Decay Rate of $\varvec{\exp (A^{-1}t)A^{-1}}$ on a Hilbert Space and the Crank–Nicolson Scheme with Smooth Initial Data

Abstract

This paper is concerned with the decay rate of $e^{A^{-1}t}A^{-1}$ for the generator A of an exponentially stable $C_0$-semigroup on a Hilbert space. To estimate the decay rate of $e^{A^{-1}t}A^{-1}$, we apply a bounded functional calculus. Using this estimate and Lyapunov equations, we also study the quantified asymptotic behavior of the Crank-Nicolson scheme with smooth initial data. A similar argument is applied to a polynomially stable $C_0$-semigroup whose generator is normal.

1 Introduction

Let A be the generator of a bounded $C_0$-semigroup $(e^{At})_{t\ge 0}$ on a Hilbert space. Suppose that the inverse $A^{-1}$ of A exists and also generates a $C_0$-semigroup $(e^{A^{-1}t})_{t\ge 0}$. The problem we study first is to estimate the decay rate of $\Vert e^{A^{-1}t}A^{-1}\Vert$ under additional assumptions on stability of $(e^{At})_{t\ge 0}$. This is a variant of the so-called inverse generator problem raised by deLaubenfels [9]: “Let A be the generator of a bounded $C_0$-semigroup on a Banach space. Assume that there exists an inverse $A^{-1}$ as a closed, densely-defined operator. Does $A^{-1}$ also generate a bounded $C_0$-semigroup?” Positive answers to the inverse generator problem have been given for bounded holomorphic $C_0$-semigroups on Banach spaces [9] and for contraction $C_0$-semigroups on Hilbert spaces [29]. Negative examples can be found in [20] with respect to the generation property of $A^{-1}$ and in [29] with respect to the boundedness property of $(e^{A^{-1}t})_{t\ge 0}$; see also [11, 16] for other counter-examples. For bounded $C_0$-semigroups on Hilbert spaces, the answer to the inverse generator problem is still unknown, but it has been shown in [28] that if A generates an exponentially stable $C_0$-semigroup on Hilbert spaces, then $\Vert e^{A^{-1}t}\Vert = O(\log t)$ as $t \rightarrow \infty$, that is, there exist $M > 0$ and $t_0 > 1$ such that $\Vert e^{A^{-1}t}\Vert \le M \log t$ for all $t \ge t_0$. This result has been extended to bounded $C_0$-semigroups with boundedly invertible generators in [5]. We refer the reader to the survey article [14] for more detailed discussion of the inverse generator problem.

It is known that if A generates an exponentially stable $C_0$-semigroup on a Banach space, then

$$\begin{aligned} \big \Vert e^{A^{-1}t} A^{-k}\big \Vert = O \left( \frac{1}{t^{k/2 - 1/4}} \right) \qquad (t \rightarrow \infty ) \end{aligned}$$

(1.1)

for all $k \in {\mathbb {N}}$. This norm-estimate has been obtained in [29] for $k=1$ and in [10] for $k \ge 2$. In this paper, we obtain an analogous estimate for the Hilbert case: if A generates an exponentially stable $C_0$-semigroup on a Hilbert space, then

$$\begin{aligned} \big \Vert e^{A^{-1}t} A^{-k}\big \Vert = O \left( \frac{1}{t^{k/2}} \right) \qquad (t \rightarrow \infty ) \end{aligned}$$

(1.2)

for all $k \in {\mathbb {N}}$. To obtain the estimate (1.1) of the Banach case, an integral representation of $(e^{A^{-1}t})_{t \ge 0}$ (see [29, Lemma 3.2] and [16, Theorem 1]) has been applied. In contrast, to obtain the estimate (1.2) of the Hilbert case, we employ a functional-calculus approach based on the ${\mathcal {B}}$-calculus introduced in [5]. Another class of $C_0$-semigroups we study are polynomially stable $C_0$-semigroups with parameter $\beta >0$ on Hilbert spaces, where $(e^{At})_{t\ge 0}$ is called a polynomially stable $C_0$-semigroup with parameter $\beta >0$ if $(e^{At})_{t\ge 0}$ is bounded and if $\Vert e^{At}(I-A)^{-1}\Vert = O(t^{-1/\beta })$ as $t \rightarrow \infty$. By a functional-calculus approach as in the case for exponentially stable $C_0$-semigroups, we show that if, in addition, A is a normal operator, then

$$\begin{aligned} \big \Vert e^{A^{-1}t} A^{-k}\big \Vert = O \left( \frac{1}{t^{k/(2+\beta )}} \right) \qquad (t \rightarrow \infty ) \end{aligned}$$

(1.3)

for all $k \in {\mathbb {N}}$. We also present simple examples for which the norm-estimates (1.2) and (1.3) cannot be improved. Moreover, we prove that the (not necessarily normal) generator A of a polynomially stable $C_0$-semigroup with any parameter on a Hilbert space satisfies $\sup _{t \ge 0} \Vert e^{A^{-1}t}A^{-1}\Vert < \infty$, inspired by the Lyapunov-equation technique developed in [28].

Next, we study the Crank–Nicolson discretization scheme with variable stepsizes. Let A be the generator of a bounded $C_0$-semigroup on a Hilbert space, and let $\tau _n$, $n \in {\mathbb {N}}_0:= \{ 0,1,2,\ldots \}$, satisfy $\tau _{\min } \le \tau _n \le \tau _{\max }$ for some $\tau _{\max } \ge \tau _{\min } >0$. Consider the time-varying difference equation

$$\begin{aligned} x_{n+1} = A_d(\tau _n)x_n,\quad n \in {\mathbb {N}}_0, \end{aligned}$$

(1.4)

where $A_d(\tau )$ is the Cayley transform of A with parameter $\tau >0$, i.e.,

$$\begin{aligned} A_d(\tau ):= \left( I + \frac{\tau }{2}A \right) \left( I - \frac{\tau }{2}A \right) ^{-1}. \end{aligned}$$

It is our aim to estimate the decay rate of the solution $(x_n)_{n \in {\mathbb {N}}}$ with smooth initial data $x_0 \in D(A)$.

For a constant stepsize $\tau >0$, upper bounds on the growth rate of $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$ have been obtained for bounded $C_0$-semigroups and exponentially stable $C_0$-semigroups on Banach spaces; see [6, 15]. The relation between $(e^{A^{-1}t})_{t \ge 0}$ and $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$ in terms of growth rates has been investigated in [15]. As in the inverse generator problem, $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$ is bounded whenever $C_0$-semigroups are bounded holomorphic on Banach spaces [7]. Furthermore, it is well known that for contraction $C_0$-semigroups on Hilbert spaces, $A_d(\tau )$ is a contraction; see [22]. It is still open whether $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$ is bounded for every bounded $C_0$-semigroup on a Hilbert space. However, it has been shown independently in [1, 13, 17] that if A and $A^{-1}$ both generate bounded $C_0$-semigroups on a Hilbert space, then $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$ is bounded. If in addition $(e^{At})_{t\ge 0}$ is strongly stable, then $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$ is strongly stable [17]. Without the assumption on $A^{-1}$, the norm-estimate $\Vert A_d(\tau )^n\Vert = O(\log n)$ as $n \rightarrow \infty$ given in [13] remains the best so far in the Hilbert case. For polynomially stable $C_0$-semigroups with parameter $\beta >0$ on Hilbert spaces, the following estimates have been obtained in [25]:

• $\Vert A_d(\tau )^nA^{-1}\Vert = O\bigg (\left( \dfrac{\log n}{n}\right) ^{1/(2+\beta )}\bigg )$ if $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$ is bounded.

• $\Vert A_d(\tau )^nA^{-1}\Vert = O\left( \dfrac{1}{n^{1/(2+\beta )}} \right)$ if A is normal.

For more information on asymptotics of $(A_d(\tau )^n)_{n \in {\mathbb {N}}_0}$, we refer to the survey [14].

Some of the above results have been extended to the case of variable stepsizes. For bounded $C_0$-semigroups on Banach spaces, the Crank–Nicolson scheme with variable stepsizes has the same growth bound as that with constant stepsizes; see [2]. It has been proved that the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the time-varying difference equation (1.4) is bounded if A generates a bounded holomorphic $C_0$-semigroups on a Banach space [21, 23, 24] or if A and $A^{-1}$ generate bounded $C_0$-semigroups on a Hilbert space [23]. The stability property, i.e., $\lim _{n \rightarrow \infty } x_n = 0$ for all initial data $x_0$, has been also obtained under some additional assumptions in [23].

We show that if A and $A^{-1}$ are the generators of an exponentially stable $C_0$-semigroup and a bounded $C_0$-semigroup on a Hilbert space, respectively, then the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the time-varying difference equation (1.4) with $x_0 \in D(A)$ satisfies

$$\begin{aligned} \Vert x_n\Vert = O\left( \sqrt{\frac{\log n}{n}} \right) \qquad (n \rightarrow \infty ). \end{aligned}$$

(1.5)

Moreover, if A is a normal operator generating a polynomially stable $C_0$-semigroup with parameter $\beta >0$ on a Hilbert space, then

$$\begin{aligned} \Vert x_n\Vert = O\left( \frac{1}{n^{1/(2+\beta )}} \right) \qquad (n \rightarrow \infty ) \end{aligned}$$

(1.6)

for all $x_0 \in D(A)$. These estimates (1.5) and (1.6) follow from the combination of the norm-estimates (1.2) and (1.3) of $e^{A^{-1}t}A^{-1}$ and the Lyapunov-equation technique developed in [23]. In the constant case $\tau _n \equiv \tau >0$, we give a simple example of an exponentially stable $C_0$-semigroup showing that the decay rate $\Vert A_d(\tau )^n A^{-1}\Vert = O(1/\sqrt{n})$ cannot in general be improved. The estimate (1.5) includes the logarithmic factor $\sqrt{\log n}$, but its necessity remains open. On the other hand, the decay rate given in (1.6) is the same as that obtained for constant stepsizes in [25], and one cannot in general replace $n^{-1/(2+\beta )}$ in (1.6) by functions with better decay rates.

This paper is organized as follows. In Sect. 2, we give long-time norm-estimates for $e^{A^{-1}t}A^{-1}$. We consider first exponentially stable $C_0$-semigroups and then polynomially stable $C_0$-semigroups. In Sect. 3, the decay rate estimates of $\Vert e^{A^{-1}t}A^{-1}\Vert$ are utilized in order to examine the quantified asymptotic behavior of the Crank–Nicolson scheme with smooth initial data.

Notation Let ${\mathbb {C}}_+:= \{ \lambda \in {\mathbb {C}}: \mathop {\textrm{Re}}\nolimits \lambda >0 \}$, $\overline{{\mathbb {C}}}_+:= \{ \lambda \in {\mathbb {C}}: \mathop {\textrm{Re}}\nolimits \lambda \ge 0 \}$, ${\mathbb {C}}_-:= \{ \lambda \in {\mathbb {C}}: \mathop {\textrm{Re}}\nolimits \lambda <0 \}$, and $i {\mathbb {R}}:= \{ i \eta : \eta \in {\mathbb {R}} \}$. Let ${\mathbb {N}}_0$ and ${\mathbb {R}}_+$ be the set of nonnegative integers and the set of nonnegative real numbers, respectively. For real-valued functions f, g on $J \subset {\mathbb {R}}$, we write

$$\begin{aligned} f(t) = O\big (g(t)\big )\qquad (t \rightarrow \infty ) \end{aligned}$$

if there exist constants $M >0$ and $t_0 \in J$ such that $f(t) \le M g(t)$ for all $t \in J$ satisfying $t \ge t_0$. The gamma function is denoted by $\Gamma$, i.e.,

$$\begin{aligned} \Gamma (\alpha ) = \int ^{\infty }_0 t^{\alpha -1} e^{-t}dt,\quad \alpha > 0. \end{aligned}$$

Let H be a Hilbert space. The inner product of H is denoted by $\langle \cdot , \cdot \rangle$. The space of bounded linear operators on H is denoted by ${\mathcal {L}}(H)$. For a linear operator A on H, let D(A) and $\sigma (A)$ denote the domain and the spectrum of A, respectively. For a densely-defined linear operator A on H, we denote by $A^*$ the Hilbert space adjoint of A. If A is the generator of a bounded $C_0$-semigroup on a Hilbert space and if A is injective, the fractional power $(-A)^{\alpha }$ of $-A$ is defined by the sectorial functional calculus for $\alpha \in {\mathbb {R}}$ as in [18, Chapter 3]. Let $\ell ^2$ be the space of complex-valued square-summable sequences $x = (x_n)_{n \in {\mathbb {N}}}$ endowed with the norm $\Vert x\Vert := \sqrt{\sum _{n=1}^{\infty } |x_n|^2}$.

2 Norm-Estimates of Semigroups Generated by Inverse Generators

The aim of this section is to derive a norm-estimate for $e^{A^{-1}t} (-A)^{-\alpha }$ with $\alpha >0$. First we assume that A is the generator of an exponentially stable $C_0$-semigroup on a Hilbert space. Next we focus on the generator A of a polynomially stable $C_0$-semigroup on a Hilbert space.

2.1 Inverses of Generators of Exponentially Stable Semigroups

Let A be the generator of an exponentially stable $C_0$-semigroup on a Hilbert space. Then A is invertible, and $A^{-1}$ generates a $C_0$-semigroup $(e^{A^{-1}t})_{t \ge 0}$. To obtain a norm-estimate for $e^{A^{-1}t} (-A)^{-\alpha }$, we employ the ${\mathcal {B}}$-calculus introduced in [5].

2.1.1 Basic Facts on $\varvec{{\mathcal {B}}}$-Calculus

Let ${\mathcal {B}}$ be the space of all holomorphic functions f on ${\mathbb {C}}_+$ such that

$$\begin{aligned} \Vert f\Vert _{{\mathcal {B}}_0}:= \int ^{\infty }_0 \sup _{\eta \in {\mathbb {R}}} |f'(\xi +i\eta )|d\xi < \infty . \end{aligned}$$

We recall elementary properties of functions f in ${\mathcal {B}}$. The proof can be found in [5, Proposition 2.2].

Proposition 2.1

For $f \in {\mathcal {B}}$, the following statements hold:

• (i) $f(\infty ):= \lim _{\mathop {\textrm{Re}}\nolimits z \rightarrow \infty } f(z)$ exists in ${\mathbb {C}}$.

• (ii) f is bounded on ${\mathbb {C}}_+$, and $\Vert f\Vert _{\infty }:= \sup _{z \in {\mathbb {C}}_+} |f(z)| \le |f(\infty )| + \Vert f\Vert _{{\mathcal {B}}_0}$.

• (iii) $f(i\eta ):= \lim _{\xi \rightarrow 0 +} f(\xi +i\eta )$ exists, uniformly for $\eta \in {\mathbb {R}}$.

A norm on ${\mathcal {B}}$ is defined by

$$\begin{aligned} \Vert f\Vert _{{\mathcal {B}}}:= \Vert f\Vert _{\infty } + \Vert f\Vert _{{\mathcal {B}}_0}, \end{aligned}$$

which is equivalent to each norm of the form

$$\begin{aligned} |f(z_0)| + \Vert f\Vert _{{\mathcal {B}}_0}\qquad (z_0 \in \overline{{\mathbb {C}}}_+ \cup \{ \infty \}). \end{aligned}$$

The space ${\mathcal {B}}$ equipped with the norm $\Vert \cdot \Vert _{{\mathcal {B}}}$ is a Banach algebra by [5, Proposition 2.3].

Let $\text {M}({\mathbb {R}}_+)$ be the Banach algebra of all bounded Borel measures on ${\mathbb {R}}_+$ under convolution, endowed with the norm $\Vert \mu \Vert _{\text {M}({\mathbb {R}}_+)}:= |\mu |({\mathbb {R}}_+)$, where $|\mu |$ is the total variation of $\mu$. For $\mu \in \text {M}({\mathbb {R}}_+)$, the Laplace transform of $\mu$ is the function

$$\begin{aligned} {\mathcal {L}}\mu :\overline{{\mathbb {C}}}_+ \rightarrow {\mathbb {C}},\quad ({\mathcal {L}}\mu )(z) = \int _{{\mathbb {R}}_+} e^{-zt} \mu (dt). \end{aligned}$$

We define

$$\begin{aligned} {{\mathcal {L}}}{{\mathcal {M}}}:= \{ {\mathcal {L}}\mu : \mu \in \text {M}({\mathbb {R}}_+) \}. \end{aligned}$$

Then the space ${{\mathcal {L}}}{{\mathcal {M}}}$ endowed with the norm $\Vert {\mathcal {L}}\mu \Vert _{\text {HP}}:= \Vert \mu \Vert _{\text {M}({\mathbb {R}}_+)}$ becomes a Banach algebra. As shown on [5, p. 42], ${{\mathcal {L}}}{{\mathcal {M}}}$ is a subspace of ${\mathcal {B}}$ with continuous inclusion.

Let $-B$ be the generator of a bounded $C_0$-semigroup $(e^{-Bt})_{t \ge 0}$ on a Hilbert space H. Define $K:= \sup _{t \ge 0} \Vert e^{-Bt}\Vert$. Using Plancherel’s theorem and the Cauchy-Scwartz inequality, we obtain

$$\begin{aligned} \int _{{\mathbb {R}}} \left| \langle (\xi + i \eta +B)^{-2}x,y \rangle \right| d\eta \le \frac{\pi K^2}{\xi } \Vert x\Vert \, \Vert y\Vert \end{aligned}$$

for all $x,y \in H$ and $\xi >0$; see [5, Example 4.1]. From this observation, we define

$$\begin{aligned}{} & {} \langle f(B)x,y \rangle \nonumber \\{} & {} := f(\infty ) \langle x,y \rangle - \frac{2}{\pi } \int ^{\infty }_0 \xi \int _{{\mathbb {R}}} \langle (\xi - i\eta + B)^{-2} x, y \rangle f'(\xi +i\eta ) d\eta d\xi \end{aligned}$$

(2.1)

for $f \in {\mathcal {B}}$ and $x,y \in H$. This definition yields a bounded functional calculus; see [5, Theorem 4.4] for the proof.

Theorem 2.2

Let $-B$ be the generator of a bounded $C_0$-semigroup $(e^{-Bt})_{t \ge 0}$ on a Hilbert space H, and let f(B) be defined as in (2.1). Then the following statements hold:

• (i) The map $\Phi _B:f \mapsto f(B)$ is a bounded algebra homomorphism from ${\mathcal {B}}$ into ${\mathcal {L}}(H)$.

• (ii) If $f = {\mathcal {L}}\mu \in {{\mathcal {L}}}{{\mathcal {M}}}$, then f(B) coincides with the operator

  $$\begin{aligned} x \mapsto \int _{{\mathbb {R}}_+} e^{-Bt}x \mu (dt) \end{aligned}$$

  as defined in the Hille-Phillips calculus.

• (iii) If $K:= \sup _{t \ge 0} \Vert e^{-Bt}\Vert$, then

  $$\begin{aligned} \Vert f(B)\Vert \le |f(\infty )| + 2K^2\Vert f\Vert _{{\mathcal {B}}_0} \le 2K^2\Vert f\Vert _{{\mathcal {B}}} \end{aligned}$$

  for all $f \in {\mathcal {B}}$.

The map

$$\begin{aligned} \Phi _B: {\mathcal {B}} \rightarrow {\mathcal {L}}(H),\quad f \mapsto f(B) \end{aligned}$$

is called the ${\mathcal {B}}$-calculus for B.

2.1.2 Norm-Estimate by $\varvec{{\mathcal {B}}}$-Calculus

By Theorem 2.2, the norm-estimate for a function $f \in {\mathcal {B}}$ implies the norm-estimate for the corresponding operator f(B). In [5, Lemma 3.4], the estimate

$$\begin{aligned} \Vert h_t\Vert _{{\mathcal {B}}} = O\left( \log t\right) \qquad (t \rightarrow \infty ),\qquad \text {where}\ h_t(z) := e^{-t/(z+1)}, \end{aligned}$$

(2.2)

has been derived for the norm-estimate of $e^{A^{-1}t}$. To estimate $\Vert e^{A^{-1}t} (-A)^{-\alpha }\Vert$ for $\alpha >0$, the following result is used.

Lemma 2.3

For $t,\alpha > 0$, define

$$\begin{aligned} f_{t,\alpha }(z):= \frac{e^{-t/(z+1)}}{(z+1)^\alpha },\quad z \in {\mathbb {C}}_+. \end{aligned}$$

Then $f_{t,\alpha } \in {\mathcal {B}}$ for all $t,\alpha >0$. Moreover,

$$\begin{aligned} \Vert f_{t,\alpha }\Vert _{{\mathcal {B}}_0} = O\left( \frac{1}{t^{\alpha /2}}\right) \qquad (t \rightarrow \infty ) \end{aligned}$$

(2.3)

for each $\alpha >0$.

Proof

For $t >0$ and $\alpha > 1$, define

$$\begin{aligned} F_\alpha (t):= \int ^{\infty }_0 \sup _{\eta \in {\mathbb {R}}} \left| \frac{e^{-t/(\xi +i\eta +1)}}{(\xi +i\eta +1)^\alpha }\right| d\xi . \end{aligned}$$

Since

$$\begin{aligned} f_{t,\alpha }'(z) = \frac{te^{-t/(z+1)} }{(z+1)^{\alpha +2}} - \frac{\alpha e^{-t/(z+1)} }{(z+1)^{\alpha +1}} \end{aligned}$$

for all $z \in {\mathbb {C}}_+$, we obtain

$$\begin{aligned} \int ^{\infty }_0 \sup _{\eta \in {\mathbb {R}}} |f_{t,\alpha }'(\xi +i\eta )|d\xi \le tF_{\alpha +2}(t) + \alpha F_{\alpha +1}(t) \end{aligned}$$

(2.4)

for all $t,\alpha > 0$.

Let $\alpha >1$. Define

$$\begin{aligned} g_{t,s,\alpha }(r):= \frac{e^{-ts/(s^2+r)}}{(s^2+r)^{\alpha /2}} \end{aligned}$$

for $t>0$, $s >1$, and $r \ge 0$. Then

$$\begin{aligned} F_{\alpha }(t) = \int ^{\infty }_1 \sup _{r \ge 0} g_{t,s,\alpha }(r)ds \end{aligned}$$

for all $t > 0$. Put $c:= \left( \alpha /(2e)\right) ^{\alpha /2}$. We have that for each $s > 1$,

$$\begin{aligned} \sup _{r \ge 0} g_{t,s,\alpha }(r) = {\left\{ \begin{array}{ll} g_{t,s,\alpha }(0) = \dfrac{e^{-t/s}}{s^\alpha }, &{} 0 < t \le \dfrac{\alpha s}{2}, \vspace{6pt}\\ g_{t,s,\alpha }\left( \dfrac{2ts}{\alpha }-s^2 \right) = \dfrac{c}{(ts)^{\alpha /2}}, &{} t \ge \dfrac{\alpha s}{2} > \dfrac{\alpha }{2}. \end{array}\right. } \end{aligned}$$

If $0 < t \le \alpha /2$, then

$$\begin{aligned} \int ^{\infty }_1 \sup _{r \ge 0} g_{t,s,\alpha }(r)ds&= \int ^{\infty }_1 \frac{e^{-t/s}}{s^\alpha }ds = \frac{1}{t^{\alpha -1}} \int ^t_0 u^{\alpha -2}e^{-u} du \le \frac{\Gamma (\alpha -1)}{t^{\alpha -1}}. \end{aligned}$$

If $t > \alpha /2$, then

$$\begin{aligned} \int ^{\infty }_1 \sup _{r \ge 0} g_{t,s,\alpha }(r)ds&= \frac{c}{t^{\alpha /2}} \int ^{2t/\alpha }_{1} \frac{1}{s^{\alpha /2}}ds + \int ^{\infty }_{2t/\alpha } \frac{e^{-t/s}}{s^\alpha } ds. \end{aligned}$$

We write the first term on the right-hand side as

$$\begin{aligned} \frac{c}{t^{\alpha /2}} \int ^{2t/\alpha }_{1} \frac{1}{s^{\alpha /2}}ds&= {\left\{ \begin{array}{ll} \dfrac{c \log t}{t}, &{}{} \alpha = 2, \\ \dfrac{c}{\alpha /2-1} \left( \dfrac{1}{t^{\alpha /2}} - \dfrac{(\alpha /2)^{\alpha /2-1}}{t^{\alpha -1}} \right) , &{}{} \alpha \not =2. \end{array}\right. }\end{aligned}$$

The second term satisfies

$$\begin{aligned} \int ^{\infty }_{2t/\alpha } \frac{e^{-t/s}}{s^\alpha } ds = \frac{1}{t^{\alpha -1}} \int ^{\alpha /2}_0 u^{\alpha -2}e^{-u} du \le \frac{\Gamma (\alpha -1)}{t^{\alpha -1}}. \end{aligned}$$

Therefore, $(0\le )\, F_\alpha (t) < \infty$ for each $t >0$, and

$$\begin{aligned} F_\alpha (t) = {\left\{ \begin{array}{ll} O\left( \dfrac{1}{t^{\alpha -1}}\right) , &{}{} 1<\alpha < 2, \\ O\left( \dfrac{\log t}{t}\right) , &{}{} \alpha = 2,\\ O\left( \dfrac{1}{t^{\alpha /2}}\right) , &{}{} \alpha >2. \end{array}\right. } \end{aligned}$$

(2.5)

From the estimate (2.4), we obtain $f_{t,\alpha } \in {\mathcal {B}}$ for all $t,\alpha >0$. Since (2.5) yields

$$\begin{aligned} tF_{\alpha +2}(t) + \alpha F_{\alpha +1}(t) = O\left( \frac{1}{t^{\alpha /2}}\right) \qquad (t \rightarrow \infty ) \end{aligned}$$

for all $\alpha >0$, we obtain the desired estimate (2.3). $\square$

In [5, Corollary 5.7], the ${\mathcal {B}}$-calculus and the norm-estimate (2.2) of $h_t(z) = e^{-t/(z+1)}$ have been used to obtain the norm-estimate for $e^{A^{-1}t}$. Analogously, the following theorem gives the norm-estimate for $e^{A^{-1}t}(-A)^{-\alpha }$ with $\alpha > 0$.

Theorem 2.4

Let A be the generator of an exponentially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$ on a Hilbert space H. Then

$$\begin{aligned} \big \Vert e^{A^{-1}t} (-A)^{-\alpha }\big \Vert = O\left( \frac{1}{t^{\alpha /2}} \right) \qquad (t \rightarrow \infty ) \end{aligned}$$

(2.6)

for all $\alpha > 0$.

Proof

There exist constants $K,\omega >0$ such that

$$\begin{aligned} \big \Vert e^{tA}\big \Vert \le Ke^{-\omega t} \end{aligned}$$

for all $t \ge 0$. We may assume that $\omega = 1$ by replacing A by $\omega ^{-1}A$ and t by $\omega t$. Note that $B:= -A-I$ generates a bounded $C_0$-semigroup $(e^{-tB})_{t \ge 0}$ with $\sup _{t \ge 0} \Vert e^{-tB}\Vert \le K$.

Take $t,\alpha > 0$ arbitrarily, and consider the functions

$$\begin{aligned} h_t(z):= e^{-t/(z+1)},\quad r_\alpha (z):= \frac{1}{(z+1)^{\alpha }},\quad z \in {\mathbb {C}}_+. \end{aligned}$$

Then $h_t$ belongs to ${{\mathcal {L}}}{{\mathcal {M}}} \subset {\mathcal {B}}$; see [5, Example 2.12]. From the definition of the gamma function, we have that for all $z >0$,

$$\begin{aligned} \Gamma (\alpha ) = \int ^{\infty }_0 t^{\alpha -1} e^{-t}dt = (z+1)^{\alpha } \int ^{\infty }_0 t^{\alpha -1} e^{-(z+1)t}dt. \end{aligned}$$

Hence, the uniqueness theorem for holomorphic functions yields

$$\begin{aligned} r_{\alpha } (z) = \frac{1}{\Gamma (\alpha )} \int ^{\infty }_0 t^{\alpha -1}e^{-(z+1)t} dt \end{aligned}$$

for all $z\in {\mathbb {C}}_+$. This means that $r_{\alpha }$ is the Laplace transform of the function

$$\begin{aligned} t\mapsto \frac{t^{\alpha -1} e^{-t}}{\Gamma (\alpha )},\quad t > 0. \end{aligned}$$

Hence $r_{\alpha }$ also belongs to ${{\mathcal {L}}}{{\mathcal {M}}} \subset {\mathcal {B}}$.

Theorem 2.2 (i) shows that the function $f_{t,\alpha } =h_t r_{\alpha }$ satisfies

$$\begin{aligned} f_{t,\alpha }(B) = h_t(B)r_{\alpha }(B). \end{aligned}$$

We obtain

$$\begin{aligned} h_t(B) = e^{A^{-1}t} \end{aligned}$$

as shown in the proof of [5, Corollary 5.7]. By Theorem 2.2 (ii), $r_{\alpha }(B)$ coincides with the operator defined as the Hille-Phillips calculus. This and [18, Proposition 3.3.5] give

$$\begin{aligned} r_{\alpha }(B) = (I+B)^{-\alpha } = (-A)^{-\alpha }. \end{aligned}$$

Therefore,

$$\begin{aligned} f_{t,\alpha }(B) = e^{A^{-1}t}(-A)^{-\alpha }. \end{aligned}$$

Using Theorem 2.2 (iii), we have

$$\begin{aligned} \big \Vert e^{A^{-1}t}(-A)^{-\alpha }\big \Vert \le |f_{t,\alpha }(\infty )| + 2K^2 \Vert f_{t,\alpha }\Vert _{{\mathcal {B}}_0} = 2K^2 \Vert f_{t,\alpha }\Vert _{{\mathcal {B}}_0}. \end{aligned}$$

Thus, the desired estimate (2.6) holds by Lemma 2.3. $\square$

From the next example, we see that the norm-estimate (2.6) cannot be improved in general.

Example 2.5

Let $\gamma >0$ and set $\lambda _k:= -\gamma + ik$ for $k \in {\mathbb {N}}$. Define an operator A on $\ell ^2$ by

$$\begin{aligned} (A\zeta _k)_{k \in {\mathbb {N}}}:= (\lambda _k \zeta _k)_{n \in {\mathbb {N}}} \end{aligned}$$

with domain

$$\begin{aligned} D(A):= \{ \zeta = (\zeta _k)_{k \in {\mathbb {N}}} \in \ell ^2: (\lambda _k \zeta _k)_{k \in {\mathbb {N}}} \in \ell ^2 \}. \end{aligned}$$

Then

$$\begin{aligned} \big \Vert e^{A^{-1}t} (-A)^{-\alpha }\big \Vert = \sup _{k \in {\mathbb {N}}} \frac{e^{t \mathop {\textrm{Re}}\nolimits \lambda _k/ |\lambda _k|^2}}{|\lambda _k|^\alpha } = \sup _{k \in {\mathbb {N}}} \frac{e^{-\gamma t/(\gamma ^2+k^2)}}{(\gamma ^2+k^2)^{\alpha /2}} \end{aligned}$$

for all $t,\alpha >0$. Put $f(w):= w^{-\alpha /2}e^{-\gamma t/w}$ for $w \ge \gamma ^2+1$. Then

$$\begin{aligned} \sup _{w \ge \gamma ^2+1} f(w) = f\left( \frac{2\gamma t}{\alpha } \right) = \left( \frac{\alpha }{2e\gamma t} \right) ^{\alpha /2} \end{aligned}$$

for all $t \ge \alpha (\gamma ^2+1)/(2\gamma )$. From this, it follows that for all $k \in {\mathbb {N}}$ and $t = \alpha (\gamma ^2+k^2)/(2\gamma )$,

$$\begin{aligned} \big \Vert e^{A^{-1}t} (-A)^{-\alpha }\big \Vert = \left( \frac{\alpha }{2e\gamma t} \right) ^{\alpha /2}. \end{aligned}$$

Hence

$$\begin{aligned} \liminf _{t \rightarrow \infty } t^{\alpha /2} \big \Vert e^{A^{-1}t} (-A)^{-\alpha }\big \Vert \ge \left( \frac{\alpha }{2e\gamma } \right) ^{\alpha /2}, \end{aligned}$$

and the estimate (2.6) cannot be improved for this diagonal operator. Moreover, we see that the rate of polynomial decay of $\Vert e^{A^{-1}t} (-A)^{-\alpha }\Vert$ does not depend on the exponential growth bound $-\gamma$ of the $C_0$-semigroup $(e^{tA})_{t \ge 0}$.

2.2 Inverses of Generators of Polynomially Stable Semigroups

We recall the definition of polynomially stable $C_0$-semigroups.

Definition 2.6

A $C_0$-semigroup $(e^{At})_{t\ge 0}$ on a Hilbert space is polynomially stable with parameter $\beta >0$ if $(e^{At})_{t\ge 0}$ is bounded and if

$$\begin{aligned} \big \Vert e^{At}(I-A)^{-1}\big \Vert = O\left( \frac{1}{t^{1/\beta }} \right) \qquad (t \rightarrow \infty ). \end{aligned}$$

(2.7)

A $C_0$-semigroup $(e^{At})_{t\ge 0}$ is simply called polynomially stable if it is polynomially stable with some parameter $\beta >0$.

Let A be the generator of a polynomially stable semigroup on a Hilbert space H. By [4, Theorem 1.1], $i{\mathbb {R}} \cap \sigma (A) = \emptyset$. Then A is invertible, and therefore $A^{-1}$ generates a $C_0$-semigroup on H. When A is normal, we obtain the rate of decay of $\Vert e^{A^{-1}t}(-A)^{-\alpha }\Vert$ as in the case of exponentially stable $C_0$-semigroups. We also show that $\sup _{t \ge 0}\Vert e^{A^{-1}t}A^{-1}\Vert < \infty$ without assuming that A is normal.

2.2.1 Case Where Generators are Normal

When the generator A is a normal operator on a Hilbert space, a spectral condition equivalent to polynomial decay is known. The proof can be found in [3, Proposition 4.1].

Proposition 2.7

Let H be a Hilbert space and let $A:D(A)\subset H \rightarrow H$ be a normal operator. Assume that $\sigma (A) \subset {\mathbb {C}}_-$. For a fixed $\beta >0$, the $C_0$-semigroup $(e^{At})_{t\ge 0}$ satisfies

$$\begin{aligned} \big \Vert e^{At}A^{-1}\big \Vert = O\left( \frac{1}{t^{1/\beta }} \right) \qquad (t \rightarrow \infty ) \end{aligned}$$

if and only if there exist $C,\delta >0$ such that

$$\begin{aligned} |\mathop {\textrm{Im}}\nolimits \lambda | \ge \frac{C}{|\mathop {\textrm{Re}}\nolimits \lambda |^{1/\beta }} \end{aligned}$$

for all $\lambda \in \sigma (A)$ with $\mathop {\textrm{Re}}\nolimits \lambda \ge -\delta$.

The next result gives an estimate for the rate of decay of $\Vert e^{A^{-1}t} (-A)^{-\alpha }\Vert$.

Proposition 2.8

Let H be a Hilbert space, and let $A:D(A)\subset H \rightarrow H$ be a normal operator generating a polynomially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$ with parameter $\beta >0$ on H. Then

$$\begin{aligned} \big \Vert e^{A^{-1}t} (-A)^{-\alpha }\big \Vert = O\left( \frac{1}{t^{\alpha /(2+\beta )}}\right) \qquad (t \rightarrow \infty ) \end{aligned}$$

(2.8)

for all $\alpha > 0$.

Proof

For all nonzero $\lambda \in {\mathbb {C}}$, we obtain

$$\begin{aligned} \lambda I - A^{-1} = \left( A - \frac{1}{\lambda } I \right) \lambda A^{-1}, \end{aligned}$$

which implies that $1/\lambda \in \sigma (A)$ if and only if $\lambda \in \sigma (A^{-1})$. The normality of A implies that of $A^{-1}$ by [27, Theorem 5.42]. Fix $t,\alpha >0$, and define

$$\begin{aligned} f(\lambda ):= {\left\{ \begin{array}{ll} e^{-\lambda t}\lambda ^\alpha , &{} \lambda \in {\mathbb {C}} {\setminus } (-\infty ,0], \\ 0, &{} \lambda = 0. \end{array}\right. } \end{aligned}$$

Then $f(-A^{-1}) = e^{A^{-1}t} (-A)^{-\alpha }$. Moreover, $f(-A^{-1})$ is normal and

$$\begin{aligned} f\big (\sigma (-A^{-1})\big ) = \sigma \big (f(-A^{-1})\big ); \end{aligned}$$

see, e.g., [19, Theorem 4.5]. Therefore,

$$\begin{aligned} \big \Vert e^{A^{-1}t}(-A)^{-\alpha }\big \Vert = \sup _{\lambda \in \sigma (-A^{-1}) } |f(\lambda )| = \sup _{\lambda \in \sigma (A) } \frac{e^{t \mathop {\textrm{Re}}\nolimits \lambda /|\lambda |^2}}{|\lambda |^{\alpha }}. \end{aligned}$$

By Proposition 2.7, there exist $C,\delta >0$ such that $|\mathop {\textrm{Im}}\nolimits \lambda | \ge C|\mathop {\textrm{Re}}\nolimits \lambda |^{-1/\beta }$ for all $\lambda \in \sigma (A)$ with $|\mathop {\textrm{Re}}\nolimits \lambda | \le \delta$. If $\lambda \in \sigma (A)$ satisfies $|\mathop {\textrm{Re}}\nolimits \lambda | \le \delta$, then

$$\begin{aligned} |\mathop {\textrm{Re}}\nolimits \lambda | \ge \frac{C^{\beta }}{|\lambda |^{\beta }}, \end{aligned}$$

and therefore

$$\begin{aligned} \frac{e^{t \mathop {\textrm{Re}}\nolimits \lambda /|\lambda |^2}}{ |\lambda |^{\alpha } } \le \sup _{s >0} \frac{e^{-C^{\beta } t/s^{2+\beta }}}{ s^{\alpha }} = \left( \frac{\alpha }{(2+\beta )eC^{\beta }t} \right) ^{\frac{\alpha }{2+\beta }}. \end{aligned}$$

On the other hand, for all $\lambda \in \sigma (A)$ with $|\mathop {\textrm{Re}}\nolimits \lambda | > \delta$, we obtain

$$\begin{aligned} \frac{e^{t \mathop {\textrm{Re}}\nolimits \lambda /|\lambda |^2}}{ |\lambda |^{\alpha } } \le \sup _{s >0} \frac{e^{-\delta t /s^2} }{ s^{\alpha } } = \left( \frac{\alpha }{2e \delta t} \right) ^{\frac{\alpha }{2}}. \end{aligned}$$

Thus, the norm-estimate (2.8) holds. $\square$

As in the case of exponentially stable $C_0$-semigroups, we present an example of an operator on $\ell ^2$ showing that the norm-estimate (2.8) cannot be in general improved.

Example 2.9

Set $\lambda _k:= -1/k+ ik$ for $k \in {\mathbb {N}}$. Define an operator A on $\ell ^2$ by

$$\begin{aligned} (A\zeta _k)_{k \in {\mathbb {N}}}:= (\lambda _k \zeta _k)_{k \in {\mathbb {N}}}\end{aligned}$$

with domain

$$\begin{aligned} D(A):= \{ \zeta = (\zeta _k)_{k\in {\mathbb {N}}} \in \ell ^2: (\lambda _k \zeta _k)_{k \in {\mathbb {N}}} \in \ell ^2 \}. \end{aligned}$$

By Proposition 2.7, A is the generator of a polynomially stable $C_0$-semigroup with parameter $\beta = 1$. Let $t,\alpha >0$. We have

$$\begin{aligned} \big \Vert e^{A^{-1}t} (-A)^{-\alpha } \big \Vert = \sup _{k \in {\mathbb {N}}} \frac{e^{t \mathop {\textrm{Re}}\nolimits \lambda _k/ |\lambda _k|^2}}{|\lambda _k|^\alpha } = \sup _{k \in {\mathbb {N}}} \frac{e^{-kt/(k^4+1)}}{(1/k^2+k^2)^{\alpha /2}}. \end{aligned}$$

Moreover,

$$\begin{aligned} \frac{e^{-kt/(k^4+1)}}{(1/k^2+k^2)^{\alpha /2}} \ge \frac{e^{-t/k^3}}{2^{\alpha /2} k^\alpha } \end{aligned}$$

for all $k \in {\mathbb {N}}$. Put $f(w):= w^{-\alpha }e^{-t/w^3}$ for $w \ge 1$. Then

$$\begin{aligned} \sup _{w \ge 1} f(w) = f\left( \left( \frac{3t}{\alpha }\right) ^{1/3} \right) = \left( \frac{\alpha }{3et}\right) ^{\alpha /3} \end{aligned}$$

for all $t \ge \alpha /3$. This implies that for all $k \in {\mathbb {N}}$ and $t = \alpha k^3 /3$,

$$\begin{aligned} \big \Vert e^{A^{-1}t} (-A)^{-\alpha }\big \Vert \ge \frac{1}{2^{\alpha /2}} \left( \frac{\alpha }{3et}\right) ^{\alpha /3}. \end{aligned}$$

Therefore

$$\begin{aligned} \liminf _{t \rightarrow \infty } t^{\alpha /3} \big \Vert e^{A^{-1}t} (-A)^{-\alpha }\big \Vert \ge \frac{1}{2^{\alpha /2}} \left( \frac{\alpha }{3e}\right) ^{\alpha /3}, \end{aligned}$$

and the estimate (2.8) cannot be improved for this operator A.

2.2.2 Norm-Estimate Through Lyapunov Equations

We show that if A is the generator of a polynomially stable $C_0$-semigroup on a Hilbert space, then $\sup _{t \ge 0}\Vert e^{A^{-1}t} A^{-1}\Vert < \infty$. To this end, we use the following results on Lyapunov equations; see [8, Theorem 4.1.3, Corollary 6.5.1, and Theorem 6.5.2] for the proofs.

Lemma 2.10

Let A be the generator of an exponentially stable semigroup $(e^{At})_{t \ge 0}$ on a Hilbert space H. Then there exists a unique self-adjoint, positive operator $P \in {\mathcal {L}}(H)$ such that $PD(A) \subset D(A^*)$ and

$$\begin{aligned} A^*P + P A = -I\quad \hbox { on}\ D(A). \end{aligned}$$

(2.9)

Moreover, the operator P is given by

$$\begin{aligned} Px = \int ^{\infty }_0 \big (e^{At} \big )^* e^{At}x dt \end{aligned}$$

(2.10)

for all $x \in H$.

To obtain $\sup _{t \ge 0}\Vert e^{A^{-1}t} A^{-1}\Vert < \infty$, we shall use the fact that $P \in {\mathcal {L}}(H)$ defined by (2.10) satisfies the Lyapunov equation (2.9).

Lemma 2.11

Let A be the generator of a $C_0$-semigroup on a Hilbert space H, and let C be a bounded linear operator from H to another Hilbert space Y. If there exists a self-adjoint, non-negative operator $P \in {\mathcal {L}}(H)$ such that $PD(A) \subset D(A^*)$ and

$$\begin{aligned} A^*P + PA = -C^*C\quad \hbox { on}\ D(A), \end{aligned}$$

then

$$\begin{aligned} \int ^{\infty }_0 \big \Vert Ce^{At}x\big \Vert ^2 dt \le \langle x, Px \rangle \end{aligned}$$

for all $x \in H$.

We shall need the estimate for adjoint operators in the following lemma; see [28, Lemma 2.1] for the proof.

Lemma 2.12

Let A be the generator of a bounded $C_0$-semigroup $(e^{At})_{t \ge 0}$ on a Hilbert space H, and let $K:= \sup _{t \ge 0} \Vert e^{At}\Vert$. Then for all $x \in H$ and $\xi ,\gamma >0$,

$$\begin{aligned} \int ^{\infty }_0 \big \Vert e^{(\gamma A-\xi I)^{-1}t}(\gamma A - \xi I)^{-1} x \big \Vert ^2 dt \le \frac{K^2 \Vert x\Vert ^2}{2\xi }. \end{aligned}$$

(2.11)

The same estimate holds for the adjoint.

We are now in a position to prove that $\sup _{t \ge 0} \Vert e^{A^{-1}t} A^{-1}\Vert < \infty$ if A is the generator of a polynomially stable $C_0$-semigroup. Actually, we will make the slightly weaker assumption that $\Vert e^{At} (I-A)^{-1}\Vert = O((\log t)^{-\beta })$ as $t\rightarrow \infty$ for some $\beta >1$. Notice that if the bounded $C_0$-semigroup $(e^{At})_{t \ge 0}$ satisfies this estimate, then $i{\mathbb {R}} \cap \sigma (A) = \emptyset$ by [4, Theorem 1.1]. The proof of the following theorem is inspired by the arguments in the proof of [28, Theorem 2.2].

Theorem 2.13

Let A be the generator of a bounded $C_0$-semigroup on a Hilbert space H. If there exists $\beta >1$ such that

$$\begin{aligned} \big \Vert e^{At} (I-A)^{-1}\big \Vert = O\left( \frac{1}{(\log t)^{\beta }} \right) \qquad (t \rightarrow \infty ), \end{aligned}$$

(2.12)

then

$$\begin{aligned} \sup _{t \ge 0} \big \Vert e^{A^{-1}t} A^{-1}\big \Vert < \infty . \end{aligned}$$

(2.13)

In particular, if A is the generator of a polynomially stable $C_0$-semigroup on H, then the estimate (2.13) holds.

Proof

The proof consists of three steps. In the first step, we prove an estimate analogous to (2.11) for bounded $C_0$-semigroups satisfying (2.12). In the next step, we obtain the norm-estimate of

$$\begin{aligned} e^{(A-\xi _1I)^{-1}t}x - e^{(A-\xi _2I)^{-1}t} x \end{aligned}$$

for $x \in D(A)$ and $0< \xi _1,\xi _2 < \delta _0$, where $\delta _0 >0$ is some sufficiently small constant. This estimate is used to compare $e^{(A-\delta I)^{-1}t}x$ and $e^{(A-\delta e^{-N}I)^{-1}t}x$ for $x \in D(A)$, $N \in {\mathbb {N}}$, and $0< \delta < \delta _0$. Letting $N \rightarrow \infty$, we show the desired estimate (2.13) in the last step.

Step 1: Let A be the generator of a bounded $C_0$-semigroup $(e^{At})_{t\ge 0}$ such that (2.12) holds for some $\beta > 1$. We write $K:= \sup _{t \ge 0} \Vert e^{At}\Vert$. By (2.12), there exist $M_0 >0$ and $t_0 > 1$ such that

$$\begin{aligned} \big \Vert e^{A t}A^{-1}\big \Vert \le \frac{M_0}{(\log t)^{\beta }} \end{aligned}$$

(2.14)

for all $t \ge t_0$. Choose $t_1 > \max \{t_0,e^{2\beta }\}$. There exists $M_1 >0$ such that

$$\begin{aligned} \int _{t_1}^{\infty } \frac{e^{-\xi t}}{(\log t)^{2\beta }} dt \le \frac{M_1}{\xi |\log \xi |^{2\beta }} \end{aligned}$$

(2.15)

for all $0< \xi < 1/t_1$; see [26, Lemma 4.2]. For $\xi >0$, define the operator $P(\xi ) \in {\mathcal {L}}(H)$ by

$$\begin{aligned} P(\xi ) x:= \int ^{\infty }_0 e^{-2\xi t}\big (e^{At} \big )^* e^{At}x dt,\quad x \in H. \end{aligned}$$

From the estimates (2.14) and (2.15), we obtain

$$\begin{aligned} \langle x, P(\xi ) x\rangle&= \int ^{t_1}_0 e^{-2\xi t} \big \Vert e^{At}x\big \Vert ^2 dt + \int ^{\infty }_{t_1} e^{-2\xi t} \big \Vert e^{A t}x\big \Vert ^2 dt \\&\le t_1 K^2 \Vert x\Vert ^2 + \frac{M_0^2 M_1 \Vert Ax\Vert ^2 }{2 \xi |\log (2\xi )|^{2 \beta }} \end{aligned}$$

for all $x \in D(A)$ and $0< \xi < 1/(2t_1) =: \delta _0$. Hence there exists $K_1 >0$ such that

$$\begin{aligned} \sup _{0< \xi < \delta _0} \xi |\log \xi |^{2\beta } \langle x,P(\xi )x \rangle \le K_1 \Vert Ax\Vert ^2 \end{aligned}$$

(2.16)

for all $x \in D(A)$.

By Lemma 2.10, $P(\xi )D(A) \subset D(A^*)$ and

$$\begin{aligned} (A- \xi I)^*P(\xi ) + P(\xi )(A - \xi I) = -I\quad \hbox { on}\ D(A) \end{aligned}$$

(2.17)

for all $\xi >0$. Since (2.17) yields

$$\begin{aligned} (A^*- \xi I)^{-1}P(\xi ) + P(\xi )(A - \xi I)^{-1} = -(A^*- \xi I)^{-1}(A - \xi I)^{-1}, \end{aligned}$$

we have from Lemma 2.11 and the estimate (2.16) that

$$\begin{aligned} \int ^{\infty }_0 \big \Vert e^{(A-\xi I)^{-1}t} (A- \xi I)^{-1}x \big \Vert ^2 dt \le \langle x,P(\xi )x \rangle \le \frac{K_1 \Vert Ax\Vert ^2}{\xi |\log \xi |^{2\beta }} \end{aligned}$$

(2.18)

for all $x \in D(A)$ and $0< \xi < \delta _0$. Moreover, by Lemma 2.12, there exists $K_2 >0$ such that

$$\begin{aligned} \int ^{\infty }_0 \big \Vert e^{(A^*-\xi I)^{-1}t} (A^*- \xi I)^{-1}y \big \Vert ^2 dt \le \frac{K_2 \Vert y\Vert ^2}{\xi } \end{aligned}$$

(2.19)

for all $y \in H$ and $\xi >0$.

Step 2: Take $0< \xi _1,\xi _2 < \delta _0$. The variation of constants formula yields

$$\begin{aligned}&e^{(A-\xi _1 I)^{-1}t} x - e^{(A-\xi _2 I)^{-1}t}x \\&\quad = \int ^t_0 e^{(A-\xi _1 I)^{-1}(t-s)} \big ( (A-\xi _1 I)^{-1} - (A-\xi _2 I)^{-1} \big ) e^{(A-\xi _2 I)^{-1}s} xds \end{aligned}$$

for all $x \in H$ and $t \ge 0$. Moreover, the resolvent equation gives

$$\begin{aligned} (A-\xi _1 I)^{-1} - (A-\xi _2 I)^{-1} = (\xi _1 - \xi _2) (A-\xi _1 I)^{-1} (A-\xi _2 I)^{-1}. \end{aligned}$$

Therefore, we obtain

$$\begin{aligned}&e^{(A-\xi _1 I)^{-1}t} x - e^{(A-\xi _2 I)^{-1}t}x \\&\quad =(\xi _1 - \xi _2) \int ^t_0 e^{(A-\xi _1 I)^{-1}(t-s)} (A-\xi _1 I)^{-1} (A-\xi _2 I)^{-1}e^{(A-\xi _2 I)^{-1}s}xds \end{aligned}$$

for all $x \in H$ and $t \ge 0$. Using the estimates (2.18) and (2.19), we have that for all $x \in D(A)$, $y \in H$, and $t \ge 0$,

$$\begin{aligned}&\left| \left\langle y,~ e^{(A-\xi _1 I)^{-1}t} x - e^{(A-\xi _2 I)^{-1}t} x \right\rangle \right| \\&\quad = |\xi _1 - \xi _2|\, \bigg | \int ^t_0 \Big \langle e^{(A^*-\xi _1 I)^{-1}(t-s)} (A^*- \xi _1 I)^{-1}y,\\&\hspace{100pt} e^{(A-\xi _2 I)^{-1}s} (A- \xi _2 I)^{-1}x \Big \rangle ds \bigg | \\&\quad \le |\xi _1 - \xi _2| \left( \int ^{\infty }_0 \big \Vert e^{(A^*-\xi _1 I)^{-1}t} (A^*- \xi _1 I)^{-1}y \big \Vert ^2 dt \right) ^{1/2} \\&\hspace{100pt} \times \left( \int ^{\infty }_0 \big \Vert e^{(A-\xi _2 I)^{-1}t} (A- \xi _2 I)^{-1}x \big \Vert ^2 dt \right) ^{1/2} \\&\quad \le \frac{ K_0|\xi _1 - \xi _2| }{ \sqrt{\xi _1\xi _2} \, |\log \xi _2|^{\beta }} \Vert Ax\Vert \, \Vert y\Vert , \end{aligned}$$

where $K_0:= \sqrt{K_1K_2}$. Hence

$$\begin{aligned} \big \Vert e^{(A-\xi _1 I)^{-1}t} x - e^{(A-\xi _2 I)^{-1}t} x \big \Vert \le \frac{K_0|\xi _1 - \xi _2| }{ \sqrt{\xi _1\xi _2} \, |\log \xi _2|^{\beta } }\Vert Ax\Vert \end{aligned}$$

(2.20)

for all $x \in D(A)$ and $t \ge 0$.

Step 3: Let $N \in {\mathbb {N}}$ and $0<\delta< \delta _0\, (< 1)$. Substituting $\xi _1 = \delta e^{-n+1}$ and $\xi _2 = \delta e^{-n}$, $n=1,2,\dots ,N$, into the estimate (2.20), we obtain

$$\begin{aligned} \big \Vert e^{(A- \delta I)^{-1}t} x - e^{(A- \delta e^{-N} I)^{-1}t} x \big \Vert&= \left\| \sum _{n=1}^N e^{(A- \delta e^{-n+1} I)^{-1}t} x - e^{(A-\delta e^{-n} I)^{-1}t} x \right\| \\&\le K_0 \Vert Ax\Vert \sum _{n=1}^N \frac{|e^{-n+1} - e^{-n}| }{\sqrt{e^{-n+1} e^{-n}}\, |\log (\delta e^{-n})|^{\beta }} \\&\le \frac{(e-1)K_0}{\sqrt{e} } \Vert Ax\Vert \sum _{n=1}^{N} \frac{1}{|n - \log \delta |^{\beta }} \end{aligned}$$

for all $x \in D(A)$ and $t \ge 0$. From the assumption $\beta > 1$, we obtain

$$\begin{aligned} K_3:= \sum _{n=1}^{\infty } \frac{1}{|n - \log \delta |^{\beta }} < \infty . \end{aligned}$$

Since

$$\begin{aligned} \lim _{N \rightarrow \infty } \big \Vert e^{(A-\delta e^{-N} I)^{-1}t} - e^{A^{-1}t} \big \Vert = 0 \end{aligned}$$

for each $t \ge 0$, it follows that

$$\begin{aligned} \big \Vert e^{(A- \delta I)^{-1}t} x - e^{A^{-1}t} x \big \Vert \le \frac{(e-1)K_0K_3}{\sqrt{e} } \Vert Ax\Vert \end{aligned}$$

for all $x \in D(A)$ and $t \ge 0$. This gives

$$\begin{aligned} \sup _{t\ge 0} \big \Vert e^{(A-\delta I)^{-1}t} A^{-1} - e^{A^{-1}t} A^{-1} \big \Vert < \infty . \end{aligned}$$

Since $A- \delta I$ generates an exponentially stable $C_0$-semigroup, Theorem 2.4 implies that

$$\begin{aligned} \sup _{t \ge 0 }\big \Vert e^{(A- \delta I)^{-1}t} A^{-1}\big \Vert < \infty . \end{aligned}$$

Thus,

$$\begin{aligned}&\sup _{t \ge 0} \big \Vert e^{A^{-1}t} A^{-1} \big \Vert \\&\quad \le \sup _{t \ge 0} \big \Vert e^{(A-\delta I)^{-1}t} A^{-1} \big \Vert + \sup _{t \ge 0}\big \Vert e^{(A- \delta I)^{-1}t} A^{-1} - e^{A^{-1}t} A^{-1} \big \Vert < \infty . \end{aligned}$$

$\square$

3 Crank–Nicolson Scheme with Smooth Initial Data

Let A be the generator of a bounded $C_0$-semigroup on a Hilbert space H. For $\tau > 0$, we define

$$\begin{aligned} A_d(\tau ):= \left( I + \frac{\tau }{2}A \right) \left( I - \frac{\tau }{2}A \right) ^{-1}. \end{aligned}$$

(3.1)

Let $(\tau _n)_{n \in {\mathbb {N}}_0}$ be a sequence of strictly positive real numbers. We consider the time-varying difference equation

$$\begin{aligned} x_{n+1} = A_d(\tau _n) x_n,\quad n \in {\mathbb {N}}_0;\qquad x_0 \in H. \end{aligned}$$

(3.2)

In this section, we study the decay rate of the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2) with smooth initial data.

3.1 Generators of Exponentially Stable Semigroups

Let H be a Hilbert space, and let $A:D(A) \subset H \rightarrow H$ be injective. Suppose that A and $A^{-1}$ generate bounded $C_0$-semigroups on H. Take $\xi >0$. By Lemma 2.10, there exist unique self-adjoint, positive operators $P(\xi ), Q(\xi ) \in {\mathcal {L}}(H)$ such that $P(\xi )D(A) \subset D(A^*)$, $Q(\xi )D(A^{-1}) \subset D((A^{-1})^*)$, and

$$\begin{aligned} (A- \xi I)^*P(\xi ) + P(\xi )(A - \xi I)= & {} -I\quad \hbox { on}\ D(A), \end{aligned}$$

(3.3a)

$$\begin{aligned} (A^{-1} - \xi I)^*Q(\xi ) + Q(\xi )(A^{-1} - \xi I)= & {} -I\quad \hbox { on}\ D(A^{-1}). \end{aligned}$$

(3.3b)

The operators $P(\xi )$ and $Q(\xi )$ are given by

$$\begin{aligned} P(\xi ) x= & {} \int ^{\infty }_0 e^{-2\xi t} (e^{At})^* e^{At}x dt, \end{aligned}$$

(3.4a)

$$\begin{aligned} Q(\xi ) x= & {} \int ^{\infty }_0 e^{-2\xi t} (e^{A^{-1}t})^* e^{A^{-1}t}x dt \end{aligned}$$

(3.4b)

for all $x \in X$. To estimate the decay rate of the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2), we shall use that the operators $P(\xi )$ and $Q(\xi )$ defined as in (3.4) solve the Lypapunov equations (3.3).

In the proofs of [23, Lemmas 2.1 and 2.2], the following result has been obtained from the Lypapunov equations (3.3).

Lemma 3.1

Let A be the generator of a bounded $C_0$-semigroup on a Hilbert space H. Suppose that A is injective and that $A^{-1}$ also generate a bounded $C_0$-semigroup on H. Let $0< \tau _{\min } \le \tau _n \le \tau _{\max } < \infty$ for all $n \in {\mathbb {N}}_0$, and let $P(\xi ), Q(\xi ) \in {\mathcal {L}}(H)$ be given by (3.4) for $\xi >0$. Define $R(r) \in {\mathcal {L}}(H)$ by

$$\begin{aligned} R(r):= \frac{2}{\tau _{\min }}P\left( \frac{\xi _r}{\tau _{\max }} \right) + 2\tau _{\max } Q(\tau _{\min } \xi _r),\quad r \in (0,1), \end{aligned}$$

where

$$\begin{aligned} \xi _r:= \frac{1-r^2}{2(r^2+1)}. \end{aligned}$$

Then there exists a constant $M >0$ such that the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2) satisfies

$$\begin{aligned} |(n+1)r^n \langle y, x_n \rangle | \le \frac{M \Vert y\Vert }{\sqrt{1-r}} \sqrt{\langle x_0, R(r)x_0 \rangle } \end{aligned}$$

for all $x_0, y \in X$, $n \in {\mathbb {N}}$, and $r \in (0,1)$.

Now we estimate $\langle x,P(\xi ) x \rangle$ and $\langle x,Q(\xi ) x \rangle$, by using the integral representations (3.4). Suppose that A is the generator of an exponentially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$. Let $K_0 \ge 1$ and $\omega >0$ satisfy $\Vert e^{At}\Vert \le K_0e^{-\omega t}$ for all $t \ge 0$. Then the operator $P(\xi )$ given by (3.4a) satisfies

$$\begin{aligned} \langle x, P(\xi )x \rangle \le K_0^2\Vert x\Vert ^2 \int ^{\infty }_0 e^{-2(\omega + \xi )t}dt \le \frac{K_0^2 \Vert x\Vert ^2}{2\omega } \end{aligned}$$

for all $x \in X$ and $\xi >0$. Hence

$$\begin{aligned} \sup _{\xi >0} \langle x, P(\xi )x \rangle \le K_1 \Vert x\Vert ^2 \end{aligned}$$

(3.5)

for all $x \in X$, where $K_1:= K_0^2/(2\omega )$. Using the norm-estimate (2.6) for $e^{A^{-1}t}(-A)^{-\alpha }$ with $0 < \alpha \le 1$, we estimate $\langle x, Q(\xi )x \rangle$ for $x \in D((-A)^{\alpha })$ in the next lemma.

Lemma 3.2

Let A be the generator of an exponentially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$ on a Hilbert space H. Suppose that $A^{-1}$ generates a bounded $C_0$-semigroup $(e^{A^{-1}t})_{t \ge 0}$ on H. Then $Q(\xi ) \in {\mathcal {L}}(H)$ given by (3.4b) for $\xi >0$ satisfies the following statements:

• (i) For each $0< \alpha < 1$, there exists $K>0$ such that

  $$\begin{aligned} \sup _{0< \xi < 1} \xi ^{1-\alpha } \langle x, Q(\xi )x \rangle \le K \Vert (-A)^{\alpha }x\Vert ^2 \end{aligned}$$

  (3.6)

  for all $x \in D((-A)^{\alpha })$

• (ii) There exists $K>0$ such that

  $$\begin{aligned} \sup _{0< \xi < 1/2} \frac{ \langle x, Q(\xi )x \rangle }{\log (1/\xi )} \le K \Vert Ax\Vert ^2 \end{aligned}$$

  (3.7)

  for all $x \in D(A)$.

Proof

Let $0 < \alpha \le 1$ and $M_1:= \sup _{t \ge 0}\Vert e^{A^{-1} t}\Vert < \infty$. By Theorem 2.4, there exist $M_2, t_0 >0$ such that

$$\begin{aligned} \big \Vert e^{A^{-1}t}x\big \Vert \le \frac{M_2 \Vert (-A)^{\alpha }x\Vert }{t^{\alpha /2}} \end{aligned}$$

for all $x \in D((-A)^{\alpha })$ and $t \ge t_0$. Hence

$$\begin{aligned} \langle x, Q(\xi ) x\rangle&= \int ^{t_0}_0 e^{-2\xi t} \big \Vert e^{A^{-1}t}x\big \Vert ^2 dt + \int ^{\infty }_{t_0} e^{-2\xi t}\big \Vert e^{A^{-1}t}x\big \Vert ^2 dt \\&\le t_0 M_1^2 \Vert x\Vert ^2 + M_2^2 \Vert (-A)^{\alpha }x\Vert ^2 \int ^{\infty }_{t_0} \frac{e^{-2\xi t} }{t^{\alpha }} dt \end{aligned}$$

for all $x \in D((-A)^{\alpha })$ and $\xi >0$. When $0< \alpha < 1$, we have

$$\begin{aligned} \int ^{\infty }_{t_0} \frac{e^{-2\xi t} }{t^{\alpha }} dt \le \int ^{\infty }_0 \frac{e^{-2\xi t} }{t^{\alpha }} dt = \frac{\Gamma (1-\alpha )}{(2\xi )^{1-\alpha }}. \end{aligned}$$

Hence there is $K>0$ such that (3.6) holds for all $x \in D((-A)^{\alpha })$.

Next we consider the case $\alpha = 1$. The exponential integral satisfies

$$\begin{aligned} \int _\tau ^{\infty } \frac{e^{-t}}{t} dt \le e^{-\tau }\log \left( 1+ \frac{1}{\tau } \right) \end{aligned}$$

for all $\tau >0$; see [12, inequality (5)]. Therefore,

$$\begin{aligned} \int _{t_0}^{\infty } \frac{e^{-2\xi t}}{t} dt = \int _{2\xi t_0}^{\infty } \frac{e^{-t}}{t} dt \le e^{-2\xi t_0} \log \left( 1 + \frac{1}{2\xi t_0} \right) \end{aligned}$$

for all $\xi >0$. There exists $c>0$ such that for all $0< \xi <1/2$,

$$\begin{aligned} \log \left( 1 + \frac{1}{2\xi t_0} \right) \le c\log \left( \frac{1}{\xi } \right) . \end{aligned}$$

Thus, (3.7) holds for all $x \in D(A)$ and some suitable constant $K >0$. $\square$

Combining Lemma 3.1 with the estimates (3.5)–(3.7), we obtain estimates for the decay rate of the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2) with smooth initial data.

Theorem 3.3

Let A be the generator of an exponentially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$ on a Hilbert space H. Suppose that $A^{-1}$ generates a bounded $C_0$-semigroup $(e^{A^{-1}t})_{t \ge 0}$ on H. If $0< \inf _{n \in {\mathbb {N}}_0} \tau _n \le \sup _{n \in {\mathbb {N}}_0} \tau _n < \infty$, then there exists $K>0$ such that the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2) satisfies the following statements:

• (i) For each $0< \alpha < 1$, there exists $K>0$ such that

  $$\begin{aligned} \Vert x_n\Vert \le \frac{K}{n^{\alpha /2}} \Vert (-A)^\alpha x_0\Vert \end{aligned}$$

  (3.8)

  for all $x_0 \in D((-A)^{\alpha })$ and $n \ge 1$.

• (ii) There exists $K>0$ such that

  $$\begin{aligned} \Vert x_n\Vert \le K \sqrt{\frac{\log n}{n}} \Vert Ax_0\Vert \end{aligned}$$

  (3.9)

  for all $x_0 \in D(A)$ and $n \ge 2$.

Proof

Set

$$\begin{aligned} \tau _{\min }:= \inf _{n \in {\mathbb {N}}_0} \tau _n >0,\quad \tau _{\max }:= \sup _{n \in {\mathbb {N}}_0} \tau _n < \infty , \end{aligned}$$

and let the operator $R(r) \in {\mathcal {L}}(H)$ be as in Lemma 3.1. Then there is $M>0$ such that the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2) satisfies

$$\begin{aligned} \Vert x_n\Vert \le \frac{M \sqrt{\langle x_0, R(r)x_0 \rangle } }{(n+1)r^n \sqrt{1-r}} \end{aligned}$$

(3.10)

for all $x_0 \in X$, $n \in {\mathbb {N}}$, and $r \in (0,1)$.

Let $0 < \alpha \le 1$, and define

$$\begin{aligned} f_{\alpha }(\xi ):= {\left\{ \begin{array}{ll} \xi ^{\alpha -1}, &{}{} 0<\alpha <1, \\ \log \left( \dfrac{1}{\xi } \right) , &{}{} \alpha = 1 \end{array}\right. } \end{aligned}$$

for $0< \xi < 1$. By the estimates (3.5)–(3.7), there exist $K_1,K_2 >0$ and $r_0\in (0,1)$ such that

$$\begin{aligned} \langle x_0, R(r)x_0 \rangle \le \frac{2K_1}{\tau _{\min }} \Vert x_0\Vert ^2 + 2\tau _{\max } K_2 \Vert (-A)^{\alpha }x_0\Vert ^2f_{\alpha }(\tau _{\min } \xi _r) \end{aligned}$$

(3.11)

for all $x_0 \in D((-A)^\alpha )$ and $r \in (r_0,1)$. Put $r = n/(n+1)$ for $n \in {\mathbb {N}}$. Then

$$\begin{aligned} \xi _r = \frac{1-r^2}{2(r^2+1)} = \frac{2n+1}{2(2n^2+2n+1)} \end{aligned}$$

and

$$\begin{aligned} \frac{1}{(n+1)r^n \sqrt{1-r}} = \frac{(1+1/n)^n}{\sqrt{n+1}}. \end{aligned}$$

Combining the estimates (3.10) and (3.11), we have that there exist $K_4 >0$ and $n_0 \in {\mathbb {N}}$ such that

$$\begin{aligned} \Vert x_n\Vert \le K_4 \sqrt{\frac{f_{\alpha } (1/n)}{n}} \Vert (-A)^{\alpha }x_0\Vert \end{aligned}$$

for all $x_0 \in D((-A)^{\alpha })$ and $n \ge n_0 + 1$. By definition,

$$\begin{aligned} \sqrt{\frac{f_{\alpha } (1/n)}{n}} = {\left\{ \begin{array}{ll} \dfrac{1}{n^{\alpha /2}}, &{} 0<\alpha <1, \\ \sqrt{\dfrac{\log n }{n}}, &{} \alpha = 1 \end{array}\right. } \end{aligned}$$

for all $n \ge 2$. Since

$$\begin{aligned} \Vert A_d(\tau )\Vert = \left\| 2\left( I - \frac{\tau }{2}A \right) ^{-1} - I \right\| \le c \end{aligned}$$

for all $\tau \in [\tau _{\min },\tau _{\max }]$ and some $c \ge 1$, it follows that $\Vert x_n\Vert \le c^{n_0} \Vert x_0\Vert$ for all $x_0 \in X$ and $0\le n \le n_0$. Thus, we obtain the desired conclusion. $\square$

We compare the norm-estimates (3.8) and (3.9) with those in the time-invariant case $\tau _n \equiv 2$, using a simple example.

Example 3.4

Let the operator A on $\ell ^2$ be as in Example 2.5. Define $A_d:= A_d(2)= (I+A)(I-A)^{-1}$ and let $0 < \alpha \le 1$. To obtain the decay rate of the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the time-invariant difference equation

$$\begin{aligned} x_{n+1} = A_d x_n,\quad n \in {\mathbb {N}}_0;\qquad x_0 \in D((-A)^{\alpha }), \end{aligned}$$

we estimate $\Vert A_d^{n}(-A)^{-\alpha }\Vert$ for $n \in {\mathbb {N}}_0$.

We have that for all $n \in {\mathbb {N}}_0$,

$$\begin{aligned} \big \Vert A_d^{n^2}(-A)^{-\alpha }\big \Vert = \sup _{k \in {\mathbb {N}}} \left| \frac{1+\lambda _k}{1-\lambda _k} \right| ^{n^2} |\lambda _k|^{-\alpha } \ge \left| \frac{1+\lambda _n}{1-\lambda _n} \right| ^{n^2} |\lambda _n|^{-\alpha }. \end{aligned}$$

Moreover,

$$\begin{aligned} \left| \frac{1+\lambda _n}{1-\lambda _n} \right| ^{n^2} \rightarrow e^{-2\gamma },\quad n^{\alpha } |\lambda _n|^{-\alpha } \rightarrow 1 \end{aligned}$$

as $n \rightarrow \infty$. Hence

$$\begin{aligned} \liminf _{n \rightarrow \infty } n^{\alpha } \Vert A_d^{n^2} (-A)^{-\alpha }\Vert \ge e^{-2\gamma }. \end{aligned}$$

This implies that the norm-estimate (3.8) for the case $0< \alpha <1$ is optimal in the sense that one cannot obtain any better rates in general.

Next we show that

$$\begin{aligned} \Vert A_d^{n}A^{-1}\Vert = O\left( \frac{1}{\sqrt{n}} \right) \qquad (n \rightarrow \infty ). \end{aligned}$$

(3.12)

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

$$\begin{aligned} \Vert A_d^{n}A^{-1}\Vert = \sup _{k \in {\mathbb {N}}} \left| \frac{1+\lambda _k}{1-\lambda _k} \right| ^{n} |\lambda _k|^{-1} \le \sqrt{\sup _{w \ge \gamma ^2 + 1} f_n(w)}, \end{aligned}$$

where

$$\begin{aligned} f_n(w):= \left( \frac{w+1-2\gamma }{w+1+2\gamma } \right) ^n \frac{1}{w}. \end{aligned}$$

A simple calculation shows that for all sufficiently large $n \in {\mathbb {N}}$,

$$\begin{aligned} \sup _{w \ge \gamma ^2 + 1} f_n(w) = f_n(w_n), \end{aligned}$$

where

$$\begin{aligned} w_n:= 2\sqrt{\gamma ^2n^2+\gamma ^2-\gamma n} + 2\gamma n - 1. \end{aligned}$$

Since

$$\begin{aligned} \left( \frac{w_n+1-2\gamma }{w_n+1+2\gamma } \right) ^n \rightarrow e^{-1},\quad \frac{n}{w_n} \rightarrow \frac{1}{4\gamma } \end{aligned}$$

as $n \rightarrow \infty$, we conclude that the estimate (3.12) holds. It is not clear, in general, whether one can remove the logarithmic term $\sqrt{\log n}$ in the estimate (3.9) for the case $\alpha =1$.

3.2 Normal Generators of Polynomially Stable Semigroups

Suppose that A is a normal operator on a Hilbert space H and generates a polynomially stable $C_0$-semigroup. As in the case of exponentially stable $C_0$-semigroups, one can obtain an estimate for the decay rate of the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2) with smooth initial data. Note that $A^{-1}$ generates a contraction $C_0$-semigroup on H; see [29, Lemma 4.1].

The following lemma gives estimates analogous to those in Lemma 3.2.

Lemma 3.5

Let H be a Hilbert space and let $A:D(A)\subset H \rightarrow H$ be a normal operator generating a polynomially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$ with parameter $\beta >0$ on H. Then $Q(\xi ) \in {\mathcal {L}}(H)$ given by (3.4b) for $\xi >0$ satisfies the following statements:

• (i) For each $0< \alpha < 1+\beta /2$, there exists $K>0$ such that

  $$\begin{aligned} \sup _{0< \xi < 1} \xi ^{1-\frac{2\alpha }{2+\beta }} \langle x, Q(\xi )x \rangle \le K \Vert (-A)^{\alpha }x\Vert ^2 \end{aligned}$$

  (3.13)

  for all $x \in D((-A)^{\alpha })$.

• (ii) There exists $K>0$ such that

  $$\begin{aligned} \sup _{0< \xi < 1/2} \frac{ \langle x, Q(\xi )x \rangle }{\log (1/\xi )} \le K \Vert (-A)^{1+\beta /2}x\Vert ^2 \end{aligned}$$

  (3.14)

  for all $x \in D((-A)^{1+\beta /2})$.

Proof

Let $0< \alpha \le 1+\beta /2$, and put

$$\begin{aligned} {{\widetilde{\alpha }}}:= \frac{2\alpha }{2+\beta } \le 1. \end{aligned}$$

By Proposition 2.8, there exist $M, t_0 >0$ such that

$$\begin{aligned} \big \Vert e^{A^{-1}t}x\big \Vert \le \frac{M \Vert (-A)^{\alpha }x\Vert }{t^{{{\widetilde{\alpha }}} / 2}} \end{aligned}$$

(3.15)

for all $x \in D((-A)^{\alpha })$ and $t \ge t_0$. The rest of the proof is quite similar to that of Lemma 3.2, and hence we omit it. $\square$

If A is the generator of a polynomially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$ with parameter $\beta >0$, then [3, Proposition 3.1] shows that for all $\alpha >0$, there exist $M,t_0 >0$ such that

$$\begin{aligned} \big \Vert e^{A t}x\big \Vert \le \frac{M \Vert (-A)^{\alpha } x\Vert }{t^{\alpha /\beta }} \end{aligned}$$

for all $x \in D((-A)^{\alpha })$ and $t \ge t_0$. The decay rate $t^{-\alpha /\beta }$ is faster than the decay rate of $\Vert e^{A^{-1}t}x\Vert$ in (3.15). Hence, the operator $P(\xi )$ given by (3.4a) satisfies the same estimates as $Q(\xi )$.

Using the estimates on $P(\xi )$ and $Q(\xi )$, we derive a norm-estimate for the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2). Since this result can be obtained by the same arguments as in the proof of Theorem 3.3, we omit the proof.

Proposition 3.6

Let H be a Hilbert space and let $A:D(A)\subset H \rightarrow H$ be a normal operator generating a polynomially stable $C_0$-semigroup $(e^{At})_{t \ge 0}$ with parameter $\beta >0$ on H. If $0< \inf _{n \in {\mathbb {N}}_0} \tau _n \le \sup _{n \in {\mathbb {N}}_0} \tau _n < \infty$, then the solution $(x_n)_{n \in {\mathbb {N}}_0}$ of the difference equation (3.2) satisfies the following statements:

• (i) For each $0< \alpha < 1 + \beta /2$, there exists $K>0$ such that

  $$\begin{aligned} \Vert x_n\Vert \le \frac{K}{n^{\alpha /(2+\beta )}} \Vert (-A)^\alpha x_0\Vert \end{aligned}$$

  (3.16)

  for all $x_0 \in D((-A)^{\alpha })$ and $n \ge 1$.

• (ii) There exists $K>0$ such that

  $$\begin{aligned} \Vert x_n\Vert \le K \sqrt{\frac{\log n}{n}} \Vert (-A)^{1+\beta /2}x_0\Vert \end{aligned}$$

  (3.17)

  for all $x_0 \in D((-A)^{1+\beta /2})$ and $n \ge 2$.

Example 3.7

Consider again the operator A on $\ell ^2$ in Example 2.9. We recall that the parameter $\beta$ of polynomial decay is given by $\beta = 1$. Define $A_d:= A_d(2)= (I+A)(I-A)^{-1}$. It has been shown in [25, Example 4.6] that the estimate $\Vert A_d^n (-A)^{-3/m}\Vert = O(n^{-1/m})$ is optimal for all $m \in {\mathbb {N}}$. Hence we see from the case $m =3$ that the estimate (3.16) cannot in general be improved. Although $\Vert A_d^n (-A)^{-3/2}\Vert = O(1/\sqrt{n})$ can be deduced from the case $m=2$, it is open whether the logarithmic term $\sqrt{\log n}$ in the norm-estimate (3.17) can be omitted.