1$, we have

$\displaystyle C_{T_{2}}(s)^{2}\|E([\lambda_{00},\infty))\phi\|^{2}$
$\displaystyle\leq
C_{T_{2}}(s)^{2}\big{(}\max_{j=1,2,\dots,N}\|E([\lambda_{00},\infty))y_{j}\|+\tfrac{\varepsilon}{\sqrt{8}C_{T_{2}}(s)}\big{)}^{2}$
$\displaystyle\leq
2C_{T_{2}}(s)^{2}\big{[}\max_{j=1,2,\dots,N}\|E([\lambda_{00},\infty))y_{j}\|^{2}+\big{(}\tfrac{\varepsilon}{\sqrt{8}C_{T_{2}}(s)})^{2}\big{]}$
$\displaystyle<\tfrac{\varepsilon^{2}}{4}+\tfrac{\varepsilon^{2}}{4}=\tfrac{\varepsilon^{2}}{2}.$

Returning to the first term on the right-hand side of (4.31) where take
$\lambda_{0}=\lambda_{00}$, and noting that $\|T_{3}(s;\tau)f\|\leq
C_{T_{3}}(s)\|f\|$ holds in view of (4.20), we see that there exists a number
$\delta=\delta(f;\varepsilon;s)>0$ such that for a fixed $s$ we have

$t\in{\mathbb{R}}\setminus\\{0\\}\;\text{\rm
with}\;|t-s|<\delta\;\Longrightarrow(|t-s|\lambda_{00})^{2}(C_{T_{3}}(s)\|f\|)^{2}<\tfrac{\varepsilon^{2}}{2},$

uniformly for all $\tau\in(0,1]$; one can be again explicit and choose

$\delta(f;\varepsilon;s):=\min\\{\tfrac{1}{\sqrt{2}\lambda_{00}(1+(C_{T_{3}}(s)\|f\|)^{2})^{1/2}}\,\varepsilon,\,\tfrac{|s|}{2}\\}\,.$

The estimate (4.31) then gives

$\|T_{2}(t,s;\tau)T_{3}(s;\tau)f\|^{2}<\tfrac{\varepsilon^{2}}{2}+\tfrac{\varepsilon^{2}}{2}=\varepsilon^{2},$

or in other words

$\|D(t,s;\tau)f\|\leq\|T_{2}(t,s;\tau)T_{3}(s;\tau)f\|<\varepsilon,$

which yields (4.25). This concludes the proof of the present lemma, and in
this way also of Lemma 3.6 as we have outlined in the opening of this section.
∎

##  5\. An example

Let us finally mention briefly a typical situation in which Zeno dynamics
occurs as indicated, for instance, in the paper [8], namely the perpetual
position ascertaining. We consider an open domain
$\Omega\subset\mathbb{R}^{d}$ with a smooth boundary, thought of as the
detector volume, and associate with it the orthogonal projection $P$ on
$L^{2}(\mathbb{R}^{d})$ acting as the multiplication operator by the indicator
function $\chi_{\Omega}$ of the set $\Omega$. Suppose that the dynamics of the
particle undisturbed by the measurement is free, that is, described by the
Hamiltonian $H=-\Delta$, i.e. the Laplacian in $\mathbb{R}^{d}$ which is a
nonnegative self-adjoint operator in $L^{2}(\mathbb{R}^{d})$. The assumption
of density of the domain of $H^{1/2}P=(-\Delta)^{1/2}\chi_{\Omega}$ is
satisfied, since it contains $C_{0}^{\infty}(\Omega)\cup
C_{0}^{\infty}({\mathbb{R}}^{d}\setminus\overline{\Omega})$, where
$\overline{\Omega}$ is the closure of $\Omega$, and this family of functions
is dense in $L^{2}({\mathbb{R}}^{d})$.

Consider further the Dirichlet Laplacian $-\Delta_{\Omega}$ in $L^{2}(\Omega)$
defined in the usual way [22, Thm XIII.15] as the Friedrichs extension of the
appropriate quadratic form. It can be checked [5, Prop. 6.1] that

$(-\Delta)_{P}:=((-\Delta)^{1/2}P)^{*}(-\Delta)^{1/2}P$

is densely defined and its restriction to the subspace $L^{2}(\Omega)$ is
nothing but the Dirichlet Laplacian $-\Delta_{\Omega}$ with the domain
$D[-\Delta_{\Omega}]=W_{0}^{1}(\Omega)\cap W^{2}(\Omega)$, and

$-\Delta_{\Omega}=(-\Delta)_{P}\restriction_{PL^{2}({\mathbb{R}}^{d})}=(-\Delta)_{P}\restriction_{L^{2}(\Omega)}\,.$

Then Theorem 1.1 says that

$\text{s\,-}\\!\\!\lim_{n\rightarrow\infty}(P\,\mathrm{e}^{-it(-\Delta/n)}P)^{n}=\mathrm{e}^{-it(-\Delta_{\Omega})}P$

holds in the strong operator topology of $\mathcal{B}(L^{2}(\mathbb{R}^{d}))$,
the Banach space of the bounded linear operators on $L^{2}(\mathbb{R}^{d})$,
in other words, that the perpetual reduction of the wave function forces the
particle to move within the region $\Omega$ as if its boundary was Dirichlet,
i.e. hard wall. This is, of course, the expected conclusion indicated, e.g. in
[7, 8], however, only Theorem 1.1 allows one to state such a claim with the
proper rigor.

### Acknowledgments

The research was supported by the Czech Science Foundation within the project
17-01706S and in part by Grant-in Aid for Scientific Research 16K05230, Japan
Society for the Promotion of Science, and by the EU project
CZ.02.1.01/0.0/0.0/16_019/0000778. The authors are grateful to Tsuyoshi Ando
for valuable discussions, to Hiroshi Tamura, Valentin Zagrebnov, and late
Hagen Neidhardt for a number of useful comments, and to Hideo Tamura for his
unceasing encouragement.

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