# Rational approximation of operator semigroups via the $\mathcal{B}$-calculus

Alexander Gomilko  Faculty of Mathematics and Computer Science
Nicolas Copernicus University
Chopin Str. 12/18
87-100 Toruń, Poland<EMAIL_ADDRESS>and  Yuri Tomilov  Institute of
Mathematics
Polish Academy of Sciences
Śniadeckich 8
00-656 Warsaw, Poland  Faculty of Mathematics and Computer Science
Nicolas Copernicus University
Chopin Str. 12/18
87-100 Toruń, Poland<EMAIL_ADDRESS>
###### Abstract.

We improve the classical results by Brenner and Thomée on rational
approximations of operator semigroups. In the setting of Hilbert spaces, we
introduce a finer regularity scale for initial data, provide sharper stability
estimates, and obtain optimal approximation rates. Moreover, we strengthen a
result due to Egert-Rozendaal on subdiagonal Padé approximations of operator
semigroups. Our approach is direct and based on the theory of the
$\mathcal{B}$\- functional calculus developed recently. On the way, we
elaborate a new and simple approach to construction of the
$\mathcal{B}$-calculus thus making the paper essentially self-contained.

###### Key words and phrases:

rational approximation, operator semigroup, functional calculus, Padé
approximation, Hilbert space

###### 2010 Mathematics Subject Classification:

Primary: 47A60, 41A20, 41A25; Secondary: 47D03, 30E10, 41A21, 65J08

##  1\. Introduction

The theory of rational approximation of $C_{0}$-semigroups on Banach spaces is
a classical chapter of semigroup theory with a multitude of important
applications, e.g. in numerical methods for the study of PDE. However, a
number of well-known results of the theory were obtained solely in the
framework of abstract Banach spaces, and the influence of geometrical
properties of the spaces was not exploited enough. One of the reasons was
apparently a lack of an appropriate functional calculus for semigroup
generators taking into account fine properties of the underlying spaces. In
this paper, using the theory of $\mathcal{B}$-calculus developed recently in
[3] and [4], we intend to fill this gap and to obtain Hilbert space versions
of several crucial results on rational approximation of semigroups.

Starting a very brief introduction to the subject, observe that if $-A$ is the
generator of a $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$ on a Banach space $X,$
then the Cauchy problem

(1.1) $x^{\prime}(t)=-Ax(t),\qquad x(0)=x\in{\rm dom}(A),$

where ${\rm dom}(A)$ stands for the domain of $A$, is well-posed. Many time-
discretisation methods of (1.1) lead to the problem of high accuracy
approximation of the propagator $(e^{-tA})_{t\geq 0}$ to (1.1) by rational
functions of $A,$ not relying on the specific structure of $A.$ Without loss
of generality, one may assume that $(e^{-tA})_{t\geq 0}$ is bounded, so in the
sequel we will deal only with bounded semigroups. Given a sequence of rational
functions $(r_{n})_{n\geq 1}$ holomorphic in the open right half-plane
$\mathbb{C}_{+},$ it is thus of substantial interest to characterise the
convergence $r_{n}(tA)x\to e^{-tA}x$ as $n\to\infty,$ $x\in X,$ in a priori
terms, and to equip this convergence with optimal approximation rates
depending on regularity of the initial data $x.$ A simple but a very natural
choice is provided by

$r_{n}(z)=r^{n}(z/n),\qquad n\geq 1,$

for a rational function $r,$ and it will be the main focus in this paper.

The study of such approximation problems even in the context of finite-
dimensional $X$ (see e.g. [23]) reveals a necessity to deal with
$\mathcal{A}$-stable rational functions and to quantify their approximation
properties. Recall that a rational function $r$ is said to be _$\mathcal{A}$
-stable_ if $r$ is holomorphic in $\mathbb{C}_{+}$ and
$\sup_{z\in\mathbb{C}_{+}}|r(z)|\leq 1,$ and $r$ is said to be an
approximation of order $q\in\mathbb{N}$ to the exponential function $e^{-z}$
if

(1.2) $e^{-z}-r(z)=\mbox{O}(|z|^{q+1}),\qquad z\to 0,$

or, in other words, $r^{(k)}(0)=(-1)^{k}$ for all $0\leq k\leq q.$ In the
following, we will often refer to $e^{-z}$ as _the exponential_.

If $r$ is an $\mathcal{A}$-stable rational approximation to the exponential of
certain order, then by the well-known Lax-Richtmyer-Chernoff theorem the
approximation property $\lim_{n\to\infty}r(tA/n)^{n}x=e^{-tA}x$ for all $x\in
X$ and $t\geq 0$ is equivalent to the stability condition
$\sup_{n\in\mathbb{N},t\geq 0}\|r(tA/n)^{n}\|<\infty,$ see e.g. [25, p. 437],
and also [13, Theorem 2.1]. (This also follows from Theorem 1.1 below.)
Moreover, $\mathcal{A}$-stable approximations $r$ of order $q$ allow one to
obtain approximation rates for $e^{-tA}x-r^{n}(tA/n)x$ as $n\to\infty$
depending on $q$, often at the price of considering the initial data $x$ with
additional regularity, in particular from ${\rm dom}(A^{s})$ with suitable
$s>0,$ and the stability condition plays a role here. Even when the condition
does not hold, it is often useful to quantify the growth of $\|r(tA/n)^{n}\|$
for fixed $t>0.$ A thorough discussion of these and related issues with a
number of pertinent references can be found in [25, Section 2], see also [12,
Section 1] and [13].

###  1.1. Rational approximation of semigroups: a glimpse at the state of the
art

Let ${\rm M}(\mathbb{R}_{+})$ be the Banach algebra of bounded Borel measures
on $\mathbb{R}_{+}=[0,\infty,)$ and let $\mathcal{LM}(\mathbb{C}_{+})$ be the
Banach algebra of Laplace transforms $\widehat{\nu}$ of $\nu\in{\rm
M}(\mathbb{R}_{+})$ with the norm
$\|\widehat{\nu}\|_{\mathcal{LM}(\mathbb{C}_{+})}:=\|\nu\|_{{\rm
M}(\mathbb{R}_{+})}.$ Note that if $r$ is an $\mathcal{A}$-stable rational
function, then $r_{n}\in\mathcal{LM}(\mathbb{C}_{+})$ for every
$n\in\mathbb{N}.$ If, moreover, $r$ is an $\mathcal{A}$-stable rational
approximation to $e^{-z}$ of order $q,$ and
$\Delta_{n,r,s}(z):=(e^{-z}-r_{n}(z))z^{-s},$ then
$\Delta_{n,r,s}\in\mathcal{LM}(\mathbb{C}_{+})$ for all $n\in\mathbb{N}$ and
integers $s$ such that $0\leq s\leq q+1.$ These observations (see [7, p. 685])
put rational approximations $(r_{n}(A))_{n\geq 1}$ of $(e^{-tA})_{t\geq 0}$
into the framework of the Hille-Phillips (HP-) functional calculus. Recall
that if $-A$ generates a bounded $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$ on a
Banach space $X,$ and $M:=\sup_{t\geq 0}\|e^{-tA}\|,$ then the Hille-Phillips
(HP-) calculus for $A$ is defined as a bounded homomorphism
$\Psi_{A}:\mathcal{LM}(\mathbb{C}_{+})\to L(X)$ given by the following
operator version of Laplace transform:

$\Psi_{A}(\widehat{\nu})x=\widehat{\nu}(A)x=\int_{0}^{\infty}e^{-tA}x\,d\nu(t),\qquad
x\in X,\quad\widehat{\nu}\in\mathcal{LM}(\mathbb{C}_{+}),$

so that $\|\hat{\nu}(A)\|\leq M\|\nu\|_{{\rm M}(\mathbb{R}_{+})}.$ More
details on the HP-calculus can be found e.g. in [20, Chapter 3.3]. The
approach to the study of rational approximations by means of the HP-calculus
was pioneered by Hersch and Kato in [24], and elaborated by Brenner and Thomee
in [7]. In particular, the next basic result in the theory of rational
approximation of semigroups was obtained in in [7, Theorem 1, Theorem 4 and
Remarks].

######  Theorem 1.1.

Let $-A$ be the generator of a $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$ on
Banach space $X$, and $M:=\sup_{t\geq 0}\|e^{-tA}\|<\infty.$

  * (i)

If $r$ is an $\mathcal{A}$-stable rational function, then there exists
$C=C(r)$ such that for all $n\in\mathbb{N}$ and $t\geq 0,$

(1.3) $\|r^{n}(tA/n)\|\leq CMn^{1/2}.$

  * (ii)

If $r$ is an $\mathcal{A}$-stable rational approximation of order
$q\in\mathbb{N}$ to the exponential, and $s\in\mathbb{N}$ satisfies
$s\in(0,q+1],s\neq(q+1)/2,$ then there exists $C=C(s,r)>0$ such that

(1.4) $\displaystyle\|e^{-tA}x-r^{n}(tA/n)x\|\leq$ $\displaystyle
CM\frac{t^{s}}{n^{s-1/2}}\|A^{s}x\|,\quad s\in(0,(q+1)/2),$ (1.5)
$\displaystyle\|e^{-tA}x-r^{n}(tA/n)x\|\leq$ $\displaystyle
CM\frac{t^{s}}{n^{sq/(q+1)}}\|A^{s}x\|,\quad s\in((q+1)/2,q+1],$

for all $x\in{\rm dom}\,(A^{s}),t\geq 0,$ and $n\in\mathbb{N}.$ If, under the
assumptions above, $s=(q+1)/2,$ then there is $C=C(s,r)>0$ such that

(1.6) $\|e^{-tA}x-r^{n}(tA/n)x\|\leq
CM\frac{t^{s}\log(n+1)}{n^{q/2}}\|A^{s}x\|,$

for all $x\in{\rm dom}\,(A^{s}),t\geq 0,$ and $n\in\mathbb{N}.$

The estimates in Theorem 1.1, are, in general, optimal, as the examples of
shift semigroups on $L^{1}(\mathbb{R})$ (or on $C_{0}(\mathbb{R})$) show. The
sharpness of stability estimate (1.3) is discussed already in [7, p. 687], see
also [8, Section 2], [27, Theorem 2.2] and [26, Theorem 3.3] for more explicit
arguments. The optimality of (1.4), (1.5) and (1.6) is elaborated in [27,
Section 5], see also [7, Remarks, p. 691].

It is natural to try to use interpolation and to extend the estimates in
Theorem 1.1 to the whole range of $s\in(0,q+1].$ This idea was realised in
[27], where for $s>0$ the Favard interpolation spaces $\mathcal{F}_{s}$
containing ${\rm dom}\,(A^{s})$ were introduced, and other, related
interpolation spaces were studied. However, the estimates obtained in [27] do
not seem to yield Theorem 1.1 with real $s$. It looks plausible that one may
just repeat the arguments in [7, Theorems 1 and 4] for any fixed
$s\in(0,q+1],$ and, using the product rule for the HP-calculus (as in the
proof of Theorem 1.3, (ii) below), obtain Theorem 1.1 as formulated above with
integer $s$ replaced by $s\in(0,q+1].$ A justification of this claim goes
beyond the scope of the present paper, and thus is omitted.

Apart from fundamental use of the HP-calculus, the arguments in [24] relied on
Fourier multiplier estimates for $L^{1}(\mathbb{R}_{+})$ provided by Carlson’s
inequality. A refinement of this technique in [7] based on appropriate
partitions of unity and local application of Carlson’s inequality led to
Theorem 1.1. These techniques with variations and adjustments were used in
virtually all subsequent papers dealing with rational approximations of (in
general, non-holomorphic) $C_{0}$-semigroups.

Comparatively recently, there appeared other approaches to semigroup rational
approximation relying on Padé approximations to the exponential. As emphasized
already in [24] the examples of rational functions satisfying the assumptions
of Theorem 1.1 can be constructed using Padé approximations. Toy examples are
provided by, for instance, Euler’s and Crank-Nicolson’s schemes corresponding
to $r(z)=(1+z)^{-1}$ and $r(z)=(1-z/2)(1+z/2)^{-1}.$ Because of their strong
approximation properties and explicit form, Padé approximations play a
distinguished role in the theory of rational approximation. Recall that if
$r=P/Q,$ where $P$ and $Q$ are polynomials with ${\rm deg}\,P=m$ and ${\rm
deg}\,Q=n,$ then $r$ satisfies (1.2) with order $q$ not exceeding $m+n,$ and
Padé approximations satisfy (1.2) with the maximal order $m+n.$ Normalising
$Q(0)=1,$ the Padé approximations $r_{[m,n]}=P_{m}/Q_{n}$ of order $m+n$ to
the exponential are unique and can be written explicitly. So they can be
thought of as an infinite $(m,n)$-table. By a famous result proved in [10] and
[35], see e.g. [23, Theorem 4.12], the approximations $r_{[m,n]}$ are
$\mathcal{A}$-stable if and only if $m\leq n\leq m+2.$ Thus, the diagonal, the
first subdiagonal, and the second subdiagonal Padé approximations
$r_{[n,n]},r_{[n,n+1]}$ and $r_{[n,n+2]},$ $n\in\mathbb{N},$ are of primary
interest from the point of view of numerical methods.

In the context of Theorem 1.1, it was observed in [28] that when approximating
the exponential it is computationally advantageous to replace the sequence
$(r(\cdot/n)^{n})_{n\geq 1}$ by the sequence $(r_{[n,n+1]})_{n\geq 1}$ of
subdiagonal Padé approximations, so that each $r_{[n,n+1]}$ is a linear
combination of simple fractions, and thus $r_{[n,n+1]}(A)$ is a linear
combination of resolvents. However, the convergence of $r_{[n,n+1]}(tA)x$ to
$e^{-tA}x$ as $n\to\infty$ for, at least, regular enough data $x$ and all
$t\geq 0$ was left as an open problem in [28], and it was explored very
thoroughly in [9] in the spirit of considerations of [7]. The key results from
[9] can be summarized as follows.

######  Theorem 1.2.

Let $-A$ generate a $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$ on a Banach space
$X,$ and $M:=\sup_{t\geq 0}\|e^{-tA}\|<\infty.$ Let
$(r_{[n,n+1]})_{n\in\mathbb{N}}$ be the first subdiagonal Padé approximations
to the exponential, and let $s>0$ and $x\in{\rm dom}\,(A^{s})$ be given.

  * (i)

If $s>1/2,$ then there exists $C=C(s)>0$ such that

$\|r_{[n,n+1]}(tA)x-e^{-tA}x\|\leq CM\frac{t^{s}}{n^{s-1/2}}\|A^{s}x\|$

for all $t\geq 0,$ and $n\in\mathbb{N}$ with $n\geq s-\frac{1}{2}$.

  * (ii)

If $X$ is a Hilbert space, $a\in(0,s),$ and $(e^{-tA})_{t\geq 0}$ satisfies
$\|e^{-tA}\|\leq Me^{-\omega t}$ for some $M\geq 1$ and $\omega>0,$ then there
is $C=C(M,\omega,s-a)$ such that

$\displaystyle\left\|r_{[n,n+1]}(tA)x-e^{-tA}x\right\|\leq
C\left(\frac{t}{n}\right)^{a}\|A^{s}x\|$

for all $t\geq 0$ and $n\in\mathbb{N}$ such that $n\geq\frac{a}{2}-1$.

  * (iii)

If $A$ admits a bounded ${\rm H}^{\infty}$-calculus on $\mathbb{C}_{+}$ with
${\rm H}^{\infty}$-bound $C$, then

$\displaystyle\left\|r_{[n,n+1]}(tA)x-e^{-tA}x\right\|\leq
2C\left(\frac{t}{n}\right)^{s}\|A^{s}x\|$

for all $t\geq 0$ and $n\in\mathbb{N}$ such that $n>\frac{s}{2}-1.$

As in the case of Theorem 1.1, the proof of (i) was based on Fourier
multiplier estimates and Carlson’s inequality, while the proof of (ii) relied
on an advanced transference technique developed in [22] and [20]. Note that
the assumption of exponential stability in (ii) is indispensable for the
methods of [9]. The assumption of boundedness of the ${\rm
H}^{\infty}$-calculus in (iii) allows one to employ direct estimates of
rational functions in the ${\rm H}^{\infty}$-norm. It is difficult to express
it in abstract terms, especially if $A$ is far from being sectorial. For
$L^{p}$ (and more general) spaces $X$ and $C_{0}$-groups
$(e^{-tA})_{t\in\mathbb{R}}$ on $X,$ decaying exponentially as $t\to\infty,$
the stability properties and approximation rates in Theorems 1.1 and 1.2 were
improved in [31]. However, the improvements are still weaker than our Hilbert
space results formulated in the next section.

###  1.2. Rational approximation via the $\mathcal{B}$-calculus: the results

Recently, the HP-calculus was used in [18] and [19] to unify and extend a
number of basic approximation formulas in semigroup theory and to equip them
with optimal approximation rates depending on smoothness of initial data. The
formulas were recast in terms of completely monotone and Bernstein functions,
and thus the arguments relied essentially on the study of Laplace transforms
of positive measures. The positivity of measures led to results which are
optimal even in the Hilbert space setting.

While the approach in [18] and [19] encompassed some rational functions, e.g.
$(1+z)^{-1}$ originating from Euler’s formula, the studies of rational
approximations for semigroups often require to deal with Laplace transforms
of, in general, non-positive measures. Obtaining sharp estimates in this case
is problematic, and the optimal bounds require a finer functional calculus.
The theory of such a calculus, called the $\mathcal{B}$-calculus in [3], was
created recently in [3] and [4] in the setting of bounded $C_{0}$-semigroups
on Hilbert spaces. Being based on a Banach algebra
$\mathcal{B}\supset\mathcal{LM}(\mathbb{C}_{+}),$ the $\mathcal{B}$-calculus
strictly extends the HP-calculus, and leads to, in general, sharper operator-
norm estimates than the ones provided by the HP-calculus. For more details on
the $\mathcal{B}$-calculus see Section 3.

In the present article, using the $\mathcal{B}$-calculus, we substantially
improve Theorems 1.1 and 1.2 if $X$ is a Hilbert space. Our first result reads
as follows.

######  Theorem 1.3.

Let $-A$ be the generator of a $C_{0}$-semigroup $(e^{-tA})_{t\geq 0},$ on a
Hilbert space $X,$ and $M:=\sup_{t\geq 0}\|e^{-tA}\|<\infty.$

  * (i)

If $r$ is an ${\mathcal{A}}$-stable rational function, then there exists
$C=C(r)$ such that for all $n\in\mathbb{N}$ and $t\geq 0,$

(1.7) $\|r^{n}(tA/n)\|\leq C\pi M^{2}(1+\log(n)).$

  * (ii)

If $r$ is an $\mathcal{A}$-stable rational approximation of order $q$ to the
exponential and $s\in(0,q+1],$ then there exists $C=C(s,r)>0$ such that

(1.8) $\|e^{-tA}x-r^{n}(tA/n)x\|\leq C\pi
M^{2}\frac{t^{s}}{n^{sq/(q+1)}}\|A^{s}x\|$

for all $x\in{\rm dom}\,(A^{s}),$ $n\in\mathbb{N},$ and $t\geq 0.$

The estimates given in (1.8) are optimal with respect to approximation rates
as we show in Theorem 3.8 below. The optimality of (1.7) is not clear, and it
is related to the long-standing open problem on power boundedness for Cayley’s
transform of $A,$ see e.g. [17] and Section 3 for more on that.

If a rational function $r$ is $\mathcal{A}$-stable, then the limit

$r(\infty):=\lim_{{\rm Re}\,z\to\infty}r(z),$

exists and $|r(\infty)|\leq 1.$ The stability part of Theorem 1.3 can be
strengthened if one assumes that $r$ is small at infinity in the sense that
$|r(\infty)|<1.$ The latter smallness condition, called sometimes strong
$\mathcal{A}$-stability or $L$-stability in the literature, is part of
condition (*) in [7, p. 687]. In the situation of Hilbert spaces it suffices
to employ this smallness condition alone. As we show below the condition
ensures convergence of rational approximations on the whole of $X,$ the
property appearing usually in the study of rational approximations for
_holomorphic_ semigroups.

######  Theorem 1.4.

Let $A$ be as in Theorem 1.3, and let $r$ be an ${\mathcal{A}}$-stable
rational function satisfying

(1.9) $|r(\infty)|<1.$

Then there exists $C=C(r)$ such that

(1.10) $\|r^{n}(tA/n)\|\leq C\pi M^{2},\qquad n\in\mathbb{N},\quad t\geq 0.$

As a consequence, if $r$ is an $\mathcal{A}$-stable rational approximation of
any order to the exponential, and (1.9) holds, then

(1.11) $\lim_{n\to\infty}r^{n}(tA/n)x=e^{-tA}x,\qquad x\in X,\quad t\geq 0.$

Observe that the first and the second subdiagonal Padé approximations to the
exponential provide natural examples of $r$ satisfying the assumptions of
Theorem 1.4.

Using the power of the $\mathcal{B}$-calculus, we also substantially
strengthen Theorem 1.2 and obtain an estimate similar to Theorem 1.2,(iii) for
_arbitrary_ bounded Hilbert space $C_{0}$-semigroups, however modulo a
logarithmic correction term. Namely, the next statement holds.

######  Theorem 1.5.

Let $-A$ be the generator of a $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$ on a
Hilbert space $X$, and let $M:=\sup_{t\geq 0}\|e^{-tA}\|<\infty.$ If
$(r_{[n,n+1]})_{n\in\mathbb{N}}$ are the first subdiagonal Padé approximations
to the exponential, then for every $s>0$ there exists $C=C(s)>0$ such that

(1.12) $\|e^{-tA}x-r_{[n,n+1]}(tA)x\|\leq\frac{\pi
M^{2}t^{s}}{n^{s}}\left(C+4\log(2n+1)\right)\|A^{s}x\|$

for all $x\in{\rm dom}\,(A^{s}),t\geq 0,$ and $n\in\mathbb{N}$ such that
$n\geq(s-1)/2.$

It would be of interest to clarify whether the logarithmic correction term in
the right hand side of (1.12) can be omitted. Recalling that the optimality of
the logarithmic bound in (1.7) has been open for a long time, this problem
does not look surprising. It is also related to another, apparently open,
problem of whether $r_{[n,n+1]}(tA)\to e^{-tA}$ as $n\to\infty$ strongly for
every $t\geq 0$ or, equivalently, whether
$\sup_{n\in\mathbb{N}}\|r_{[n,n+1]}(tA)\|<\infty.$

Finishing this section, we fix some relevant notations and conventions. For a
Hilbert space $X,$ we denote by $\langle\cdot,\cdot\rangle$ the inner product
of $X$, and we let $L(X)$ stand for the Banach algebra of all bounded linear
operators on $X.$ The domain and the spectrum of a linear operator $A$ on $X$
are denoted by ${\rm dom}(A)$ and $\sigma(A),$ respectively. If $A$ a densely
defined linear operator $A$ on $X,$ then $A^{*}$ denotes the Hilbert space
adjoint of $A.$

Denote by ${\rm Hol}(\Omega)$ the space of functions holomorphic on a domain
$\Omega\subset\mathbb{C}.$ For any function $f:\mathbb{C}_{+}\to\mathbb{C}$ we
write $f(\infty):=\lim_{{\rm Re}\,z\to\infty}f(z)$ if the latter limit exists
in $\mathbb{C}.$

For a subset $S$ of the complex plane $\mathbb{C}$ we denote by $\partial S$
the topological boundary of $S$, and by $\overline{S}$ the closure of $S.$ The
open right half-plane $\\{z\in\mathbb{C}:{\rm Re}\,z>0\\}$ is denoted by
$\mathbb{C}_{+}.$

Writing $C=C(r)$ for a rational function $r$ we will mean that $C$ depends on
$r$ and other parameters associated exclusively with $r,$ such as order
approximation $q$ or its domain of holomorphicity. If $C$ will depend on other
parameters, then they will be mentioned explicitly. Nevertheless, sometimes we
emphasize the dependence of $C$ on certain parameters associated to given $r,$
and then indicate the corresponding relations.

##  2\. Function-theoretical estimates

It was shown in [3, Sections 2.2 and 2.4], see also [34, Section 1.5], that if
$\mathcal{B}$ is the space of functions $f$ holomorphic in $\mathbb{C}_{+}$
and satisfying

(2.1)
$\|f\|_{\mathcal{B}_{0}}:=\int_{0}^{\infty}\sup_{y\in\mathbb{R}}|f^{\prime}(x+iy)|\,dx<\infty,$

then $\mathcal{LM}(\mathbb{C}_{+})\subset\mathcal{B}\subset{\rm
H}^{\infty}(\mathbb{C}_{+}),$ where ${\rm H}^{\infty}(\mathbb{C}_{+})$ is the
Banach algebra of bounded holomorphic functions on $\mathbb{C}_{+}.$ Moreover,
$\mathcal{B}$ equipped with the norm $\|f\|_{\mathcal{B}}:=\|f\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}+\|f\|_{\mathcal{B}_{0}},f\in\mathcal{B}$, is a
Banach algebra, and if $\widehat{\nu}\in\mathcal{LM}(\mathbb{C}_{+}),$ then
$\|\widehat{\nu}\|_{\mathcal{B}}\leq\|\widehat{\nu}\|_{\mathcal{LM}(\mathbb{C}_{+})}.$
For every $f\in\mathcal{B}$ the limit $f(\infty)=\lim_{{\rm
Re}\,z\to\infty}f(z)$ exists, $f(\infty)\in\mathbb{C},$ and
$\mathcal{B}_{0}:=\\{f\in\mathcal{B}:f(\infty)=0\\}$ is the closed subalgebra
of $\mathcal{B}.$ Note that (2.1) defines a seminorm on $\mathcal{B},$ and
since by [3, Proposition 2.2, (2)]

$\|f\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}\leq|f(\infty)|+\|f\|_{\mathcal{B}_{0}},\qquad
f\in\mathcal{B},$

(2.1) defines a norm on $\mathcal{B}_{0}$ equivalent to the norm on
$\mathcal{B}.$

Given $s>0$ and an appropriate rational function $r$ or the first subdiagonal
Padé approximations $(r_{[n,n+1]})_{n\geq 1}$ to the exponential, we obtain
sharp $\mathcal{B}$-norm estimates for sequences of the form
$(r(z/n)^{n})_{n\geq 1},$ $((e^{-z}-r^{n}(z/n))z^{-s})_{n\geq 1}$ and
$((e^{-z}-r_{[n,n+1]}(z))z^{-s})_{n\geq 1},$ arising in the functional calculi
approach to Theorems 1.3, 1.4 and 1.5. Using the $\mathcal{B}$-calculus, these
estimates will then be transferred to the corresponding operator norm-
estimates in (1.7), (1.8), (1.10) and (1.12). Note that if $f\in\mathcal{B},$
$a>0,$ and $f_{a}(z):=f(az),\,z\in\mathbb{C}_{+},$ then $f_{a}\in\mathcal{B}$
and

(2.2)
$\|f_{a}\|_{\mathcal{B}_{0}}=\|f\|_{\mathcal{B}_{0}},\qquad\|f_{a}\|_{\mathcal{B}}=\|f\|_{\mathcal{B}}.$

This very simple invariance property of the $\mathcal{B}$-norm will be useful
in our studies.

###  2.1. Stability estimates in terms of $\mathcal{B}$-norms

In this section, given an $\mathcal{A}$-stable rational function $r,$ we will
study the $\mathcal{B}$-norm estimates for $r^{n}$ and related functions
mentioned above. Since $|r(\infty)|\leq 1$ and $r_{[n,n+1]}(\infty)=0$ it will
suffice to concentrate on estimates of their $\mathcal{B}_{0}$-(semi)norms.
The next elementary lemma will be crucial.

######  Lemma 2.1.

If a rational function $r$ is $\mathcal{A}$-stable, different from constant,
and satisfies $|r(it)|=1$ for every $t\in\mathbb{R},$ then $r$ is a finite
Blaschke product:

(2.3) $r(z)=a\prod_{j=1}^{m}\frac{z_{j}-z}{\overline{z}_{j}+z},$

for some $\\{z_{j}:1\leq j\leq m\\}\subset\mathbb{C}_{+}$ and
$a\in\mathbb{C},|a|=1.$

###### Proof.

Indeed, let $|r|=1$ on $i\mathbb{R},$ and $r=P/Q,$ where $P$ and $Q$ are
polynomials. Then ${\rm deg}\,P={\rm deg}\,Q,$ and therefore
$\lim_{|z|\to\infty}|r(z)|=1.$ Denote by $B_{m}$ the Blaschke product in
$\mathbb{C}_{+}$ whose zeros are the same as the zeros of $r$ in
$\mathbb{C}_{+},$ counting multiplicities. Observe that both, $r/B_{m}$ and
$B_{m}/r,$ are holomorphic and bounded in $\mathbb{C}_{+},$ and have moduli
identically equal to $1$ on $i\mathbb{R}.$ From the maximum principle it
follows that $r/B_{m}$ is a constant in $\mathbb{C}_{+}$ (and then in
$\mathbb{C}$) of modulus $1$. ∎

The direct argument above can essentially be found e.g. in [16, Chapter 1.2,
p. 6], and we gave it here in view of its importance for the sequel.

Thus, if a rational function $r$ is $\mathcal{A}$-stable and different from a
constant, then either $|r|\not\equiv 1$ on $i\mathbb{R},$ or, otherwise $r$ is
a finite Blaschke product. We will split the study of powers of $r$ into
considering these two cases. First, we consider the situation when $r$ is a
Blaschke product. This assumption will be replaced by a more general
assumption on $r$ to have at least one zero in $\mathbb{C}_{+}.$ The arguments
are essentially the same under both assumptions, while the technical details
become somewhat clearer in a more general set-up. We will need a family of
auxiliary functions. For $n\in\mathbb{N}$ and $s\geq 0,$ define

$F_{n,s}(x,t):=\frac{((x-1)^{2}+t)^{n-1}}{((x+1)^{2}+t)^{n+1+s}},\qquad
x,t>0,$

and denote

$a_{n,s}:=\frac{(2n+s)-2\sqrt{(n+1+s)(n-1)}}{(2+s)},$

and

$b_{n,s}:=a_{n,s}^{-1}=\frac{(2n+s)+2\sqrt{(n+1+s)(n-1)}}{(2+s)}.$

Observe that

(2.4) $1\leq\frac{2n+s}{2+s}\leq b_{n,s}\leq\frac{2(2n+s)}{2+s}\leq 2n.$

The next elementary technical lemma will simplify our estimates of powers of
rational functions having at least one zero in $\mathbb{C}_{+}.$

######  Lemma 2.2.

For $n\in\mathbb{N}$ and $s\geq 0,$ let

$G_{n,s}(x):=\sup_{t>0}\,F_{n,s}(x,t),\qquad x>0.$

Then

$G_{n,s}(x)=\begin{cases}\frac{(x-1)^{2(n-1)}}{(x+1)^{2(n+1+s)}},&\qquad
x\not\in[a_{n,s},b_{n,s}],\\\
\frac{(n-1)^{n-1}}{(n+1+s)^{n+1+s}}\cdot\frac{(2+s)^{2+s}}{4^{2+s}x^{2+s}},&\qquad
x\in[a_{n,s},b_{n,s}].\end{cases}$

for all $n\in\mathbb{N}$ and $s\geq 0.$

###### Proof.

If $x>0$ and $n\in\mathbb{N}$ are fixed, then

$\frac{((x+1)^{2}+t)^{n+2+s}}{(x-1)^{2}+t)^{n-2}}\frac{d}{dt}F_{n,s}(x,t)=-(2+s)t+w_{n,s}(x),$

where

$w_{n,s}(x)=-(2+s)x^{2}+2(2n+s)x-(2+s).$

Since the roots of $w_{n,s}(x)=0$ are precisely $a_{n,s}$ and $b_{n,s}$, we
have $w_{n,s}(x)>0$ if and only if $x\in(a_{n,s},b_{n,s}).$ So, if
$x\not\in[a_{n,s},b_{n,s}]$, then

$G_{n,s}(x)=F_{n,s}(x,0)=\frac{(x-1)^{2(n-1)}}{(x+1)^{2(n+1+s)}}.$

On the other hand, if $x\in[a_{n,s},b_{n,s}]$ and
$t_{n,s}(x)={w_{n,s}(x)}(2+s)^{-1},$ then

$G_{n,s}(x)=F_{n,s}(x,t_{n,s}(x))=\frac{(n-1)^{n-1}}{(n+1+s)^{n+1+s}}\cdot\frac{(2+s)^{2+s}}{4^{2+s}x^{2+s}}.\qed$

Recall that if $v$ is the Cayley transform, i.e. $v(z)=\frac{1-z}{1+z},$ then
by [3, Lemma 3.7] (and its proof),

(2.5) $\|v^{n}\|_{\mathcal{B}_{0}}\leq 2+2\log(2n),\qquad n\in\mathbb{N}.$

The logarithmic bound in (2.5) is optimal by [4, Lemma 5.1]. For a rational
function $r$ with at least one zero in $\mathbb{C}_{+},$ our estimates will be
reduced to the particular case when $r=v.$ For $s>0$ let

(2.6) $\eta_{s}(z):=(1+z)^{-s},\qquad z\in\mathbb{C}_{+}.$

######  Lemma 2.3.

Let $r$ be an $\mathcal{A}$-stable rational function on $\mathbb{C}_{+}$
satisfying $r(\lambda)=0$ for some $\lambda\in\mathbb{C}_{+},$ and let $s>0.$
Then there exist $C=C(r)>0$ and $C_{1}=C_{1}(r,s)>0$ such that

(2.7) $\|r^{n}\|_{\mathcal{B}_{0}}\leq C(1+\log(n))$

and

(2.8) $\|r^{n}\eta_{s}\|_{\mathcal{B}_{0}}\leq C_{1}(r,s)$

for all $n\in\mathbb{N}.$

###### Proof.

Let $n\in\mathbb{N}$ be fixed. First observe that since $r(\lambda)=0,$ the
$\mathcal{A}$-stability of $r$ and the maximum principle imply that

(2.9) $r(z)=\left(\frac{\lambda-z}{\overline{\lambda}+z}\right)r_{0}(z),\qquad
z\in\mathbb{C}_{+},$

where a rational function $r_{0}$ satisfy $\|r_{0}\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}\leq 1.$ Clearly, $r_{0}=r(\infty)+P/Q,$ where $P$
and $Q$ are polynomials with ${\rm deg}\,P<{\rm deg}\,Q,$ and
$r(\infty)=\lim_{z\to\infty}r(z).$ So there exists $C=C(r_{0})>0$ such that

(2.10) $|r^{\prime}_{0}(z)|\leq\frac{C}{|\overline{\lambda}+z|^{2}},\qquad
z\in\mathbb{C}_{+}.$

Hence, using (2.9), (2.10) and $\|r_{0}\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}\leq 1,$

(2.11) $|(r^{n}(z))^{\prime}|\leq
n\frac{|\lambda-z|^{n-1}}{|\overline{\lambda}+z|^{n-1}}\frac{C+2{\rm
Re}\lambda}{|\overline{\lambda}+z|^{2}},\qquad z\in\mathbb{C}_{+}.$

Denoting $C_{\lambda}=C+2{\rm Re}\,\lambda,$ $\lambda=a+ib$, and $z=x+iy,$ we
infer from (2.11) that

(2.12) $\displaystyle\|r^{n}\|_{\mathcal{B}_{0}}\leq$ $\displaystyle
nC_{\lambda}\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{((x-a)^{2}+(y+b)^{2})^{(n-1)/2}}{((x+a)^{2}+(y+b)^{2})^{(n+1)/2}}\,dx$
$\displaystyle=$ $\displaystyle
nC_{\lambda}\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{((x-1)^{2}+y^{2})^{(n-1)/2}}{((x+1)^{2}+y^{2})^{(n+1)/2}}\,dx$
$\displaystyle=$
$\displaystyle\frac{C_{\lambda}}{2}\,\|v^{n}\|_{\mathcal{B}_{0}}.$

Hence (2.12) and (2.5) yield (2.7).

Next, let $s>0$ be fixed. By (2.11) and the $\mathcal{A}$-stability of $r$,

(2.13) $\displaystyle|(r^{n}(z)\eta_{s}(z))^{\prime}|\leq$
$\displaystyle\frac{|(r^{n}(z))^{\prime}|}{|1+z|^{s}}+s\frac{|r^{n}(z)|}{|1+z|^{s+1}}$
$\displaystyle\leq$ $\displaystyle
n\frac{|\lambda-z|^{n-1}}{|\overline{\lambda}+z|^{n-1}}\frac{C_{\lambda}}{|\overline{\lambda}+z|^{2+s}}+\frac{s}{|1+z|^{s+1}}.$

Since

(2.14)
$\int_{0}^{\infty}\sup_{y>0}\,\frac{s}{((x+1)^{2}+y^{2})^{(s+1)/2}}\,dx=\int_{0}^{\infty}\frac{s\,dx}{(1+x)^{s+1}}=1,$

we have

$\|r^{n}\eta_{s}\|_{\mathcal{B}_{0}}\leq C_{\lambda}I_{n,s}+1,$

where, in view of Lemma 2.2,

$\displaystyle I_{n,s}=$ $\displaystyle
n\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{((x-a)^{2}+(y+b)^{2})^{(n-1)/2}}{((x+a)^{2}+(y+b)^{2})^{(n+1+s)/2}}\,dx$
$\displaystyle=$ $\displaystyle
n\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{((x-a)^{2}+y^{2})^{(n-1)/2}}{((x+a)^{2}+y^{2})^{(n+1+s)/2}}\,dx$
$\displaystyle=$ $\displaystyle n\int_{0}^{\infty}(G_{n,s}(x))^{1/2}\,dx,$

and $G_{n,s}$ is given by Lemma 2.2.

Thus,

$\displaystyle I_{n,s}=$ $\displaystyle
n\int_{0}^{a_{n,s}}\frac{(1-x)^{n-1}}{(x+1)^{n+1}}\,dx+n\int_{b_{n,s}}^{\infty}\frac{(x-1)^{n-1}}{(x+1)^{n+1}}\,dx$
$\displaystyle+$ $\displaystyle
n\frac{(n-1))^{(n-1)/2}}{(n+1+s)^{(n+1+s)/2}}\frac{(2+s)^{1+s/2}}{2^{2+s}}\int_{a_{n,s}}^{b_{n,s}}\frac{dx}{x^{1+s/2}}.$

Recalling that $a_{n,s}=b^{-1}_{n,s},$ observe that

$n\int_{0}^{a_{n,s}}\frac{(1-x)^{n-1}}{(x+1)^{n+1}}\,dx+n\int_{b_{n,s}}^{\infty}\frac{(x-1)^{n-1}}{(x+1)^{n+1}}\,dx=2\int_{b_{n,s}}^{\infty}d\left(\frac{(x-1)^{n}}{(x+1)^{n}}\right)\leq
2.$

Therefore, taking into account (2.4),

$\displaystyle I_{n,s}\leq
2+\frac{2}{n^{s/2}}\frac{(2+s)^{1+s/2}}{2^{2+s}s}b_{n,s}^{s/2}\leq
2+\frac{(1+s/2)^{s/2+1}}{s},$

and then

(2.15)
$\|r^{n}\eta_{s}\|_{\mathcal{B}_{0}}\leq\left(2+\frac{(1+s/2)^{s/2+1}}{s}\right)C_{\lambda}+1.$

Thus, (2.8) holds with $C(r,s)$ given by the right-hand side of (2.15). ∎

Now we turn to the case when $|r|$ is not identically $1$ on $i\mathbb{R}$
(but $r$ may not have zeros in $\mathbb{C}_{+}$). As above, we will first need
a technical lemma facilitating our estimates of of $r^{n},n\in\mathbb{N},$
under this assumption.

For $a>0,$ $s\geq 0,$ and $n\in\mathbb{N}$ define

$S_{n,s}(x,t):=\frac{e^{-anx/(x^{2}+t)}}{(x^{2}+t)^{1+s/2}},\qquad x,t>0.$

######  Lemma 2.4.

For all $n\in\mathbb{N},$ $a,b>0,$ and $s\geq 0$ there are $C=C(a,b,s)>0$ such
that

(2.16)
$\int_{b/n}^{\infty}\sup_{t>0}\,S_{n,0}(x,t)\,dx\leq\begin{cases}C\frac{\log(n+1)}{n},&\,\,\text{if}\,\,s=0,\\\
\frac{C}{n},&\,\,\text{if}\,\,s>0.\end{cases}$

###### Proof.

Let $n\in\mathbb{N},$ $s\geq 0,$ and $x>0$ be fixed. Then for all $t>0,$

$e^{-anx/(x^{2}+t)}(x^{2}+t)^{3+s/2}\frac{d}{dt}S_{n,s}(x,t)=anx-(1+s/2)(x^{2}+t),$

and $S^{{}^{\prime}}_{n,s}(x,\cdot)$ has the unique zero at
$t_{0}=a_{s}nx-x^{2},$ where we denoted $a_{s}=\frac{a}{1+s/2}$ for shorthand.
Therefore,

$\sup_{t>0}\,S_{n,s}(x,t)=S_{n,s}(x,t_{0})=\frac{e^{-(2+s)/2}}{(a_{s}nx)^{1+s/2}},\qquad\text{if}\quad
x\in(0,a_{s}n),$

and

$\sup_{t>0}\,S_{n,s}(x,t)=S_{n,s}(x,0)=\frac{e^{-a}}{x^{2+s}},\qquad\text{if}\quad
x\in(a_{s}n,\infty).$

Then for large enough $n,$ we have

$\int_{b/n}^{\infty}\sup_{t>0}\,S_{n,0}(x,t)\,dx\leq\frac{1}{a_{s}n}\int_{b/n}^{an}\frac{dx}{x}+\int_{an}^{\infty}\frac{dx}{x^{2}}\leq
C\frac{\log(n+1)}{n}$

and, if $s>0$,

$\int_{b/n}^{\infty}\sup_{t>0}\,S_{n,s}(x,t)\,dx\leq\frac{1}{(a_{s}n)^{1+s/2}}\int_{b/n}^{a_{s}n}\frac{dx}{x^{1+s/2}}+\int_{a_{s}n}^{\infty}\frac{dx}{x^{2+s}}\leq\frac{C}{n},$

for appropriate constants $C=C(a,b,s)>0.$ This finishes the proof. ∎

The next lemma is a counterpart of Lemma 2.3 under complementary assumptions
on $r$. For $R>0$ denote

$\mathbb{D}_{R}^{+}:=\\{z\in\mathbb{C}_{+}:\,|z|<R\\}.$

######  Lemma 2.5.

Let $r$ be an $\mathcal{A}$-stable rational function satisfying $|r|\not\equiv
1$ on $i\mathbb{R}.$ Then there exist $C=C(r)>0$ and $C_{1}=C_{1}(r,s)$ such
that (2.7) and (2.8) hold for all $n\in\mathbb{N}$ and $s>0.$

###### Proof.

Since $|r|$ is not identically $1$ on $i\mathbb{R},$ the set $\\{z\in
i\mathbb{R}:|r(z)|=1\\}$ is at most finite. Therefore, taking into account
that $r$ is $\mathcal{A}$-stable, there exists $R>0$ such that $|r(\pm
iR)|<1.$ Since, by the maximum principle, $|r(z)|<1$ for all
$z\in\mathbb{C}_{+},$ we have

$\sup_{z\in\mathbb{C}_{+},|z|=R}|r(z)|=:\omega<1.$

If $a=\frac{|\log\omega|}{R}$ and

$f(z):=e^{az}r(z),\qquad z\in\overline{\mathbb{C}}_{+},$

then

$|f(z)|\leq 1,\qquad z\in\partial\mathbb{D}_{R}^{+},$

and the maximum principle yields $|f(z)|\leq 1$ for all
$z\in\mathbb{D}_{R}^{+}.$ Thus

(2.17) $|r(z)|\leq e^{-a{\rm Re}\,z},\qquad z\in\mathbb{D}_{R}^{+}.$

By applying the argument above to $r^{*}(z):=r(1/z)$ in place of $r,$ we infer
that

(2.18) $|r(z)|\leq e^{-a{\rm Re}\,(1/z)},\qquad
z\in\mathbb{C}_{+}\setminus\mathbb{D}_{R}^{+}.$

Moreover, as in the proof of Lemma 2.3,

(2.19) $|r^{\prime}(z)|\leq\frac{C}{|1+z|^{2}},\qquad z\in\mathbb{C}_{+},$

for some $C=C(r)>0.$

Thus, for every $n\in\mathbb{N}$, the estimates (2.17), (2.18) and (2.19)
combined with the $\mathcal{A}$-stability of $r$ imply that

(2.20) $|(r^{n}(z))^{\prime}|\leq\begin{cases}Cne^{-a(n-1){\rm Re}\,z},&\qquad
z\in\mathbb{D}_{R}^{+},\\\ Cne^{-a(n-1){\rm Re}\,(1/z)}|1+z|^{-2},&\qquad
z\in\mathbb{C}_{+}\setminus\mathbb{D}_{R}^{+}.\end{cases}$

Fix $s\geq 0$ and $n>\max(R,2),$ and let $z=x+iy\in\mathbb{C}_{+}.$ Writing

(2.21)
$(r^{n}\eta_{s}(z))^{\prime}=\frac{(r^{n}(z))^{\prime}}{(z+1)^{s}}-s\frac{r^{n}(z)}{(z+1)^{s+1}},\qquad
z\in\mathbb{C}_{+},$

and employing (2.20), we have

$\displaystyle\|r^{n}\eta_{s}\|_{\mathcal{B}_{0}}\leq$
$\displaystyle\int_{0}^{\frac{R}{n}}\sup_{y\in\mathbb{R}}\,\frac{|(r^{n}(z))^{\prime}|}{|1+z|^{s}}\,dx+\int_{\frac{R}{n}}^{\infty}\sup_{|z|>1,\,y\in\mathbb{R}}\,\frac{|(r^{n}(z))^{\prime}|}{|1+z|^{s}}\,dx$
$\displaystyle+$
$\displaystyle\int_{\frac{R}{n}}^{1}\sup_{|z|<1,\,y\in\mathbb{R}}\,\frac{|(r^{n}(z))^{\prime}|}{|1+z|^{s}}\,dx+s\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{|r^{n}(z)|}{|1+z|^{s+1}}\,dx$
$\displaystyle\leq$ $\displaystyle
C+Cn\int_{\frac{R}{n}}^{\infty}\sup_{t>0}\,S_{n-1,s}(x,t)\,dx+Cn\int_{\frac{R}{n}}^{1}\frac{e^{-a(n-1)x}}{(x+1)^{2+s}}\,dx$
$\displaystyle+s$
$\displaystyle\int_{0}^{\infty}\frac{dx}{(x^{2}+1)^{(1+s)/2}}\leq
C_{1}+Cn\int_{\frac{R}{n}}^{\infty}\sup_{t>0}\,S_{n-1,s}(x,t)\,dx$

for some $C_{1}=C_{1}(s)>0.$ The estimates (2.7) and (2.8) follow from
(2.16).∎

Observe that the estimate given by Lemma 2.5 is optimal, as the following
example shows.

######  Example 2.6.

Let $z=x+iy\in\overline{\mathbb{C}}_{+},$ and consider the rational function

$r(z)=\frac{(z+2)^{2}}{(z+1)(z+4)}.$

Noting that

$r^{\prime}(z)=\frac{z^{2}-4}{(z+1)^{2}(z+4)^{2}},$

by a simple but somewhat tedious calculation, we conclude that

$|r(z)|\geq\frac{|z-1|}{|z+1|},\qquad|r^{\prime}(z)|\geq\frac{1}{4|z+1|^{2}},\quad{\rm
Re}\,z\geq 8.$

Hence, for all $n\geq 2,$

$|(r^{n}(z))^{\prime}|\geq\frac{n}{4}\frac{|z-1|^{n-1}}{|z+1|^{n+1}},\qquad{\rm
Re}\,z\geq 8.$

so that, by Lemma 2.3 with $s=0,$

$\|r^{n}\|_{\mathcal{B}_{0}}\geq\int_{8}^{n}\sup_{y\in\mathbb{R}}\,|(r^{n}(z))^{\prime}|\,dx\geq\frac{n(n-1)^{(n-1)/2}}{8(n+1)^{(n+1)/2}}\int_{8}^{n}\frac{dx}{x}\geq
C\log(n+1)$

for some $C>0.$

Moreover, for all $t\in\mathbb{R},$

$|r(it)|^{2}=\frac{(t^{2}+4)^{2}}{(t^{2}+1)(t^{2}+16)}\leq 1,$

and

$|r(it)|<1,\qquad t\neq 0,\quad r(0)=1,\quad\text{and}\quad r(\infty)=1.$

Combining now Lemmas 2.3 and 2.5, we obtain (2.7) and (2.8) for any
$\mathcal{A}$-stable $r.$

######  Corollary 2.7.

Let $r$ be an $\mathcal{A}$-stable rational function. Then there exist
$C=C(r)>0$ and $C_{1}=C_{1}(r,s)>0$ such that (2.7) and (2.8) hold for all
$n\in\mathbb{N}$ and $s>0.$ Moreover, if $r_{n}(z)=r^{n}(z/n),$
$z\in\mathbb{C}_{+},$ then

(2.22) $\|r_{n}\eta_{s}\|_{\mathcal{B}_{0}}\leq C_{1}(r,s)+1$

for all $n\in\mathbb{N}$ and $s>0.$

###### Proof.

If $r$ is a constant function, then the statement is obvious. Otherwise, as
noted in the beginning of this section, either $|r|$ is not identically $1$ on
$i\mathbb{R},$ or $r$ is a finite Blaschke product. Thus, by combining Lemmas
2.3 and 2.5, we obtain (2.7) and (2.8) for some $C=C(r)>0,$
$C_{1}=C_{1}(r,s)>0$ and any $\mathcal{A}$-stable rational function $r.$

To obtain (2.22) it suffices to note that by (2.21) and (2.14),

$\displaystyle\|r_{n}\eta_{s}\|_{\mathcal{B}_{0}}\leq$ $\displaystyle
1+\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{|(r_{n}(z))^{\prime}|}{|1+z|^{s}}\,dx$
$\displaystyle=$ $\displaystyle
1+\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{|(r^{n}(z))^{\prime}|}{|1+nz|^{s}}\,dx\leq
1+\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{|(r^{n}(z))^{\prime}|}{|1+z|^{s}}\,dx$
$\displaystyle=$ $\displaystyle 1+\|r^{n}\eta_{s}\|_{{\mathcal{B}}_{0}}.\qed$

The bound (2.7) can be substantially strengthened if $|r(\infty)|<1.$

######  Lemma 2.8.

Let $r$ be an $\mathcal{A}$-stable rational function such that
$|r(\infty)|<1.$ Then

(2.23) $\sup_{n\in\mathbb{N}}\|r^{n}\|_{\mathcal{B}_{0}}<\infty.$

###### Proof.

Let $n\in\mathbb{N}$ be fixed. Observe that there exists $C=C(r)>0$ such that

(2.24) $|(r^{n}(z))^{\prime}|\leq\frac{Cn}{|1+z|^{2}}|r(z)|^{n-1},\qquad
z\in\mathbb{C}_{+}.$

Next, fix $\omega\in(|r(\infty)|,1).$ By assumption, there exists $R>0$ such
that

(2.25) $|r(z)|\leq\omega<1,\qquad
z\in\overline{\mathbb{C}}_{+}\setminus\mathbb{D}_{R}^{+}.$

Arguing as in the proof of Lemma 2.5 and letting $a=\frac{|\log\omega|}{R},$
we infer that by the maximum principle,

(2.26) $|r(z)|\leq e^{-a{\rm Re}\,z},\qquad z\in\mathbb{D}_{R}^{+}.$

From (2.24), (2.25) and (2.26), taking into account that $\|r\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}\leq 1,$ it then follows that

$|(r^{n}(z))^{\prime}|\leq\begin{cases}Cne^{-a(n-1){\rm Re}\,z},&\quad
z\in\mathbb{D}_{R}^{+},\\\ {Cn\omega^{n-1}}{|1+z|^{-2}},&\quad
z\in\mathbb{C}_{+}\setminus\mathbb{D}_{R}^{+}.\end{cases}$

Hence, letting $z=x+iy\in\mathbb{C}_{+},$ we have

$\displaystyle\|r^{n}\|_{\mathcal{B}_{0}}\leq$
$\displaystyle\int_{0}^{R}\sup_{y\in\mathbb{R},\,z\in\mathbb{D}_{R}^{+}}\,|(r^{n}(z))^{\prime}|\,dx+\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,z\in\mathbb{C}_{+}\setminus\mathbb{D}^{+}_{R}}\,|(r^{n}(z))^{\prime}|\,dx$
$\displaystyle\leq$ $\displaystyle
Cn\int_{0}^{R}e^{-a(n-1)x}\,dx+Cn\omega^{n-1}\int_{0}^{\infty}\frac{dx}{(1+x)^{2}}$
$\displaystyle\leq$
$\displaystyle\frac{Cn}{a(n-1)}+Cn\omega^{n-1}\leq\frac{2CR}{|\log\omega|}+\frac{C}{e\omega|\log\omega|},$

so that (2.23) holds. ∎

###  2.2. Estimates for approximation rates in terms of
$\mathcal{B}_{0}$-norms

For an $\mathcal{A}$-stable rational function $r$ and $n\in\mathbb{N}$ let

$\Delta_{r,n}(z):=e^{-z}-r^{n}(z/n),\qquad\Delta_{r,n,s}(z):=z^{-s}\Delta_{r,n}(z),\qquad
z\in\mathbb{C}_{+}.$

If a rational function $r$ approximates the exponential with order
$q,s\in[0,q+1),$ and $R>0$ is small enough, then $\Delta_{r,n}\in{\rm
Hol}(\mathbb{D}_{R})$ and in this case we write $\Delta_{r,n}(z)$ also for
$z\in\mathbb{D}_{R}.$ Similarly, $\Delta_{r,n,s}\in{\rm
Hol}(\mathbb{D}^{+}_{Rn}),$ and $\Delta_{r,n,s}$ extends continuously to
$(-iR,iR).$

Given a rational approximation $r$ to the exponential, we turn to obtaining
estimates for the $\mathcal{B}_{0}$-norms of $\Delta_{r,n,s}.$ In Section 3,
these estimates will be converted into semigroup approximation rates by means
of the $\mathcal{B}$-calculus.

The next lemma is an important tool for obtaining the $\mathcal{B}_{0}$-norm
bounds for $\Delta_{r,n,s}$ which are optimal with respect to approximation
order of $r$.

######  Lemma 2.9.

Let $r$ be an $\mathcal{A}$-stable rational approximation of order
$q\in\mathbb{N}$ to the exponential. If

$s\in(0,q+1],\quad\,\delta=q/(q+1),\quad\,\text{and}\quad\,a=-\frac{\Delta^{(q+1)}_{r,n}(0)}{(q+1)!},$

then there exist $R=R(r)\in(0,1)$ and $d>0$ such that the following hold.

  * (i)

One has

(2.27)
$|\Delta_{r,n,s}^{\prime}(z)|\leq\left((q+1-s)|a|\frac{|z|^{q-s}}{n^{q}}+\frac{d}{n^{\delta
s}}\right)e^{-{\rm Re}\,z}$

for all $z\in\mathbb{D}^{+}_{Rn^{\delta}}$ and $n\in\mathbb{N}.$

  * (ii)

If $r^{(q+1)}(0)\neq(-1)^{q+1},$ then $a\neq 0,$ and there exist
$\\{\xi(n,\cdot):n\geq 1\\}\subset{\rm Hol}(\mathbb{D}^{+}_{Rn})$ such that

(2.28)
$\Delta_{r,n,s}(z)=\left(1-e^{az^{q+1}/n^{q}}e^{\xi(n,z)}\right){e^{-z}}{z^{-s}},\qquad|\xi(n,z)|\leq
d\frac{|z|^{q+2}}{n^{q+1}},$

for all $z\in\overline{\mathbb{D}}^{+}_{Rn^{\delta}}$ and $n\in\mathbb{N}.$

The proof of Lemma 2.9 is rather technical, and is postponed to Appendix. The
assumption $r^{(q+1)}(0)\neq(-1)^{q+1}$ in Lemma 2.9, (ii) means that $r$
approximates $e^{-z}$ _precisely_ with order $q,$ and is equivalent to $a\neq
0.$ Note also, that if $r$ is not so, then as the proof of Lemma 2.9 will
reveal one may put $d_{1}=0$ in (2.27).

If a rational function $r$ approximates the exponential with some
$q,s\in(0,q+1],$ and $R_{0}>0$ is small enough, then $r$ extends
holomorphically to $\mathbb{D}_{R_{0}},$ and for $n\in\mathbb{N}$ we let

$\Delta_{r,n}(z):=e^{-z}-r^{n}(z/n),\qquad z\in\mathbb{D}_{R_{0}},$

and

$\Delta_{r,n,s}(z):=z^{-s}\Delta_{r,n}(z),\qquad z\in\mathbb{D}^{+}_{R_{0}}.$

Clearly, $\Delta_{r,n},\Delta_{r,n,s}\in{\rm Hol}(\mathbb{D}_{R_{0}}),$ and
moreover $\Delta_{r,n,s}$ extends continuously to $(-iR_{0},iR_{0}).$ If, in
addition, $r$ is $\mathcal{A}$-stable, then both $\Delta_{r,n}$ and
$\Delta_{r,n,s}$ extend holomorphically to $\mathbb{C}_{+},$ and continuously
to $\overline{\mathbb{C}}_{+}.$ We denote the extensions by the same symbols.

The following result is behind Theorem 1.3,(ii).

######  Lemma 2.10.

Let $r$ be an $\mathcal{A}$-stable rational approximation of order $q$ to the
exponential. If $\delta=q/(q+1),$ then for every $s\in(0,q+1]$ there exists
$C=C(r,s)>0$ such that

(2.29) $\|\Delta_{r,n,s}\|_{{\mathcal{B}}_{0}}\leq Cn^{-{\delta s}},\qquad
n\in\mathbb{\mathbb{N}}.$

###### Proof.

Let $s\in(0,q+1]$ and $n\in\mathbb{N}$ be fixed. From Lemma 2.9 it follows
that there is $d>0$ (depending on $r$) such that

(2.30)
$|\Delta_{r,n,s}^{\prime}(z)|\leq|a|(q+1-s)\frac{|z|^{q-s}}{n^{q}}+\frac{d}{n^{\delta
s}}e^{-{\rm Re}\,z},\qquad z\in\mathbb{D}^{+}_{R_{0}n^{\delta}}.$

For $z\in\mathbb{C}_{+}$ write $z=x+iy.$ We have

(2.31) $\|\Delta_{r,n,s}\|_{{\mathcal{B}}_{0}}\leq
B_{n,s}+U^{(1)}_{n,s}+U^{(2)}_{n,s},$

where

$\displaystyle B_{n,s}=$
$\displaystyle\int_{0}^{n^{\delta}}\sup_{y\in\mathbb{R},\,|z|\leq
n^{\delta}}\,|\Delta_{r,n,s}^{\prime}(z)|\,dx,$ $\displaystyle U^{(1)}_{n,s}=$
$\displaystyle\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|>R_{0}n^{\delta}}\,|\Delta_{r,n}^{\prime}(z)||z|^{-s}\,dx,$

and

$U^{(2)}_{n,s}=s\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|>R_{0}n^{\delta}}\,|\Delta_{r,n}(z)||z|^{-s-1}\,dx.$

We estimate each of the terms in the right hand side of (2.31) separately.
First, note that by Lemma 2.9, (i),

$B_{n,s}\leq|a|B_{n,s}^{(1)}+dB_{n,s}^{(2)},$

where

$B_{n,s}^{(1)}=\frac{(q+1-s)}{n^{q}}\int_{0}^{n^{\delta}}\,\sup_{|y|\leq
n^{\delta}}\,(x^{2}+y^{2})^{(q-s)/2}\,dx,$

if $s\in(0,q+1),$ $B_{n,q+1}^{(1)}=0$, and

$B_{n,s}^{(2)}=\frac{1}{n^{\delta
s}}\int_{0}^{n^{\delta}}e^{-x}\,dx\leq\frac{1}{n^{\delta s}}.$

Furthermore, if $s\in(0,q],$ then

$\displaystyle B_{n,s}^{(1)}=$
$\displaystyle\frac{(q+1-s)}{n^{q}}\int_{0}^{n^{\delta}}(x^{2}+n^{2\delta})^{(q-s)/2}\,dx$
$\displaystyle=$ $\displaystyle\frac{(q+1-s)}{n^{\delta
s}}\int_{0}^{1}(t^{2}+1)^{(q-s)/2}\,dt\leq\frac{q+1}{n^{\delta s}}2^{q/2},$

and, for $s\in(q,q+1),$

$B_{n,s}^{(1)}=\frac{(q+1-s)}{n^{q}}\int_{0}^{n^{\delta}}\,\frac{dx}{x^{s-q}}\leq\frac{2^{q+1}}{n^{\delta
s}}.$

So, combining the estimates for $B^{(1)}_{n,s}$ and $B^{(2)}_{n,s}$ above, we
obtain that

(2.32) $B_{n,s}\leq\frac{|a|(q+1)2^{q/2}+d}{n^{\delta s}}.$

Next, since $r$ is $\mathcal{A}$-stable, we have $\|\Delta_{r,n}\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}\leq 2$. Using this we infer that

(2.33) $\displaystyle U^{(2)}_{n,s}\leq$ $\displaystyle
2s\int_{Rn^{\delta}}^{\infty}\sup_{y\in\mathbb{R}}\,|z|^{-s-1}\,dx+2s\int_{0}^{Rn^{\delta}}\sup_{|y|\geq
Rn^{\delta}}\,|z|^{-s-1}\,dx$ $\displaystyle=$ $\displaystyle
2s\int_{Rn^{\delta}}^{\infty}\frac{dx}{x^{s+1}}+2s\int_{0}^{Rn^{\delta}}\frac{dx}{(x^{2}+R^{2}n^{2\delta})^{(s+1)/2}}\leq\frac{2(2+q)}{R^{s}n^{\delta
s}}.$

Finally, to estimate $U^{(1)}_{n,s},$ note that

$\displaystyle U^{(1)}_{n,s}\leq$
$\displaystyle\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|\geq
Rn^{\delta}}|z|^{-s}e^{-x}\,dx+\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|\geq
Rn^{\delta}}\,\frac{|((r^{n}(z/n))^{\prime}|}{|z|^{s}}\,dx$
$\displaystyle\leq$ $\displaystyle\frac{1}{R^{s}n^{\delta
s}}+\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|\geq
Rn^{\delta-1}}\,\frac{|(r^{n}(z))^{\prime}|}{n^{s}|z|^{s}}\,dx.$

If $|z|\geq Rn^{\delta-1},$ then, recalling that $\delta\in(0,1)$ and
$R\in(0,1),$ we have

$\displaystyle 2n|z|\geq n|z|+Rn^{\delta}\geq Rn^{\delta}(|z|+1)\geq
Rn^{\delta}|1+z|.$

Hence, if $C_{1}(r,s)$ is the constant given by Corollary 2.7, then

(2.34) $U^{(1)}_{n,s}\leq\frac{1}{R^{s}n^{\delta
s}}+\frac{2^{s}}{R^{s}n^{\delta
s}}\int_{0}^{\infty}\,\sup_{y\in\mathbb{R}}\,\frac{|(r^{n}(z))^{\prime}|}{|1+z|^{s}}\,dx\leq\frac{1+2^{s}C_{1}(r,s)}{R^{s}n^{\delta
s}}.$

Using (2.31) and taking into account (2.32), (2.33), and (2.34), one obtains
(2.29). ∎

###  2.3. Estimates for subdiagonal Padé approximations

Following the ideology of the previous subsections and having in mind Theorem
1.5, we proceed with obtaining fine $\mathcal{B}_{0}$-norm estimates for
$z\to(e^{-z}-r_{[n,n+1]}(z))z^{-s},$ where $(r_{[n,n+1]})_{n\geq 1}$ are the
first subdiagonal Padé approximations of $e^{-z}.$

Recall that if $P_{n}$ and $Q_{n+1}$ are polynomials such that
$r_{[n,n+1]}=P_{n}/Q_{n+1},$ $n\in\mathbb{N},$ $Q_{n+1}(0)=1,$ then for all
$n\in\mathbb{N},$

(2.35) $\displaystyle P_{n}(z)=$
$\displaystyle\sum_{j=0}^{n}\frac{(2n+1-j)!n!}{(2n+1)!j!(n-j)!}(-z)^{j},$
(2.36) $\displaystyle Q_{n+1}(z)=$
$\displaystyle\sum_{j=0}^{n+1}\frac{(2n+1-j)!(n+1)!}{(2n+1)!j!(n+1-j)!}z^{j},$

and $r_{[n,n+1]}$ are $\mathcal{A}$-stable rational approximations of order
$2n+1$ to $e^{-z},$ see e.g. [23, Theorem 3.11]. Fix $n\in\mathbb{N}$ and
define the approximation error $\Delta_{r_{[n,n+1]}}$ by

$\Delta_{r_{[n,n+1]}}(z):=r_{[n,n+1]}(z)-e^{-z},\qquad z\in\mathbb{C}_{+}.$

The next classical Perron representation for $\Delta_{r_{[n,n+1]}}:$

(2.37)
$\Delta_{r_{[n,n+1]}}(z)=\frac{1}{Q_{n+1}(z)}\frac{z^{2n+2}}{(2n+1)!}\int_{0}^{1}(1-t)^{n}t^{n+1}e^{-zt}\,dt,\qquad
z\in\mathbb{C}_{+},$

is behind several basic properties of Padé approximations to $e^{-z}.$ It can
be found in e.g. [2, p. 643], see also [9, p. 880] and [29, p. 3566]. Note
that

(2.38) $|Q_{n+1}(z)|\geq
1\qquad\text{and}\qquad\frac{|Q_{n+1}^{\prime}(z)|}{|Q_{n+1}(z)|}\leq 1,$

for all $z\in\mathbb{C}_{+}$ and $n\in\mathbb{N}.$ For $z=it,t\in\mathbb{R},$
the bounds in (2.38) were noted in [29, Propositions 1 and 2] and [9, Lemma
3.1]. They extend to $\mathbb{C}_{+}$ by the maximum principle applied to
$1/Q_{n+1}$ and $Q^{\prime}_{n+1}/Q_{n+1}.$ Via Perron’s representation,
(2.38) is instrumental in various estimates for the size of
$\Delta_{r_{[n,n+1]}}.$ In particular, combining [9, Lemma 3.2] with the next
simple inequality ([9, Lemma 3.5]):

(2.39) $\frac{n!}{(2n+1)!}\leq\frac{1}{(n+1)^{n+1}},\qquad n\in\mathbb{N},$

we have

(2.40) $|\Delta_{r_{[n,n+1]}}(z)|\leq\frac{|z|^{2n+2}}{2(n+1)^{2(n+1)}},\qquad
z\in\mathbb{C}_{+}.$

To facilitate the subsequent estimates, we will need several additional
properties of $\Delta_{r_{[n,n+1]}}$ given in the lemma below.

######  Lemma 2.11.

Let $r_{[n,n+1]}=P_{n}/Q_{n+1},n\in\mathbb{N},$ be the first subdiagonal Padé
approximations to $e^{-z},$ where $P_{n}$ and $Q_{n+1}$ are given by (2.35)
and (2.36), respectively.

  * (i)

For all $n\in\mathbb{N},$

(2.41) $\|\Delta_{r_{[n,n+1]}}\|_{{\rm H}^{\infty}(\mathbb{C}_{+})}\leq
2,\qquad\|\Delta_{r_{[n,n+1]}}^{\prime}\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}\leq 2.$

  * (ii)

For all $n\in\mathbb{N}$ and $z\in\mathbb{D}^{+}_{n},$

(2.42) $|\Delta_{r_{[n,n+1]}}^{\prime}(z)|\leq
2\frac{|z|^{2n+1}}{(2n)!}\int_{0}^{1}(1-t)^{n}t^{n+1}e^{-t{\rm Re}\,z}\,dt.$

###### Proof.

Fix $z\in\mathbb{C}_{+}$ and $n\in\mathbb{N}.$ To prove (i) it suffices to
recall that by e.g. [23, Theorem 4.12] the function $r_{[n,n+1]}$ is
$\mathcal{A}$-stable, and, moreover, by [15, p. 334], its derivative
$r_{[n,n+1]}^{\prime}$ is $\mathcal{A}$-stable as well. (A weaker property
$\|r_{[n,n+1]}^{\prime}\|_{{\rm H}^{\infty}(\mathbb{C}_{+})}\leq 2$ was also
noted in [29, Proposition 2])

To deduce (ii), note that by (2.37) and (2.38),

$\displaystyle\Delta_{r_{[n,n+1]}}^{\prime}(z)=$
$\displaystyle-\frac{Q_{n+1}^{\prime}(z)}{Q^{2}_{n+1}(z)}\frac{z^{2n+2}}{(2n+1)!}\int_{0}^{1}(1-t)^{n}t^{n+1}e^{-zt}\,dt$
$\displaystyle+$
$\displaystyle\frac{1}{Q_{n+1}(z)}\frac{(2n+2)z^{2n+1}}{(2n+1)!}\int_{0}^{1}(1-t)^{n}t^{n+1}e^{-zt}\,dt$
$\displaystyle-$
$\displaystyle\frac{1}{Q_{n+1}(z)}\frac{z^{2n+2}}{(2n+1)!}\int_{0}^{1}(1-t)^{n}t^{n+2}e^{-zt}\,dt,$

and, letting $z=x+iy,$

$\displaystyle|\Delta_{r_{[n,n+1]}}^{\prime}(z)|\leq$
$\displaystyle\frac{|z|^{2n+2}}{(2n+1)!}\int_{0}^{1}(1-t)^{n}t^{n+1}e^{-xt}\,dt$
$\displaystyle+$
$\displaystyle\frac{(2n+2)|z|^{2n+1}}{(2n+1)!}\int_{0}^{1}(1-t)^{n}t^{n+1}e^{-xt}\,dt$
$\displaystyle+$
$\displaystyle\frac{|z|^{2n+2}}{(2n+1)!}\int_{0}^{1}(1-t)^{n}t^{n+2}e^{-xt}\,dt.$

Hence, if $z\in\mathbb{D}^{+}_{n},$ then

$|\Delta_{r_{[n,n+1]}}^{\prime}(z)|\leq
2\frac{|z|^{2n+1}}{(2n)!}\int_{0}^{1}(1-t)^{n}t^{n+1}e^{-xt}\,dt.\qed$

For $s>0$ and $n\in\mathbb{N}$ let now

$\Delta_{r_{[n,n+1]},s}(z):=z^{-s}\Delta_{r_{[n,n+1]}}(z),\qquad
z\in\mathbb{C}_{+}.$

The following statement provides an estimate for
$\|\Delta_{r_{[n,n+1]},s}\|_{\mathcal{B}_{0}}$ leading to the operator-norm
error bound in Theorem 1.5 via the $\mathcal{B}$-calculus. In view of possible
improvements, we formulate the result with an explicit constant.

######  Lemma 2.12.

For all $s>0$ and $n\in\mathbb{N}$ such that $n\geq(s-1)/2,$ one has

(2.43)
$\|\Delta_{r_{[n,n+1]},s}\|_{\mathcal{B}_{0}}\leq\left\\{\frac{1}{2}+\frac{4}{s}+\frac{s}{2}+\frac{4\pi
s}{(s+1)\sin(\pi/(1+s))}+4\log(2n+1)\right\\}\frac{1}{n^{s}}.$

###### Proof.

Fix $n\in\mathbb{N}$ and $s\in(0,\infty)$ satisfying $2n+1\geq s>0.$ We write

(2.44)
$\Delta_{r_{[n,n+1]},s}^{\prime}(z)=-s\frac{\Delta_{r_{[n,n+1]}}(z)}{z^{s+1}}+\frac{\Delta_{r_{[n,n+1]}}^{\prime}(z)}{z^{s}},\qquad
z\in\mathbb{C}_{+},$

and estimate the $\mathcal{B}_{0}$-norm of each of the terms in the right hand
side of (2.44) separately.

Let $z=x+iy\in\mathbb{C}_{+}.$ Then, observing that $|z|\geq n$ implies

$2|z|^{s+1}\geq|z|^{s+1}+n^{s+1}\geq x^{s+1}+n^{s+1},$

and using (2.40), we have

(2.45)
$\displaystyle\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{|\Delta_{r_{[n,n+1]}}(z)|}{|z|^{s+1}}\,dx$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|\leq
n}\,\frac{|\Delta_{r_{[n,n+1]}}(z)|}{|z|^{s+1}}\,dx+\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|\geq
n}\,\frac{|\Delta_{r_{[n,n+1]}}(z)|}{|z|^{s+1}}\,dx$ $\displaystyle\leq$
$\displaystyle\frac{1}{2n^{s+1}}\int_{0}^{n}\,dx+4\int_{0}^{\infty}\frac{dx}{x^{s+1}+n^{s+1}}$
$\displaystyle\leq$
$\displaystyle\left(\frac{1}{2}+\frac{4\pi}{(s+1)\sin(\pi/(1+s))}\right)n^{-s},$

since by [30, p. 295],

$\int_{0}^{\infty}\frac{dx}{x^{s+1}+1}=\frac{\pi}{(s+1)\sin(\pi/(1+s))}.$

Next, in view of (2.41), we infer that

$|(\Delta_{r_{[n,n+1]}}(z))^{\prime}|\leq x^{-1}\|\Delta_{r_{[n,n+1]}}\|_{{\rm
H}^{\infty}(\mathbb{C}_{+})}\leq\min\left\\{2,x^{-1}\right\\}\leq\frac{2}{x+1/2}$

for all $z\in\mathbb{C}_{+}.$ Hence, employing (2.42),

(2.46)
$\displaystyle\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{|\Delta_{r_{[n,n+1]}}^{\prime}(z)|}{|z|^{s}}\,dx$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|\leq
n}\,\frac{|\Delta_{r_{[n,n+1]}}^{\prime}(z)|}{|z|^{s}}\,dx+\int_{0}^{\infty}\sup_{y\in\mathbb{R},\,|z|\geq
n}\,\frac{|\Delta_{r_{[n,n+1]}}^{\prime}(z)|}{|z|^{s}}\,dx$
$\displaystyle\leq$
$\displaystyle\frac{2n^{2n+1-s}}{(2n)!}\int_{0}^{1}(1-t)^{n}t^{n+1}\int_{0}^{n}e^{-xt}\,dx\,dt+\int_{0}^{\infty}\frac{4\,dx}{(n^{s}+x^{s})(1/2+x)}$
$\displaystyle\leq$
$\displaystyle\frac{2(n!)^{2}n^{2n+1-s}}{(2n)!(2n+1)!}+\frac{4}{n^{s}}\int_{0}^{\infty}\frac{dx}{(1+x^{s})(1/(2n)+x)},$

where we also used that

$\int_{0}^{1}(1-t)^{n}t^{n}\,dt=\frac{(n!)^{2}}{(2n+1)!}.$

Since

$\displaystyle\int_{0}^{\infty}\frac{dx}{(1+x^{s})(1/(2n)+x)}\leq$
$\displaystyle\int_{0}^{1}\frac{dx}{x+1/(2n)}+\int_{1}^{\infty}\frac{dx}{x^{s+1}}$
$\displaystyle=$ $\displaystyle\log(2n+1)+\frac{1}{s},$

and

(2.47) $\frac{2(n!)^{2}n^{2n+1-s}}{(2n)!(2n+1)!}\leq\frac{1}{2n^{s}},$

from (2.46) it follows that

(2.48)
$\displaystyle\int_{0}^{\infty}\sup_{y\in\mathbb{R}}\,\frac{|\Delta_{r_{[n,n+1]}}^{\prime}(z)|}{|z|^{s}}\,dx\leq\left(\frac{1}{2}+\log(2n+1)+\frac{1}{s}\right)\frac{4}{n^{s}}.$

Now (2.44), (2.45) and (2.48) imply (2.43). ∎

We do not know whether $\log(n+1)$ in (2.43) can be omitted. This would follow
from the estimate $\sup_{n\geq 1}\|r_{[n,n+1]}\|_{\mathcal{B}_{0}}<\infty$
(leading to the strong convergence of the first subdiagonal Padé
approximations of operator semigroups). Unfortunately, at the moment, this
bound is out of reach as well.

##  3\. The $\mathcal{B}$-calculus revisited and operator-norm estimates

###  3.1. Construction of the $\mathcal{B}$-calculus

As far as this paper relies on the $\mathcal{B}$-calculus we provide a
streamlined approach to its construction, different from the one developed in
[3] and [4], and thus make the paper essentially self-contained. In contrast
to [3] and [4], the construction is based on the reproducing formula for the
space $\mathcal{B},$ and does not depend on any advanced function-theoretical
machinery.

To start the construction, for $\lambda,z\in\mathbb{C}_{+},$ write
$\lambda=\alpha+i\beta$, and let

$K(z,\lambda):=-\frac{2}{\pi(z+\lambda)^{2}},\qquad
D_{\lambda}:=\frac{d}{d\lambda},\qquad\text{and}\qquad
dS(\lambda):=\alpha\,d\beta\,d\alpha.$

Recall that if $f\in\mathcal{B}$ then by [3, Proposition 2.20] $f$ can be
recovered by the reproducing formula

(3.1)
$f(z)=f(\infty)+\int_{\mathbb{C}_{+}}K(z,\overline{\lambda})D_{\lambda}f(\lambda)\,dS(\lambda),\qquad
z\in\mathbb{C}_{+},$

where the integral converges absolutely for every $z\in\mathbb{C}_{+}$ in view
of

$\sup_{\alpha>0}\alpha\int_{\mathbb{R}}|K(z,\alpha-i\beta)|\,d\beta<\infty$

and Fubini’s theorem. The formula (3.1) was obtained initially in [3,
Proposition 2.20] in a complicated manner, and it was reproved by elementary
means in [4, Theorem 2.3] and also in [5, Corollary 3.10 and Proposition
3.15].

Following an established route in the case of classical Riesz-Dunford
calculus, given an operator $A$ on a Hilbert space $X,$ it is natural to try
to define $f(A)$ for $f\in\mathcal{B}$ by plugging $A$ instead of independent
variable $z$ into (3.1), and thus creating an operator counterpart of (3.1).
In what follows, we formalise this procedure and show that it leads indeed to
a well-defined functional calculus. We say that an operator $A$ admits a
$\mathcal{B}$-calculus $\Phi$ if $A$ is densely defined,
$\sigma(A)\subset\overline{\mathbb{C}}_{+},$ and there is a bounded algebra
homomorphism $\Phi:\mathcal{B}\to L(X)$ such that
$\Phi((\lambda+\cdot)^{-1})=(\lambda+A)^{-1}$ for all
$\lambda\in\mathbb{C}_{+}.$ If $A$ admits a $\mathcal{B}$-calculus $\Phi,$
then we set $f(A):=\Phi(f),f\in\mathcal{B}.$

Note that if $-A$ generates a bounded $C_{0}$-semigroup $(e^{-tA})_{t\geq 0}$
on $X,$ with $\sup_{t\geq 0}\|{e^{-tA}}\|:=M,$ then Plancherel’s theorem
implies that for every $x\in X$ the $L^{2}(\mathbb{R},X)$-norms of
$\sqrt{\alpha}(\alpha+i\cdot+A)^{-1}x$ and
$\sqrt{\alpha}\,(\alpha-i\cdot+A^{*})^{-1}x$ are bounded by $\sqrt{\pi}M\|x\|$
uniformly in $\alpha>0.$ Using this fact, one infers easily that

(3.2) $\sup_{\alpha>0}\int_{\mathbb{R}}\alpha|\langle
K(A,\alpha-i\beta)x,x^{*}\rangle|\,d\beta\leq\pi M^{2}\|x\|\|x^{*}\|$

for all $x,x^{*}\in X.$ (See [3, Example 4.1] for a discussion of (3.2) and
relevant references.)

From (3.2) it follows that the formula

(3.3) $\langle f(A)x,x^{*}\rangle:=f(\infty)I+\int_{\mathbb{C}_{+}}\langle
K(A,\overline{\lambda})x,x^{*}\rangle\,D_{\lambda}f(\lambda)\,dS(\lambda),$

for all $x\in X$ and $x^{*}\in X,$ defines a bounded linear operator $f(A)$ on
$X,$ and moreover

(3.4) $\|f(A)\|\leq|f(\infty)|+\pi M^{2}\|f\|_{\mathcal{B}_{0}}\leq\pi
M^{2}\|f\|_{\mathcal{B}}.$

Thus, we have a well-defined, bounded linear mapping $\Phi_{A}:\mathcal{B}\to
L(X),$ given by $\Phi_{A}(f):=f(A).$ Moreover, as a simple calculation shows
(see [3, p. 42 and Lemma 4.2]), if $f=\widehat{\nu}$ with $\nu\in{\rm
M}(\mathbb{R}_{+}),$ and $f(A)$ is given by (3.3), then

$f(A)x=\int_{0}^{\infty}e^{-tA}x\,d\nu(t),\qquad x\in X,$

so that $\Phi_{A}$ extends the HP-calculus, $\|\widehat{\nu}(A)\|\leq\pi
M^{2}\|\widehat{\nu}\|_{\mathcal{B}},$ and
$\Phi_{A}((\cdot+\lambda)^{-1})=(\lambda+A)^{-1},\lambda\in\mathbb{C}_{+}$
(the latter can also be established by Cauchy’s formula). This fact was a
starting point in [3], and it will also be useful below.

If $\Phi_{A}$ is multiplicative, then $\Phi_{A}$ is a functional calculus by
the definition above. It would clearly coincide with the
$\mathcal{B}$-calculus constructed in [3], since both calculi are given by the
same formula (3.3). Having defined $f(A)$ by (3.3), the argument in [3]
proceeds with showing the homomorphism property of $\Phi_{A}.$ This is the
main step of the construction, and it is rather involved. Note that
$\mathcal{LM}(\mathbb{C}_{+})$ is not dense in $\mathcal{B}.$ So the approach
in [3] relies on finding convenient dense sets $\mathcal{G}$ in $\mathcal{B}$
and using them to show that $\mathcal{LM}(\mathbb{C}_{+})$ is dense in
$\mathcal{B}$ in a suitable weak topology. Then the homomorphism property of
$\Phi_{A}$ is established via several approximation unit arguments. The
formula (3.1) is not used explicitly, and $\mathcal{G}$ is created via
Arveson’s spectral theory for isometric groups applied to a $C_{0}$-group of
vertical shifts on $\mathcal{B}.$

We give a simple alternative proof for the homomorphism property of
$\Phi_{A},$ where a mere validity of (3.1) will allow us to set-up a
functional calculus for $A.$ The next identity, allowing one to separate the
roles of $f$ and $g$ in $(fg)(A),$ is the heart matter of our approach.

######  Lemma 3.1.

Let $f,g\in\mathcal{B}_{0}.$ Then for all $x\in X$ and $x^{*}\in X,$

(3.5)
$\langle(fg)(A)x,x^{*}\rangle=\int_{\mathbb{C}_{+}}(D_{\mu}g)(\mu)\,\langle[K(\cdot,\overline{\mu})f](A)x,x^{*}\rangle\,dS(\mu).$

###### Proof.

Applying (3.3) to $fg$ and (3.1) to $g,$ and interchanging the integration
order formally, we infer that for all $x\in X$ and $x^{*}\in X,$

(3.6) $\displaystyle\langle(fg)(A)x,x^{*}\rangle$ $\displaystyle=$
$\displaystyle\int_{\mathbb{C}_{+}}\langle
K(A,\overline{\lambda})x,x^{*}\rangle
D_{\lambda}\left(f(\lambda)\int_{\mathbb{C}_{+}}K(\lambda,\overline{\mu})D_{\mu}g(\mu)dS(\mu)\right)dS(\lambda)$
$\displaystyle=$ $\displaystyle\int_{\mathbb{C}_{+}}\langle
K(A,\overline{\lambda})x,x^{*}\rangle\left(\int_{\mathbb{C}_{+}}D_{\lambda}(K(\lambda,\overline{\mu})f(\lambda))D_{\mu}g(\mu)dS(\mu)\right)dS(\lambda)$
$\displaystyle=$
$\displaystyle\int_{\mathbb{C}_{+}}D_{\mu}g(\mu)\int_{\mathbb{C}_{+}}\langle
K(A,\overline{\lambda})x,x^{*}\rangle
D_{\lambda}\left(K(\lambda,\overline{\mu})f(\lambda)\right)\,dS(\lambda)\,dS(\mu).$

If the integrals in (3.6) converge absolutely, then Fubini’s theorem makes the
above argument rigorous, and the relation (3.5) follows from (3.6) and (3.3).
Thus it remains to show the absolute convergence.

To this aim, letting $\mu=t+is$ with $t>0$ and $s\in\mathbb{R},$ and assuming
that $\|x\|=\|x^{*}\|=1,$ observe that

(3.7) $\displaystyle\int_{\mathbb{C}_{+}}|D_{\mu}g(\mu)|$
$\displaystyle\int_{\mathbb{C}_{+}}|\langle
K(A,\overline{\lambda})x,x^{*}\rangle||D_{\lambda}\left(K(\lambda,\overline{\mu})f(\lambda)\right)|\,dS(\lambda)\,dS(\mu)$
$\displaystyle\leq$
$\displaystyle\int_{0}^{\infty}\sup_{s\in\mathbb{R}}\,|g^{\prime}(t+is)|tL(t)\,dt,$

where, for every $t>0,$

$\displaystyle L(t)=$
$\displaystyle\int_{\mathbb{R}}\int_{\mathbb{C}_{+}}|\langle
K(A,\overline{\lambda})x,x^{*}\rangle||D_{\lambda}\left(K(\lambda,t-is)f(\lambda)\right)|\,dS(\lambda)\,ds$
$\displaystyle=$ $\displaystyle\int_{\mathbb{C}_{+}}|\langle
K(A,\overline{\lambda})x,x^{*}\rangle|\left(\int_{\mathbb{R}}|D_{\lambda}\left(K(\lambda,t-is)f(\lambda)\right)|ds\right)\,dS(\lambda).$

Next, recalling that $\lambda=\alpha+i\beta,$ where
$\alpha>0,\beta\in\mathbb{R},$ we have

$\displaystyle\int_{\mathbb{R}}|D_{\lambda}(K(\lambda,t-is)f(\lambda))|\,ds$
$\displaystyle\leq
2|f(\lambda)|\int_{\mathbb{R}}\frac{ds}{|\alpha+i\beta+t-is|^{3}}+|f^{\prime}(\lambda)|\int_{\mathbb{R}}\frac{ds}{|\alpha+i\beta+t-is|^{2}}$
$\displaystyle\leq
4\left(\frac{|f(\lambda)|}{(\alpha+t)^{2}}+\frac{|f^{\prime}(\lambda)|}{\alpha+t}\right),\qquad
t>0.$

So, for all $t>0,$

$\displaystyle\frac{L(t)}{4}\leq$
$\displaystyle\int_{0}^{\infty}\alpha\sup_{\beta\in\mathbb{R}}\,\left(\frac{|f(\lambda)|}{(\alpha+t)^{2}}+\frac{|f^{\prime}(\lambda)|}{\alpha+t}\right)\int_{\mathbb{R}}|\langle
K(A,\alpha-i\beta)x,x^{*}\rangle|d\beta d\alpha$ $\displaystyle\leq$
$\displaystyle\pi
M^{2}\int_{0}^{\infty}\left(\frac{1}{(\alpha+t)^{2}}\sup_{\beta\in\mathbb{R}}\,|f(\alpha+i\beta)|+\frac{1}{\alpha+t}\sup_{\beta\in\mathbb{R}}\,|f^{\prime}(\alpha+i\beta)|\right)d\alpha$
$\displaystyle=$ $\displaystyle t^{-1}\pi M^{2}\|f\|_{\mathcal{B}}.$

This yields the absolute convergence of the left-hand side of (3.7) and
finishes the proof. ∎

Now we are ready to prove the multiplicativity of $\Phi_{A}.$

######  Proposition 3.2.

Let $f\in\mathcal{B}$ and $g\in\mathcal{B}$. Then

(3.8) $f(A)g(A)=(fg)(A).$

###### Proof.

Assume that $f,g\in\mathcal{B}_{0}.$ The proof of (3.8) will then be done in
three steps.

a) Let first $f\in\mathcal{LM}(\mathbb{C}_{+})$ and $g\in\mathcal{B}_{0}$.
Since $\Phi_{A}$ extends the HP-calculus, we use the product rule for the HP-
calculus to obtain that

$\langle[K(\cdot,\overline{\mu})f](A)x,x^{*}\rangle=\langle
K(A,\overline{\mu})f(A)x,x^{*}\rangle$

for all $\mu\in\mathbb{C}_{+},$ $x\in X$ and $x^{*}\in X.$ Hence, by (3.5),

$\displaystyle\langle(f\cdot g)(A)x,x^{*}\rangle=$
$\displaystyle\int_{\mathbb{C}_{+}}(D_{\mu}g)(\mu)\langle
K(A,\overline{\mu})f(A)x,x^{*}\rangle\,dS(\mu)$ $\displaystyle=$
$\displaystyle\langle f(A)g(A)x,x^{*}\rangle,\qquad x\in X,\,\,x^{*}\in X,$

and (3.8) follows. Moreover, by symmetry, (3.8) holds if
$g\in\mathcal{LM}(\mathbb{C}_{+})$ and $f\in\mathcal{B}_{0}$.

b) Let $\mu\in\mathbb{C}_{+}$ be fixed. By a), applying (3.8) to
$f\in\mathcal{B}_{0}$ and
$g=K(\cdot,\overline{\mu})\in\mathcal{LM}(\mathbb{C}_{+}),$ we conclude that

(3.9) $\langle[K(\cdot,\overline{\mu})f](A)x,x^{*}\rangle=\langle
K(A,\overline{\mu})f(A)x,x^{*}\rangle$

for all $x\in X$ and $x^{*}\in X$.

c) Let now $f,g\in\mathcal{B}_{0}$. Then, using (3.5) and (3.9) and arguing as
in a), we obtain (3.8).

Finally, to deduce (3.8) for $f,g\in\mathcal{B},$ it suffices to apply (3.8)
to $f-f(\infty)$ and $g-g(\infty)$ from $\mathcal{B}_{0}.$ ∎

Thus we arrive at a Hilbert space part of the result established in [3,
Theorem 4.4].

######  Corollary 3.3.

If $-A$ is the generator of a bounded $C_{0}$-semigroup on a Hilbert space
$X,$ then the mapping $\Phi_{A}:\mathcal{B}\to L(X),$ $\Phi_{A}(f):=f(A),$
given by (3.3), is a functional calculus for $A,$ extending the HP-calculus.
Moreover, (3.4) holds.

Since $\Phi_{A}$ coincides with the $\mathcal{B}$-calculus for $A$ constructed
in [3], we will refer to $\Phi_{A}$ as the $\mathcal{B}$-calculus in the
sequel. It is instructive to note that if a linear operator $A$ on $X$ admits
a $\mathcal{B}$-calculus $\Phi,$ then $\Phi$ is unique by [4, Theorem 6.2].

######  Remark 3.4.

The construction of the $\mathcal{B}$-calculus given above can be adjusted to
Banach spaces for a wider class of $A$ satisfying the so-called GSF resolvent
condition. This goes however beyond the scope of the present paper. Yet
another simple and more powerful approach to construction of the
$\mathcal{B}$-calculus covering the case of arbitrary number of commuting
Hilbert space semigroup generators is developed in [6].

###  3.2. Applications of the $\mathcal{B}$-calculus to rational
approximations

Now we turn to the proofs of the main results of this paper. The arguments are
straightforward and based on function-theoretical estimates obtained in
Section 2.1 and the $\mathcal{B}$-calculus. We also rely on the fact that the
$\mathcal{B}$-calculus (strictly) extends the HP-calculus. While the HP-
calculus will be used to keep a standard meaning for operator functions, the
$\mathcal{B}$-calculus will produce fine operator-norm estimates. We employ
the standard functional calculi theory, see e.g [20, Chapter 1], and also [9,
Section 2.3] for a discussion close to our context.

The next lemma will be instrumental in the proof of Theorem 1.3. It shows, in
particular, that estimating the $\mathcal{B}$-norms of $\Delta_{r,n,s}$ we, in
fact, estimate the size of functions in $\mathcal{LM}$ in a finer way provided
by the $\mathcal{B}$-norm. (As we mentioned in the introduction, for integer
$s\geq 0$ the statement was noted in [7, p. 685]).

######  Lemma 3.5.

Let $r$ be an $\mathcal{A}$-stable rational approximation of order $q$ to the
exponential. Then for all $s>0$ and $n\in\mathbb{N}$ one has
$\Delta_{r,n,s}\in\mathcal{LM}(\mathbb{C}_{+}).$

###### Proof.

Let $n\in\mathbb{N}$ and $s>0$ be fixed. Write

$\Delta_{r,n,s}(z)=\Delta_{r,n}(z)\eta_{s}(z)+(\eta_{s}(z-1)-\eta_{s}(z))\Delta_{r,n}(z)$

for all $z\in\mathbb{C}_{+}.$ Recalling that
$\eta_{s}(z)=(1+z)^{-s},z\in\mathbb{C}_{+},$ note that
$\eta_{s}\in\mathcal{LM}(\mathbb{C}_{+})$ (see e.g. [20, Lemma 3.3.4]), and by
partial fraction expansion we have
$\Delta_{r,n}\in\mathcal{LM}(\mathbb{C}_{+})$ as well. Hence,
$\Delta_{r,n}\eta_{s}\in\mathcal{LM}(\mathbb{C}_{+}).$ Letting

$\nu_{s}(z):=\eta_{s}(z-1)-\eta_{s}(z),\qquad z\in\mathbb{C}_{+},$

we prove next that $\Delta_{r,n}\nu_{s}\in\mathcal{LM}(\mathbb{C}_{+}),$ and
thus $\Delta_{r,n,s}\in\mathcal{LM}(\mathbb{C}_{+}),$ as required. To this
end, by the mean value inequality for $\eta_{s}$, observe that

(3.10)
$|\nu_{s}(z)|\leq\frac{s}{|z|^{s+1}}\qquad\text{and}\qquad|(\nu_{s}(z))^{\prime}|\leq\frac{s(s+1)}{|z|^{s+2}}$

for all $z\in\mathbb{C}_{+}.$ Moreover, since $r$ is $\mathcal{A}$-stable, we
have $\Delta_{r,n},\Delta_{r,n}^{\prime}\in{\rm H}^{\infty}(\mathbb{C}_{+})$
(using e.g. an argument leading to (2.10)). Hence, taking account Lemma 2.9,
we conclude that $\Delta_{r,n}\nu_{s}\in{\rm Hol}(\mathbb{C}_{+})$ and

$\sup_{x>0}\|\left(\nu_{s}\Delta_{r,n}\right)^{\prime}(x+i\cdot)\|_{L^{1}(\mathbb{R})}<\infty,$

i.e. $(\Delta_{r,n}\nu_{s})^{\prime}$ belongs to the Hardy space ${\rm
H}^{1}(\mathbb{C}_{+}).$ In view of [5, Theorem 4.12], this implies that
$\Delta_{r,n}\nu_{s}\in\mathcal{LM}(\mathbb{C}_{+}),$ and then
$\Delta_{r,n,s}\in\mathcal{LM}(\mathbb{C}_{+}).$ ∎

######  Remark 3.6.

It is instructive to note that if $r$ is $\mathcal{A}$-stable, then since
$\Delta_{r,n,s}\in\mathcal{LM}(\mathbb{C}_{+})$ for integer $s\in[0,q+1]$ and
$n\in\mathbb{N}$ by e.g. [7, p. 685] (or by a more elementary partial fraction
expansion), Lemma 3.5 follows from an interpolation result in [18, Corollary
4.3]. However, the result from [18] is involved, and we preferred to give a
direct, function-theoretical argument.

Now we are ready to prove Theorem 1.3.

###### Proof of Theorem 1.3.

Fix $n\in\mathbb{N}.$ Using the partial fraction expansion of $r,$ define
$r^{n}(A/n)$ by the HP-calculus. Since $-A$ generates a bounded
$C_{0}$-semigroup on a Hilbert space, $A$ admits the $\mathcal{B}$-calculus.
So Theorem 1.3,(i) follows directly from Corollary 2.7, the invariance
property (2.2), and the norm-bound (3.4) for the $\mathcal{B}$-calculus.
(Instead of invoking (2.2) one may observe that $-tA/n$ generates a semigroup
with the uniform norm bound $M$ for all $t\geq 0$.)

To prove Theorem 1.3,(ii), fix $s\in(0,q+1]$ and $n\in\mathbb{N}.$ We define
$\Delta_{r,n}(A)$ by the HP-calculus. If $h_{s}(z):=z^{s},z\in\mathbb{C}_{+},$
then in view of $\eta_{s}\in\mathcal{LM}(\mathbb{C}_{+}),$ we have
$h_{s}\eta_{s+1}\in\mathcal{LM}(\mathbb{C}_{+})$, see e.g. [20, Lemma 3.3.1]
(or [18, Lemma 4.1]). Thus $h_{s}$ belongs to the extended HP-calculus with
the regulariser $\eta_{s+1}$. By Lemma 3.5, $\Delta_{r,n,s}$ belongs to the
HP-calculus, and thus $\Delta_{r,n,s}(A)$ and $A^{s}$ are defined by the
extended HP-calculus.

Using the product rule for the (extended) $\mathcal{B}$-calculus (see e.g.
[20, Proposition 1.2.2]), we have

(3.11) $\Delta_{r,n}(A)x=\Delta_{r,n,s}(A)A^{s}x,\qquad x\in{\rm dom}(A^{s}).$

In view of $\Delta_{r,n,s}(\infty)=0,$ Lemma 2.10 and (3.4) yield

(3.12) $\|\Delta_{r,n}(A)x\|\leq\frac{C\pi M^{2}}{n^{qs/(q+1)}}\|A^{s}x\|$

for all $x\in{\rm dom}(A^{s}).$ Replacing $A$ by $tA$ in (3.12), we obtain
(1.8). ∎

######  Remark 3.7.

Arguing as in the proof of Theorem 1.3,(ii), and using (2.22), we conclude
that, under the assumptions of Theorem 1.3, if $s>0$ then there exists
$C=C(s,r)>0$ such that

(3.13) $\|r^{n}(tA/n)x\|\leq C\pi M^{2}\|(1+tA)^{s}x\|$

for all $x\in{\rm dom}\,(A^{s})$ and $n\in\mathbb{N}.$ Thus the stability
properties of $(r(tA/n))_{n\geq 1}$ improve for fixed $t>0$. However, one can
show easily that a version of (3.13) with $tA$ replaced by $A$ in the left
hand side of (3.13) does not, in general, hold.

The proof of Theorem 1.4 is similar to the proof Theorem 1.3,(i), although it
relies on stronger assumptions on $r.$

###### Proof of Theorem 1.4.

Since $A$ admits the $\mathcal{B}$-calculus, the estimate (1.10) is a direct
implication of (2.23), (2.2) and (3.4). As noted in the introduction, if
(1.10) holds, then (1.11) follows from general theory. Alternatively, using
Theorem 1.3 (or Theorem 1.1), we note that

$\lim_{n\to\infty}\Delta_{r,n}(tA)x=0,\qquad x\in{\rm dom}(A),\quad t\geq 0,$

and (1.11) is then an immediate corollary of (1.10). ∎

It is crucial to emphasize that the estimates for approximation rates in
Theorem 1.3,(ii) are sharp in a sense clarified in Theorem 3.8 below. To this
end, recall that if $f\\\ in\mathcal{B}$ (in particular,
$f\in\mathcal{LM}(\mathbb{C}_{+})$) and $-A$ generates a bounded
$C_{0}$-semigroup on $X,$ then the next spectral inclusion theorem holds:

(3.14) $\\{f(\lambda):\lambda\in\sigma(A)\\}\subset\sigma(f(A)).$

See e.g. [3, Theorem 4.17, (3)] for (3.14) and more general statements.

######  Theorem 3.8.

Let $r$ be an $\mathcal{A}$-stable rational approximation of order $q$ to the
exponential, with $r^{(q+1)}(0)\neq(-1)^{q+1}.$ If $-A$ generates a bounded
$C_{0}$-semigroup on a Hilbert space $X,$ and
$i\mathbb{R}_{+}\subset\sigma(A),$ then there exist $C=C(r)>0$ and
$n_{0}=n_{0}(r)\in\mathbb{N}$ such that for every $s\in(0,q+1],$

$\|\Delta_{r,n,s}(tA)\|\geq\frac{Ct^{s}}{n^{sq/(q+1)}},\qquad n\geq
n_{0},\quad t\geq 0.$

###### Proof.

Let $s\in(0,q+1]$ be fixed, and as above $\delta=q/(q+1).$ Using Lemma
2.9,(ii) we infer that there exist $R\in(0,1),$ $d>0,$ $a\in\mathbb{C}_{+}$
different from zero, and a family of functions $\\{\xi(n,\cdot):n\geq 1\\}$
holomorphic on $\mathbb{D}_{Rn}$ such that

$\Delta_{r,n,s}(z)=\left(1-e^{az^{q+1}/n^{q}}e^{\xi(n,z)}\right){e^{-z}}{z^{-s}},\qquad|\xi(n,z)|\leq
d\frac{|z|^{q+2}}{n^{q+1}},$

for all $z\in\overline{\mathbb{D}}^{+}_{Rn^{\delta}}$ and $n\in\mathbb{N}.$

Choose $c\in(0,1)$ satisfying also $c\in(0,\min(R,\frac{2\pi}{|a|}))$ and set

$z_{n}=icn^{\delta},\qquad n\in\mathbb{N},$

so that
$(z_{n})_{n=1}^{\infty}\subset\overline{\mathbb{D}}^{+}_{Rn^{\delta}}.$ If
$b=a(ic)^{q+1},$ then

$|\Delta_{r,n,s}(z_{n})|=|1-e^{b}e^{\xi(n,icn^{\delta})}|c^{-s}n^{-\delta
s},\qquad n\in\mathbb{N},$

and, taking into account (4.13),

(3.15) $\displaystyle n^{\delta s}|\Delta_{r,n,s}(z_{n})|\geq$
$\displaystyle|1-e^{b}|-e^{b}|\xi(n,icn^{\delta})|e^{|\xi(n,icn^{\delta})|}$
$\displaystyle=$
$\displaystyle|1-e^{b}|+\mbox{O}\left(\frac{1}{n^{1/(q+1)}}\right),\qquad
n\to\infty,$

with $|1-e^{b}|\neq 0.$

Since $(z_{n}t^{-1})_{n\geq 1}\subset\sigma(A),$ using the spectral inclusion
in (3.14) we have

$\|\Delta_{r,n,s}(tA)\|\geq\sup_{z\in\sigma(A)}|\Delta_{r,n,s}(tz)|\geq
t^{s}|\Delta_{r,n,s}(z_{n})|,\qquad n\in\mathbb{N},$

and the estimate (3.15) implies the claim. ∎

As a simple example satisfying the assumptions of Theorem 3.8, one may
consider a selfadjoint operator $A$ with $\sigma(A)=[0,\infty).$

We finish the section with indicating formally the argument for the proof of
Theorem 1.5.

###### Proof of Theorem 1.5.

Fix $s>0$ and $n\geq(s-1)/2.$ The proof follows the same lines as the proof of
Theorem 1.3,(ii), with $\Delta_{r,n}$ and $\Delta_{r,n,s}$ replaced by
$\Delta_{r_{[n,n+1]}}$ and $\Delta_{r_{[n,n+1]},s}$, respectively. By invoking
the HP- and the $\mathcal{B}$\- calculi, we have the identity

$\Delta_{r_{[n,n+1]}}(A)x=\Delta_{r_{[n,n+1]},s}(A)A^{s}x,\qquad x\in{\rm
dom}(A^{s}).$

The statement is then a direct implication of Lemma 2.12, the fact that
$\Delta_{r_{[n,n+1]},s}(\infty)=0,$ and (3.4). ∎

##  4\. Appendix: Rates for rational approximation of exponential function

Here we prove Lemma 2.9 formulated in Section 2.1. Its proof is simple but
rather technical, and it is divided into several steps.

###### Proof of Lemma 2.9..

We will show that if $r$ is a rational approximation of order $q$ to the
exponential, i.e.

(4.1) $e^{-z}-r(z)=\mbox{O}(|z|^{q+1}),\qquad z\to 0,$

then, setting $s\in(0,q+1],$ $\delta=q/(q+1),$ and
$a=-\Delta^{(q)}_{r,n}((q+1)!)^{-1},$ there exist $R=R(r)\in(0,1)$ and
$d=d_{(}r)>0$ such that

(4.2)
$|(\Delta_{r,n,s}(z))^{\prime}|\leq\left(|a|(q+1-s)\frac{|z|^{q-s}}{n^{q}}+\frac{d}{n^{\delta
s}}\right)e^{-{\rm Re}\,z}$

for all $z\in\overline{\mathbb{D}}^{+}_{Rn^{\delta}}.$ This will provide the
proof of Lemma 2.9,(i), and on the way will prove Lemma 2.9,(ii) as well. The
proof will be done in several steps.

By (4.1) there are $R_{0}=R_{0}(r)\in(0,1)$ and $a\in\mathbb{C}$ such that
$r\in{\rm Hol}(\mathbb{D}_{R_{0}}),$ and

(4.3) $r(z)=e^{-z}(1+\psi(z)),\qquad z\in\mathbb{D}_{R_{0}},$

where

(4.4) $\psi(z)=az^{q+1}+m(z),$

with $m\in{\rm Hol}(\mathbb{D}_{R_{0}})$ and

(4.5) $|m(z)|\leq b|z|^{q+2},\qquad|m^{\prime}(z)|\leq b|z|^{q+1},\qquad
z\in\mathbb{D}_{R_{0}},$

for some $b>0.$ Note that for all $z\in\mathbb{D}_{R_{0}},$ we clearly have

(4.6) $|\psi(z)|\leq C|z|^{q+1}$

and

(4.7) $|\psi^{\prime}(z)|\leq C|z|^{q}$

for some $C=C(a,b,R_{0}).$ Moreover, …

To simplify further presentation, given $\gamma>0$ and positive functions $f$
and $g$ on $\mathbb{D}_{R}$ we use the notation $f(z)\lesssim g(z)$ for all
$z\in\mathbb{D}_{\gamma}$ if $f(z)\leq Cg(z),z\in\mathbb{D}_{\gamma},$ where a
constant $C>0$ depends only on parameters associated to $r$ (and independent
of $z$).

Observe that from (4.6) it follows that there exists $R\in(0,R_{0})$
satisfying

(4.8) $|\psi(z)|\leq\frac{1}{2},\qquad z\in\mathbb{D}_{R}.$

Using the elementary inequality

$|\log(1+z)-z|\leq|z|^{2},\qquad|z|\leq 1/2,$

the bound (4.8) and taking into account (4.5) and (4.6), we infer that for all
$z\in\mathbb{D}_{R},$

$\displaystyle|\log(1+\psi(z))-az^{q+1}|\leq$
$\displaystyle|m(z)|+|\psi(z)|^{2}$ $\displaystyle\lesssim$
$\displaystyle|z|^{q+2}+|z|^{2(q+1)}\lesssim|z|^{q+2}.$

Hence, if $z\in\mathbb{D}_{R}$ then e.g. by power series expansion there is
$m_{0}\in{\rm Hol}(\mathbb{D}_{R})$ such that

(4.9) $\log(1+\psi(z))=az^{q+1}+m_{0}(z),\qquad|m_{0}(z)|\lesssim|z|^{q+2},$

and, in view of (4.3),

(4.10) $r(z)=e^{-z}e^{az^{q+1}}e^{m_{0}(z)}.$

Hence if $n\in\mathbb{N}$ is fixed, and

$r_{n}(z)=r^{n}(z/n),\qquad z\in\mathbb{D}_{Rn},$

then, by (4.10), we have

(4.11) $r_{n}(z)=e^{-z}e^{az^{q+1}/n^{q}}e^{m_{0,n}(z)},$

where $m_{0,n}\in{\rm Hol}(\mathbb{D}_{Rn})$ is defined by $m_{0,n}(z):=n\cdot
m_{0}(z/n),z\in\mathbb{D}_{Rn},$ and satisfies

(4.12) $|m_{0,n}(z)|\lesssim\frac{|z|^{q+2}}{n^{q+1}},\qquad
z\in\mathbb{D}_{Rn^{\delta}}.$

If $r^{(q+1)}(0)\neq(-1)^{q+1}$, then $a\neq 0.$ Letting
$\xi(n,z)=m_{0,n}(z),z\in\mathbb{D}_{Rn^{\delta}},$ and noting that
$\Delta_{r,n,s}$ extends continuously to $[-iRn^{\delta},iRn^{\delta}],$ we
obtain Lemma 2.9,(ii).

Next, using the estimates

(4.13)
$|e^{z}-1|\leq|z|e^{|z|},\qquad|e^{z}-1-z|\leq\frac{|z|^{2}}{2}e^{|z|},\qquad
z\in\mathbb{C},$

and (4.11), we define $m_{1,n}\in{\rm Hol}(\mathbb{D}_{Rn})$ by

(4.14) $r_{n}(z)=(1+m_{1,n}(z))e^{-z},$

and note that

(4.15)
$|m_{1,n}(z)|\leq(|az^{q+1}/n^{q}|+|m_{0,n}(z)|)e^{|a||z|^{q+1}/n^{q}+|m_{0,n}(z)|}\lesssim\frac{|z|^{q+1}}{n^{q}}$

for all $z\in\mathbb{D}_{Rn^{\delta}}.$ Similarly, if $m_{2,n}\in{\rm
Hol}(\mathbb{D}_{Rn})$ is given by

(4.16) $r_{n}(z)=(1+az^{q+1}/n^{q}+m_{2,n}(z))e^{-z},$

then for every $z\in\mathbb{D}_{Rn^{\delta}},$

$\displaystyle|m_{2,n}(z)|\leq$
$\displaystyle|m_{0,n}(z)|+\left(|az^{q+1}/n^{q}|^{2}+|m_{0,n}(z)|^{2}\right)e^{|a||z|^{q+1}/n^{q}+|m_{0,n}(z)|}$
$\displaystyle\lesssim$
$\displaystyle\frac{|z|^{q+2}}{n^{q+1}}+\frac{|z|^{2(q+1)}}{n^{2q}}+\frac{|z|^{2(q+2)}}{n^{2(q+1)}}\lesssim\frac{|z|^{q+2}}{n^{q}}.$

Furthermore, recalling that $\Delta_{r,n}(z)=e^{-z}-r_{n}(z),$ and using
(4.14) and (4.16), note that

(4.17) $\Delta_{r,n}(z)=-e^{-z}m_{0,n}(z),$

and

(4.18)
$\Delta_{r,n}(z)=-e^{-z}\left(a\frac{z^{q+1}}{n^{q}}+m_{2,n}(z)\right).$

Since by (4.3),

$\frac{r^{\prime}(z)}{r(z)}=-1+\frac{\psi^{\prime}(z)}{1+\psi(z)},$

we have

$\displaystyle\frac{r_{n}^{\prime}(z)}{r_{n}(z)}=$
$\displaystyle-1+\psi^{\prime}(z/n)\left(1-\frac{\psi(z/n)}{1+\psi(z/n)}\right).$

Thus from (4.4), (4.14) and (4.17) it follows that

(4.19) $\displaystyle(\Delta_{r,n}(z))^{\prime}$ $\displaystyle=$
$\displaystyle-\Delta_{r,n}(z)-r_{n}(z)\psi^{\prime}(z/n)\left(1-\frac{\psi(z/n)}{1+\psi(z/n)}\right)$
$\displaystyle=$ $\displaystyle
m_{1,n}(z)e^{-z}-(1+m_{1,n}(z))\left(a(q+1)\frac{z^{q}}{n^{q}}+m^{\prime}(z/n)\right)\left(1-\frac{\psi(z/n)}{1+\psi(z/n)}\right)e^{-z}$
$\displaystyle=$
$\displaystyle-a(q+1)\frac{z^{q}}{n^{q}}e^{-z}+m_{3,n}(z)e^{-z},$

where $m_{3,n}\in{\rm Hol}(\mathbb{D}_{Rn})$ is defined by

$\displaystyle m_{3,n}(z):=$ $\displaystyle
m_{1,n}(z)-m_{1,n}(z)\left(a(q+1)\frac{z^{q}}{n^{q}}+m^{\prime}(z/n)\right)\frac{1}{1+\psi(z/n)}$
$\displaystyle+$
$\displaystyle\left(a(q+1)\frac{z^{q}}{n^{q}}+m^{\prime}(z/n)\right)\frac{\psi(z/n)}{1+\psi(z/n)}-m^{\prime}(z/n),\qquad
z\in\mathbb{D}_{Rn}.$

Taking into account $|z|\lesssim n$, $z\in\mathbb{D}_{Rn^{\delta}}$, and
employing (4.5), (4.6), (4.8), and (4.15), we have

(4.20) $\displaystyle|m_{3,n}(z)|\lesssim$
$\displaystyle\frac{|z|^{q+1}}{n^{q}}\left(1+\frac{|z|^{q}}{n^{q}}+\frac{|z|^{q+1}}{n^{q+1}}\right)$
$\displaystyle+$
$\displaystyle\left(\frac{|z|^{q}}{n^{q}}+\frac{|z|^{q+1}}{n^{q+1}}\right)\frac{|z|^{q+1}}{n^{q+1}}+\frac{|z|^{q+1}}{n^{q+1}}$
$\displaystyle\lesssim$ $\displaystyle\frac{|z|^{q+1}}{n^{q}},\qquad
z\in\mathbb{D}_{Rn^{\delta}}.$

Therefore, using (4.16), (4.19), and (4.20), we infer that for every
$z\in\mathbb{D}^{+}_{Rn},$

(4.21) $\displaystyle(\Delta_{r,n,s}(z))^{\prime}=$
$\displaystyle\frac{\Delta_{r,n}(z)^{\prime}}{z^{s}}-s\frac{\Delta_{r,n}(z)}{z^{s+1}}$
$\displaystyle=$ $\displaystyle
a(s-(q+1))\frac{z^{q-s}}{n^{q}}e^{-z}+\frac{m_{3,n}(z)}{z^{s}}e^{-z}+m_{2,n}(z)\frac{s}{z^{s+1}}e^{-z}$
$\displaystyle=$ $\displaystyle
a(s-(q+1))\frac{z^{q-s}}{n^{q}}e^{-z}+m_{4,n}(z)e^{-z},$

with $m_{4,n}\in{\rm Hol}(\mathbb{D}^{+}_{Rn})$ satisfying

$|m_{4,n}(z)|\lesssim\frac{|z|^{q+1-s}}{n^{q}},\qquad
z\in\mathbb{D}^{+}_{Rn^{\delta}}.$

Recalling the relation $(q+1)\delta=q$, observe that

(4.22) $\frac{|z|^{q+1-s}}{n^{q}}\leq
R^{q+1-s}\frac{n^{\delta(q+1-s)}}{n^{q}}=\frac{R^{q+1-s}}{n^{\delta s}}.$

Hence, taking into account that $|e^{-z}|\leq 1$ for $z\in\mathbb{C}_{+},$
(4.21) and (4.22) imply (4.2). ∎

##  5\. Acknowledgment

We would like to thank the referee for various useful suggestions and remarks
that led to improvement of the paper.

## References

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