Abstract
Let \(-A\) be the generator of a bounded \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) on a Hilbert space. First we study the long-time asymptotic behavior of the Cayley transform \(V_{\omega }(A) :=(A-\omega I) (A+\omega I)^{-1}\) with \(\omega >0\). We give a decay estimate for \(\Vert V_{\omega }(A)^nA^{-1}\Vert \) when \((e^{-tA})_{t \ge 0}\) is polynomially stable. Considering the case where the parameter \(\omega \) varies, we estimate \(\Vert (\prod _{k=1}^n V_{\omega _k}(A))A^{-1}\Vert \) for exponentially stable \(C_0\)-semigroups \((e^{-tA})_{t \ge 0}\). Next we show that if the generator \(-A\) of the bounded \(C_0\)-semigroup has a bounded inverse, then \(\sup _{t \ge 0} \Vert e^{-tA^{-1}} A^{-\alpha } \Vert < \infty \) for all \(\alpha >0\). We also present an estimate for the rate of decay of \(\Vert e^{-tA^{-1}} A^{-1} \Vert \), assuming that \((e^{-tA})_{t \ge 0}\) is polynomially stable. To obtain these results, we use operator norm estimates offered by a functional calculus called the \(\mathcal {B}\)-calculus.
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1 Introduction
When solving differential equations numerically, we employ time-discretization methods. The first topic is the long-time asymptotic behavior of numerical solutions. Let \(-A\) be a bounded \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) on a Banach space X, and consider the differential equation
The Crank–Nicolson discretization scheme with parameter \(\omega >0\) transforms the differential equation (1) into the following difference equation:
where \(V_{\omega }(A)\) is defined by \(V_{\omega }(A) :=(A - \omega I)(A+ \omega I)^{-1}\) and is called the Cayley transform of A with parameter \(\omega \). Assume that A has a bounded inverse. To obtain a uniform rate of decay of numerical solutions \(x_d\) starting in D(A), we investigate the quantitative behavior of the operator norm \(\Vert V_{\omega }(A)^nA^{-1}\Vert \) as \(n \rightarrow \infty \).
For simplicity of notation, let \(V(A) :=V_{1}(A)\). The asymptotic behavior of \((V(A)^n )_{n \in \mathbb {N}_0}\) has been extensively studied; see the survey [15] and the book [10, Chapter 5]. Some particular relevant studies will be cited below. For the generator \(-A\) of a bounded \(C_0\)-semigroup on a Banach space, \(\Vert V(A)^n\Vert = O(\sqrt{n})\) as \(n \rightarrow \infty \) holds, i.e., there exist constants \(M>0\) and \(n_0 \in \mathbb {N}\) such that \(\Vert V(A)^n\Vert \le M\sqrt{n}\) for all \(n \ge n_0\), and this estimate cannot be improved in general; see [6, 11, 17, 22]. In the Hilbert space setting, it is still unknown whether the boundedness of \((e^{-tA})_{t \ge 0}\) implies that of \((V(A)^n )_{n \in \mathbb {N}_0}\). For bounded \(C_0\)-semigroups on Hilbert spaces, the estimates
and
remain the best so far. The former estimate (3) has been obtained in [14], and one can derive the latter estimate (4) by combining [16, Lemma 2.3] and the theory of the \(\mathcal {B}\)-calculus established in [4, 5]. If both A and \(A^{-1}\) generate a bounded \(C_0\)-semigroup on a Hilbert space, then \((V(A)^n)_{n \in \mathbb {N}_0}\) is bounded, which has been independently proved in [1, 14, 19]. Under the additional assumption that \((e^{-tA})_{t \ge 0}\) is strongly stable, \((V(A)^n)_{n \in \mathbb {N}_0}\) is also strongly stable; see [19]. The approach developed in [19] is based on Lyapunov equations and has been extended in [22] from the case of constant parameters \(V(A)^n\) to the case of variable parameters \(\prod _{k=1}^n V_{\omega _k}(A)\), where the sequence \((\omega _k)_{k \in \mathbb {N}_0}\) satisfies \(0 < \inf _{k \in \mathbb {N}}\omega _k\) and \(\sup _{k \in \mathbb {N}}\omega _k< \infty \).
In this paper, we consider bounded \(C_0\)-semigroups on Hilbert spaces with certain decay properties. In particular, we focus on polynomial stability and exponential stability, which are defined as follows:
Definition 1
A \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) on a Banach space is called
-
(a)
polynomially stable with parameter \(\beta >0\) if \((e^{-tA})_{t \ge 0}\) is bounded and satisfies \(\Vert e^{-tA}(I+A)^{-1}\Vert = O(t^{-1/\beta })\) as \(t \rightarrow \infty \), or simply polynomially stable if it is polynomially stable with some parameter; and
-
(b)
exponentially stable if \(\lim _{t \rightarrow \infty } \Vert e^{\varepsilon t}e^{-tA}\Vert = 0\) for some \(\varepsilon >0\).
Note that if the \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) is polynomially stable or exponentially stable, then the negative generator A has a bounded inverse; see [3, Theorem 1.1]. For more information on these stability notions, we refer to the survey article [7].
Several results on the rate of decay of \(\Vert V(A)^nA^{-1}\Vert \) have been developed in the Hilbert space setting. If \((V(A)^n)_{n \in \mathbb {N}_0}\) is bounded, then
for all polynomially stable \(C_0\)-semigroups with parameter \(\beta >0\); see [23]. It has been also proved in [23] that if \(-A\) is normal and generates a polynomially stable \(C_0\)-semigroup with parameter \(\beta >0\), then \(\Vert V(A)^nA^{-1}\Vert = O(n^{-1/(2+\beta )})\) as \(n \rightarrow \infty \) and the decay rate \(n^{-1/(2+\beta )}\) cannot be replaced by a better one in general. In the case where \((e^{-tA})_{t \ge 0}\) is exponentially stable and \((e^{-tA^{-1}})_{t \ge 0}\) is bounded, the following estimate has been derived in [24]:
where \((\omega _k)_{k \in \mathbb {N}}\) satisfies \(0 < \inf _{k \in \mathbb {N}}\omega _k\) and \(\sup _{k \in \mathbb {N}}\omega _k< \infty \). In the proof of the estimates (5) and (6), Lyapunov equations on V(A) and \(-A^{-1}\) play an important role, as in [19, 22]. Hence, the assumptions on the boundedness of \((V(A)^n)_{n \in \mathbb {N}_0}\) and \((e^{-tA^{-1}})_{t \ge 0}\) are placed.
Our aim is to estimate the decay rate of \(\Vert V(A)^nA^{-1}\Vert \) without assuming that \((V(A)^n)_{n \in \mathbb {N}_0}\) or \((e^{-tA^{-1}})_{t \ge 0}\) is bounded. To this end, we apply the \(\mathcal {B}\)-calculus. First, we derive a new estimate for operator norms in the context of the \(\mathcal {B}\)-calculus, which is tailored to polynomially stable \(C_0\)-semigroups on Hilbert spaces. Using this estimate, we next show that
for all polynomially stable \(C_0\)-semigroups \((e^{-tA})_{t \ge 0}\) on Hilbert spaces. The estimate (7) is less sharp than (5) because it has an additional logarithmic term, and we do not know whether the logarithmic term can be omitted. However, it should be emphasized that the boundedness of \((V(A)^n)_{n \in \mathbb {N}_0}\) is not assumed. When \(-A\) is the generator of an exponentially stable \(C_0\)-semigroup on a Hilbert space, we obtain the estimate
where \((\omega _k)_{k \in \mathbb {N}}\) satisfies \(0 < \inf _{k \in \mathbb {N}}\omega _k\) and \(\sup _{k \in \mathbb {N}}\omega _k< \infty \). This result implies that not only the assumption on the boundedness of \((e^{-tA^{-1}})_{t \ge 0}\) but also the logarithmic term \(\sqrt{\log n}\) in the previous result (6) can be omitted. Moreover, we show that the estimate (8) cannot be improved in the case \(A :=iB+I\), where B is a self-adjoint operator with a certain spectral property. A similar improved estimate is provided for a polynomially stable \(C_0\)-semigroup on a Hilbert space such that the generator is normal.
We now turn our attention to the long-time asymptotic behavior of \((e^{-t A^{-1}})_{t \ge 0}\). Let \((e^{-tA})_{t \ge 0}\) be a bounded \(C_0\)-semigroup on a Banach space, and assume that there exists a densely defined algebraic inverse \(A^{-1}\) of A. The inverse generator problem raised by deLaubenfels [8] asks whether \(-A^{-1}\) also generates a \(C_0\)-semigroup, and a survey can be found in [15]. For bounded \(C_0\)-semigroups on Banach spaces, a negative answer to this problem has been provided implicitly in [21, pp. 343–344], and other counterexamples can be found in [13, 18, 26]. The inverse generator problem in the Hilbert space setting remains open. It has been shown in [26] that if the answer is positive for all generators and all Hilbert spaces, then the \(C_0\)-semigroup \((e^{-tA^{-1}})_{t \ge 0}\) is bounded. For exponentially stable \(C_0\)-semigroups on Hilbert spaces, the estimate \(\Vert e^{-tA^{-1}}\Vert = O(\log t)\) as \(t \rightarrow \infty \) has been obtained by using Lyapunov equations in [25] and by connecting the growth bounds of \((e^{-tA^{-1}})_{t \ge 0}\) and \((V(A)^n )_{n \in \mathbb {N}_0}\) in [17]. This estimate has been extended to bounded \(C_0\)-semigroups on Hilbert spaces with invertible generators by means of the \(\mathcal {B}\)-calculus in [4, Corollary 5.7].
Since we are interested in the asymptotics of \(e^{-tA^{-1}} x\) that is uniform with respect to \(x \in D(A)\), the problem we study is to estimate \(\Vert e^{-tA^{-1}}A^{-1}\Vert \). If \((e^{-t A^{-1}})_{t \ge 0}\) is an exponentially stable \(C_0\)-semigroup on a Banach space, then \(\Vert e^{-tA^{-1}} A^{-k}\Vert = O(t^{-k/2+1/4})\) for all \(k \in \mathbb {N}\), which has been established for the case \(k=1\) in [26] and for the case \(k \ge 2 \) in [9]. The \(\mathcal {B}\)-calculus has been applied in [24] in order to obtain the estimate
with \(\alpha >0\) for all exponentially stable \(C_0\)-semigroups on Hilbert spaces. If the generator \(-A\) of a polynomially stable \(C_0\)-semigroup with parameter \(\beta >0\) on a Hilbert space is normal, then
for all \(\alpha >0\); see [24]. In [24], simple examples have been also provided to show that the estimates (9) and (10) cannot be improved in general. For polynomially stable \(C_0\)-semigroups with any parameters on Hilbert spaces, the estimate
has been derived in [24] by the Lyapunov-based approach.
Our second contribution is to improve the estimate (11) in two ways by means of the \(\mathcal {B}\)-calculus. First, we show that \(\sup _{t \ge 0}\Vert e^{-tA^{-1}} A^{-\alpha }\Vert < \infty \) for all \(\alpha >0\) when \(-A\) is the generator of a bounded \(C_0\)-semigroup on a Hilbert space and has a bounded inverse. Second, we give the estimate of the decay rate
for all polynomially stable \(C_0\)-semigroups with parameter \(\beta >0\) on Hilbert spaces. For this, the operator norm estimate exploiting polynomial stability is again applied in the setting of the \(\mathcal {B}\)-calculus. We see from the estimate (12) that a decay estimate for \((e^{-tA})_{t \ge 0}\) is transferred to that for \((e^{-tA^{-1}})_{t \ge 0}\) as in the estimate (7) for \((V(A)^n)_{n \in \mathbb {N}_0}\). There is a question whether the logarithmic term in the estimate (12) can be omitted, and it is open as in other estimates.
This paper is organized as follows: In Sect. 2, we present the notion and some properties of the \(\mathcal {B}\)-calculus. Section 3 is devoted to estimating the rate of decay of the Cayley transform. In Sect. 4, we study the asymptotic behavior of \(\Vert e^{-tA^{-1}} A^{-\alpha }\Vert \) with \(\alpha >0\).
1.1 Notation
Let \(\mathbb {C}_+ :=\{ z \in \mathbb {C}: \mathop {\textrm{Re}}\nolimits z > 0 \}\) and let \(\mathbb {R}_+ :=\{ \xi \in \mathbb {R}: \xi \ge 0 \}\). Let \(\mathbb {Z}\), \(\mathbb {N}\), and \(\mathbb {N}_0\) denote the set of integers, the set of positive integers, and the set of nonnegative integers, respectively. For a real number \(\xi \), let \(\lfloor \xi \rfloor :=\max \{k \in \mathbb {Z}: k \le \xi \}\). Given functions \(f,g:[t_0,\infty ) \rightarrow (0,\infty )\), we write
if there exist constants \(M>0\) and \(t_1 \ge t_0\) such that \(f(t) \le M g(t)\) for all \(t \ge t_1\). When \(0< \omega _p \le \omega _q < \infty \), we denote by \(\mathcal {S}(\omega _p,\omega _q)\) the set of sequences \((\omega _k)_{k \in \mathbb {N}}\) of positive real numbers satisfying \( \omega _p\le \omega _k \le \omega _q\) for all \(k \in \mathbb {N}\).
Let X be a Banach space. We denote by \(\mathcal {L}(X)\) the Banach algebra of bounded linear operators on X. For a linear operator A on X, let D(A), \(\sigma (A)\), and \(\varrho (A)\) denote the domain, the spectrum, and the resolvent set of A, respectively. Let \(-A\) be the generator of a bounded \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) on X. We define the Cayley transform \(V_\omega (A)\) of A with parameter \(\omega >0\) by \(V_\omega (A) :=(A - \omega I )(A + \omega I)^{-1}\). For simplicity of notation, we set \(V(A) :=V_1(A)\). If A is injective, the fractional power \(A^{\alpha }\) of A is defined by the sectoral functional calculus for \(\alpha \in \mathbb {R}\); see [20, Chapter 3].
Let H be a Hilbert space. We denote the inner product on H by \(\langle \cdot , \cdot \rangle \). For a densely defined linear operator A on H, the Hilbert space adjoint of A is denoted by \(A^*\).
2 \(\mathcal {B}\)-calculus
In this section, first we recall the notion of the \(\mathcal {B}\)-calculus and its basic properties. Then, we provide a new operator norm estimate in the \(\mathcal {B}\)-calculus for the negative generator of a polynomially stable \(C_0\)-semigroup on a Hilbert space.
2.1 Definition and basic facts
We provide some background material on the \(\mathcal {B}\)-calculus and refer the reader to [4, Sections 2 and 4] for more details. Let \(\mathcal {B}\) be the algebra of holomorphic functions f on \(\mathbb {C}_+\) satisfying
For all \(f \in \mathcal {B}\), the limit \(f(\infty ) :=\lim _{\mathop {\textrm{Re}}\nolimits z \rightarrow \infty } f(z)\) exists in \(\mathbb {C}\) and
Moreover, \(\mathcal {B}\) equipped with the norm
is a Banach algebra.
Let \(\textrm{M}(\mathbb {R}_+)\) be the space of all bounded Borel measures \(\mu \) on \(\mathbb {R}_+\), endowed with the total variation norm \(\Vert \mu \Vert _{\textrm{M}(\mathbb {R}_+)} :=|\mu |(\mathbb {R}_+)\). Then \(\textrm{M}(\mathbb {R}_+)\) is a Banach algebra with respect to the convolution product. We identify \(L^1(\mathbb {R}_+)\) with a subalgebra of \(\textrm{M}(\mathbb {R}_+)\). The Laplace transform of \(\mu \in \textrm{M}(\mathbb {R}_+)\) is the function on \(\mathbb {C}_+\) defined by
Define \(\mathcal{L}\mathcal{M} :=\{ \mathcal {L}\mu : \mu \in \textrm{M}(\mathbb {R}_+) \}\). Then \(\mathcal{L}\mathcal{M}\) equipped with the norm
is a Banach algebra. If \(f \in \mathcal{L}\mathcal{M}\), then \(f \in \mathcal {B}\) and \( \Vert f\Vert _{\mathcal {B}} \le 2 \Vert f\Vert _{ \textrm{HP}}\).
We present some examples of \(\mathcal{L}\mathcal{M}\)-functions, which will be used to obtain decay estimates for the Cayley transform and the inverse of semigroup generators.
Example 1
-
(a)
Let \(c >0\) and \(d \in \mathbb {R}\). Define
$$\begin{aligned} u_{c,d}(z) :=\frac{z+d}{z+c},\quad z \in \mathbb {C}_+. \end{aligned}$$Then \(u_{c,d}\) is the Laplace transform of \(\delta _0 - (c-d)e_{c} \in \textrm{M}(\mathbb {R}_+)\), where \(\delta _0\) is the Dirac measure concentrated at 0 and \(e_{c}(t) :=e^{-ct}\) for \(t \ge 0\). Hence, the n-th power \((u_{c,d})^n\) belongs to \(\mathcal{L}\mathcal{M}\) for every \(n \in \mathbb {N}\).
-
(b)
Let \(\alpha ,c >0\). Define
$$\begin{aligned}v_{\alpha ,c} (z) :=\frac{1}{(z+c)^{\alpha }},\quad z \in \mathbb {C}_+.\end{aligned}$$By the definition of the gamma function \(\Gamma \), we obtain
$$\begin{aligned} \Gamma (\alpha ) = \int ^{\infty }_0 t^{\alpha -1}e^{-t}\textrm{d}t = (z+c)^{\alpha } \int ^{\infty }_0 t^{\alpha -1} e^{-(z+c)t}\textrm{d}t \end{aligned}$$for all positive real numbers \(z >0\). Together with the uniqueness theorem for holomorphic functions, this implies that \(v_{\alpha ,c}\) is the Laplace transform of the function \(\phi _{\alpha ,c} \in L^1(\mathbb {R}_+)\) defined by
$$\begin{aligned} \phi _{\alpha ,c}(t) :=\frac{t^{\alpha -1} e^{-ct}}{\Gamma (\alpha )},\quad t >0. \end{aligned}$$Therefore, \(v_{\alpha ,c} \in \mathcal{L}\mathcal{M}\).
-
(c)
For a fixed \(t >0\), define
$$\begin{aligned} w_t (z) :=\frac{z}{z+1} e^{-t/z},\quad z \in \mathbb {C}_+. \end{aligned}$$Then, \(w_t\) is the Laplace transform of some \(L^1(\mathbb {R}_+)\)-function; see the proof of [9, Theorem 3.3]. Hence, \(w_t \in \mathcal{L}\mathcal{M}\).
Now we introduce a functional calculus for \(\mathcal {B}\). Let \(-A\) be the generator of a bounded \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) on a Hilbert space H. We define \(K :=\sup _{t \ge 0} \Vert e^{-tA}\Vert \). For all \(x\in H\), the Plancherel theorem gives
and hence
Similarly,
for all \(y \in H\). Combining these estimates with the Cauchy–Schwartz inequality, we obtain
for all \(x,y \in H\). Let \(f \in \mathcal {B}\), and define the linear operator f(A) on H by
for \(x,y \in H\). By the estimate (15), f(A) is bounded on H and satisfies
The map
is a bounded algebra homomorphism. We refer to \(\Phi _A\) as the \(\mathcal {B}\)-calculus for A.
Define the map \(\Pi _A:\mathcal{L}\mathcal{M} \rightarrow \mathcal {L}(H)\) by
for \(f=\mathcal {L}\mu \in \mathcal{L}\mathcal{M}\) and \(x \in H\). The map \(\Pi _A\) is a bounded algebra homomorphism and is called the Hille–Phillips calculus for A. The \(\mathcal {B}\)-calculus extends the Hille–Phillips calculus in the sense that \(\Phi _A(f) = \Pi _A(f)\) for all \(f \in \mathcal{L}\mathcal{M}\).
We apply the \(\mathcal {B}\)-calculus to the functions presented in Example 1.
Example 2
Let \(-A\) be the generator of a bounded \(C_0\)-semigroup on a Hilbert space such that \(0 \in \varrho (A)\). We define the functions \(u_{c,d},v_{\alpha ,c},w_t \in \mathcal{L}\mathcal{M}\) as in Example 1.
-
(a)
Since \(\Pi _A(u_{c,d}) = (A+dI) (A+cI)^{-1}\), we have
$$\begin{aligned} (u_{c,d})^n(A) = ((A+dI) (A+cI)^{-1})^n. \end{aligned}$$ -
(b)
One has \(\Pi _A(v_{\alpha ,c} ) = (A+cI)^{-\alpha }\) by [20, Proposition 3.3.5], and hence \(v_{\alpha ,c}(A) = (A+cI)^{-\alpha }\). Moreover, since
$$\begin{aligned} A^{-\alpha } = (A+cI)^{-\alpha } (I+c A^{-1})^{\alpha }, \end{aligned}$$(18)the estimate (17) yields
$$\begin{aligned} \Vert f(A) A^{-\alpha }\Vert \le 2 K^2\Vert (I+c A^{-1})^{\alpha }\Vert \, \Vert fv_{\alpha ,c}\Vert _{\mathcal {B}_0} \end{aligned}$$for all \(f \in \mathcal {B}\). We shall use this estimate frequently without comment.
-
(c)
The argument in the proof of [4, Corollary 5.8] shows that
$$\begin{aligned} w_{t}(A) = A(A+I)^{-1} e^{-tA^{-1}}. \end{aligned}$$
2.2 Operator norm estimate for polynomially stable semigroups
The norm estimate (17) can be applied to every negative generator A of a bounded \(C_0\)-semigroup on a Hilbert space. Here we present a sharper norm estimate for polynomially stable \(C_0\)-semigroups on Hilbert spaces. First we recall a useful property on rates of polynomial decay for bounded \(C_0\)-semigroups on Banach spaces; see [2, Proposition 3.1] for the proof.
Proposition 1
Let \(-A\) be the generator of a bounded \(C_0\)-semigroup \((e^{-tA})_{t\ge 0}\) on a Banach space X such that \(0 \in \varrho (A)\). Then, the following two statements are equivalent for a fixed \(\beta >0\):
-
1.
\(\Vert e^{-tA}A^{-1}\Vert = O(t^{-1/\beta })\) as \(t \rightarrow \infty \).
-
2.
\(\Vert e^{-tA}A^{-\beta q}\Vert = O(t^{-q})\) as \(t \rightarrow \infty \) for some/all \(q>0\).
Next we establish a resolvent estimate analogous to (13) for polynomially stable \(C_0\)-semigroups on Hilbert spaces. The proof is similar to that of [23, Proposition 3.1], but in order to make this paper self-contained, we give a short argument.
Lemma 1
Let \(-A\) be the generator of a bounded \(C_0\)-semigroup \((e^{-tA})_{t\ge 0}\) on a Hilbert space H such that \(0 \in \varrho (A)\). Then, the following two statements are equivalent for fixed \(\beta >0\) and \(q \in (0,1/2)\):
-
1.
\(\Vert e^{-tA}A^{-1}\Vert = O(t^{-1/\beta })\) as \(t \rightarrow \infty \).
-
2.
There exists \(M>0\) such that for all \(x \in H\),
$$\begin{aligned} \sup _{0<\xi <1} \xi ^{1-2q } \int _{-\infty }^{\infty } \Vert (\xi +i\eta +A)^{-1}A^{-\beta q } x\Vert ^2 \textrm{d}\eta \le M\Vert x\Vert ^2. \end{aligned}$$(19)
Proof
Let \(q \in (0,1/2)\) and let \(K_1 :=\sup _{t \ge 0} \Vert e^{-tA}\Vert \). By Proposition 1, the statement 1 holds if and only if there exist constants \(K_2,t_0 >0\) such that
(Proof of 1 \(\Rightarrow \) 2) Let \(x \in H\). The Plancherel theorem gives
for all \(\xi >0\). Using the estimate (20), we obtain
for all \(\xi >0\). Since
where \(\Gamma \) is a gamma function, we obtain
From (21) and (22), we conclude that the statement 2 holds.
(Proof of 2 \(\Rightarrow \) 1) Using the inverse formula given in [12, Corollary III.5.16], we have that for all \(x \in D(A^2)\), \(y \in H\), and \(t,\xi >0\),
see also [5, Corollary 8.14] for the inversion formula. Together with (14), (19), and the Cauchy–Schwartz inequality, this estimate implies that there exists a constant \(M_1>0\) such that
for all \(x \in D(A^2)\), \(t>0\), and \(0< \xi < 1\). Since \(D(A^2)\) is dense in H, setting \(\xi = 1/t\) yields
for all \(t > 1\). Thus, \(\Vert e^{-tA}A^{-1}\Vert = O(t^{-1/\beta })\) is obtained. \(\square \)
Let \(q >0\), and define
where
Since \(0 < \psi _q(\xi ) \le 1\) for all \(\xi >0\), we have \(\Vert f\Vert _{\mathcal {B}_0,q} \le \Vert f\Vert _{\mathcal {B}_0}\) for all \(f \in \mathcal {B}\). When \(-A\) is the generator of a polynomially stable \(C_0\)-semigroup with parameter \(\beta >0\), the operator norm for \(f(A)A^{-\beta q }\) can be upper-bounded by using \(\Vert f\Vert _{\mathcal {B}_0,q }\).
Proposition 2
Let \(-A\) be the generator of a polynomially stable \(C_0\)-semigroup with parameter \(\beta >0\) on a Hilbert space H. Then, for all \(q \in (0,1/2)\), there exists \(M>0\) such that
for all \(f \in \mathcal {B}\).
Proof
Let \(q \in (0,1/2)\) and let \(K :=\sup _{t \ge 0} \Vert e^{-tA}\Vert \). For all \(x, y \in H\) and \(f\in \mathcal {B}\),
To the right-hand side, we apply Lemma 1 for \(0<\xi < 1\) and the estimate (13) for \(\xi \ge 1\). For \(x \in D(A^{\beta q })\), we replace x by \(A^{\beta q}x\) in Lemma 1 and use the inequality \(\Vert x\Vert \le \Vert A^{-\beta q}\Vert \, \Vert A^{\beta q} x\Vert \) in (13). Then, we see that there exists \(M_1>0\) such that
for all \(x \in D(A^{\beta q })\). The estimates (14) and (26) together with the Cauchy–Schwartz inequality imply
for all \(x \in D(A^{\beta q })\) and \(y \in H\). Applying the estimates (25) and (27) to the definition (16) of f(A), we derive
for all \(x \in D(A^{\beta q })\), \(y \in H\), and \(f \in \mathcal {B}\). Thus,
for all \(f \in \mathcal {B}\). \(\square \)
In Sects. 3 and 4, we will derive operator norm estimates for the Cayley transform and the inverse of a semigroup generator from estimates of \(\Vert f\Vert _{\mathcal {B}_0}\) or \(\Vert f\Vert _{\mathcal {B}_0,q}\) for the corresponding \(\mathcal {B}\)-functions f, by using the inequalities (17) and (24). The inequality (24) will be applied in the following way.
Example 3
Let \(q \in (0, 1/2)\) and let \(-A\) be the generator of a polynomially stable \(C_0\)-semigroup with parameter \(\beta >0\) on a Hilbert space. For \(\alpha ,c >0\), we define \(v_{\alpha ,c} \in \mathcal{L}\mathcal{M}\) as in Example 1.(b). Using (18), we obtain
for all \(f \in \mathcal {B}\). Therefore, Proposition 2 shows that there exists \(M>0\) such that
for all \(f \in \mathcal {B}\).
3 Estimates for Cayley transforms
In this section, first we study the asymptotic behavior of \(\Vert V(A)^n A^{-\alpha }\Vert \) for \(\alpha >0\) when \(-A\) is the generator of a polynomially stable \(C_0\)-semigroup on a Hilbert space. Next we turn our attention to exponentially stable \(C_0\)-semigroups on Hilbert spaces and the case of variable parameters \(\Vert ( \prod _{k=1}^nV_{\omega _k}(A) )A^{-\alpha } \Vert \).
3.1 Case of constant parameters
When \(-A\) is the generator of a polynomially stable \(C_0\)-semigroup on a Hilbert space, the following estimate for \(\Vert V(A)^n A^{-\alpha }\Vert \) holds without the assumption that \((V(A)^n)_{n \in \mathbb {N}_0}\) is bounded. Recall that \(\lfloor \xi \rfloor \) is the largest integer less than or equal to \(\xi \in \mathbb {R}\).
Theorem 1
Let \(-A\) be the generator of a polynomially stable \(C_0\)-semigroup with parameter \(\beta >0\) on a Hilbert space H. Then, for all \(\alpha >0\),
Let \(n \in \mathbb {N}\) and \(p >0\). Define
Then \(f_{n,2p} \in \mathcal{L}\mathcal{M}\); see Example 1. To prove Theorem 1, we need a preliminary estimate for \(\Vert f_{n,2p}\Vert _{\mathcal {B}_0, q }\).
Proposition 3
For \(n \in \mathbb {N}\) and \(p >0\), define the function \(f_{n,2p}\) by (29). Then, for all \(q \in (0,1/2)\),
Proof
Step 1: Let \(n \in \mathbb {N}\). Fix \(p >0\) and \( q \in (0,1/2)\). Since
we have
where
for \(\xi >0\) and \(s \ge 0\). In Steps 2 and 3 below, we will show that
and
Combining the estimates (31)–(33), we obtain the desired conclusion (30).
Step 2: The aim of this step is to show the estimate (32) for \(g_{n,p}\). Fix \(\xi > 0\), and define
for \(s \ge 0\). Then,
where
Since
the equation \(\chi (s) =0\) has the following solution:
Moreover, the solution \(s_1\) satisfies \(s_1 \ge 0\) if and only if \( \xi _0 \le \xi \le \xi _1, \) where
Note that \(\xi _1 >1\). Hence \(\xi _0 = 1/\xi _1 < 1\). We also have
Since routine calculations show that
we obtain
Combining this with (34), we obtain
By the definition (23) of \(\psi _{ q }\),
If \(p \not =1/2\) and \(p\not = q+1/2 \), then
and hence we have from \(\xi _0(n) = O(n^{-1})\) and \(\xi _1(n) = O(n)\) as \(n \rightarrow \infty \) that
as \(n \rightarrow \infty \). Similarly, if \(p = 1/2\) or \(p= q + 1/2\), then
as \(n \rightarrow \infty \). On the other hand, we derive from (34) that
and
Therefore,
and
as \(n \rightarrow \infty \). Thus, the estimate (32) for \(g_{n,p}\) holds, and it is actually less sharp than the estimate obtained from the above argument.
Step 3: We immediately obtain the estimate (33) for \(h_{n,p}\), by replacing n with \(n-1\) and p with \(p+1/2\) in the estimates (35)–(38). \(\square \)
We are well prepared to obtain the estimate (28) for \(\Vert V(A)^n A^{-\alpha }\Vert \).
Proof of Theorem 1
Let \(p \in (0,1/2)\) and define the function \(f_{n,2p}\) by (29) for \(n \in \mathbb {N}\). By Example 3 in the case \(\alpha = 2p\), \(c=1\), \(q = p\), and
there exists a constant \(M>0\) such that
for all \(n \in \mathbb {N}\). From the estimate (30) with \(p = q\), we have
as \(n \rightarrow \infty \). Hence, the desired estimate (28) holds in the case \(k = \lfloor 2\alpha /(2+\beta ) \rfloor =0\).
Next we consider the case \(k = \lfloor 2\alpha /(2+\beta ) \rfloor \ge 1\). To this end, choose \(\alpha \ge (2+\beta )/2\) arbitrarily. Let \(k \in \mathbb {N}\) and \(0 \le \delta _0 < (2+\beta )/2\) satisfy
Then there exists \(\varepsilon >0\) such that
By definition,
and hence
for all \(n \in \mathbb {N}\). Using the estimate (39), we obtain
and
as \(n \rightarrow \infty \). Thus, the desired estimate (28) holds. \(\square \)
Remark 1
In the proof of Theorem 1, we have used the estimate (30) with \(p = q \) for \(\Vert f_{n,2p}\Vert _{\mathcal {B}_0, q }\). This is because the estimate (30) with \(p \not = q \) yields a worse decay rate. To see this, assume that the \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) is polynomially stable with parameter \(\beta >0\). The estimate (30) with \(p< q \, (< 1/2) \) gives
If \(2p+\beta q = 1\) and \(p < q \), then
Hence, the estimate (40) with \(2p+\beta q = 1\) is worse than the estimate (28) with \(\alpha = 1\). A similar argument can be applied to the estimate (30) with \(p > q \).
3.2 Case of variable parameters
Here we study the case where the parameter \(\omega \) of the Cayley transform \(V_{\omega }(A)\) varies. First we improve the previous estimate developed in [24, Theorem 3.3] for exponentially stable \(C_0\)-semigroups. Next we consider polynomially stable \(C_0\)-semigroups with normal generators and present a similar improved estimate over the one given in [24, Proposition 3.6].
3.2.1 Exponentially stable semigroups
When the \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) is exponentially stable, we give an estimate for \(\Vert (\prod _{k=1}^n V_{\omega _k}(A) ) A^{-\alpha } \Vert \) without assuming the boundedness of \((e^{-tA^{-1}})_{t \ge 0}\). Before that, we define the function \(F_{\alpha }\) on \(\mathbb {N}\) by
Recall that for \(0<\omega _p\le \omega _q < \infty \), we denote by \(\mathcal {S}(\omega _p,\omega _q)\) the set of sequences \((\omega _k)_{k \in \mathbb {N}}\) of positive real numbers satisfying \( \omega _p\le \omega _k \le \omega _q\) for all \(k \in \mathbb {N}\).
Theorem 2
Let \(-A\) be the generator of an exponentially stable \(C_0\)-semigroup on a Hilbert space H. Then, for all \(\alpha \ge 0\) and \(0< \omega _p \le \omega _q < \infty \), there exist constants \(M>0\) and \(n_0 \in \mathbb {N}\) such that
for all \(n \ge n_0\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\), where the function \(F_{\alpha }\) is as in (41).
When \((e^{-tA})_{t \ge 0}\) is an exponentially stable \(C_0\)-semigroup, \(-A + cI\) generates a bounded \(C_0\)-semigroup for some sufficiently small constant \(c>0\). To prove Theorem 2 by means of the \(\mathcal {B}\)-calculus, we consider the function \(f_{n,\alpha ,(\omega _k)}\) defined by
where \(0< \omega _p \le \omega _q < \infty \) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\). As seen in Example 1, we have \(f_{n,\alpha ,(\omega _k)} \in \mathcal{L}\mathcal{M}\). The following result gives an estimate for \(\Vert f_{n,\alpha ,(\omega _k)}\Vert _{\mathcal {B}_0}\).
Proposition 4
Let \(\alpha \ge 0\), \(c> 0\), and \(0< \omega _p \le \omega _q < \infty \). Then there exist constants \(M>0\) and \(n_0 \in \mathbb {N}\) such that the function \(f_{n,\alpha ,(\omega _k)} \) defined by (43) satisfies
for all \(n \ge n_0\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\), where the function \(F_{\alpha }\) is as in (41).
The difficulty in estimating \(\Vert f_{n,\alpha ,(\omega _k)}\Vert _{\mathcal {B}_0}\) is that the parameter \(\omega _k\) can vary on \([\omega _p,\omega _q]\), which complicates the computation of \( \sup _{\eta \in \mathbb {R}} |f_{n,\alpha ,(\omega _k)}'(\xi +i \eta )|. \) To circumvent this difficulty, we show that the function on \([\omega _p,\omega _q]\) defined by
takes the maximum value at \(\omega = \omega _p\) for all \(\xi > 0\) and \(\eta \in \mathbb {R}\) under a suitable condition on the constant c.
Lemma 2
Let \(c> 0\) and \(0< \omega _p \le \omega _q < \infty \) satisfy \(c^2 \ge \omega _p \omega _q\). If \(\omega _p \le \omega \le \omega _q\), then
for all \(\xi >0\) and \(s \ge 0\).
Proof
Fix \(\xi >0\) and \(s \ge 0\). Set \(\zeta :=\xi + c\). Define
Then
Therefore, \(\omega _0 :=\sqrt{\zeta ^2+s}\) satisfies \(\phi _{\zeta ,s}'(\omega _0) = 0\), and \(\phi _{\zeta ,s}\) decreases on the interval \((0,\omega _0)\) and increases on the interval \((\omega _0,\infty )\). This implies that for all \(\omega \in [\omega _p, \omega _q]\),
It remains to show that if \(c^2 \ge \omega _p \omega _q\), then
This can be proved by a direct calculation. We have
where
One can rewrite \(\chi _{\zeta ,s}\) as
The coefficient of s satisfies
Moreover, a routine calculation shows that
Substituting \(\zeta = \xi + c\), we have from the condition \(c^2 \ge \omega _p \omega _q\) that
Thus, the inequality (45) holds. \(\square \)
Next we give auxiliary estimates to be used after Lemma 2 is applied.
Lemma 3
Let \(n \in \mathbb {N}\), \(\alpha \ge 0\), and \(\omega ,c >0\). Define
for \(\xi >0\) and \(s \ge 0\). Then
and
Proof
It suffices to show the estimate (48) for \(g_{n,\alpha ,\omega }\). Indeed, the estimate (49) for \(h_{n,\alpha ,\omega }\) is obtained by replacing n with \(n-1\) and \(\alpha \) with \(\alpha +1\) in the estimate (48) for \(g_{n,\alpha ,\omega }\).
Let \(n \in \mathbb {N}\), \(\alpha \ge 0\), and \(\omega >0\). To obtain the estimate for \(g_{n,\alpha ,\omega }\), fix \(\xi >0\) and define
for \(s \ge 0\). Then,
where
We have
Therefore, \(\chi (s_1) = 0\), where \(s_1\) is defined by
Since \(\omega , c >0\), we obtain
for all sufficiently large \(n \in \mathbb {N}\). Hence, there exists \(n_1 \in \mathbb {N}\) such that the equation
has a unique positive solution \(\xi _1 = \xi _1(n)\) for all \(n \ge n_1\), and
Let \(n \ge n_1\). We obtain
Routine calculations show that
Hence we obtain
This and (50) give
Noting that \(\xi _1(n) = O(n)\) as \(n \rightarrow \infty \), we have
as \(n \rightarrow \infty \). We also see from (50) that
Hence,
as \(n \rightarrow \infty \). Combining the estimates (51) and (52), we obtain the estimate (48) for \(g_{n}\). \(\square \)
Now we are in a position to prove Proposition 4 and Theorem 2.
Proof of Proposition 4
Define \({\tilde{\omega }}_p :=\min \{ \omega _p,\,c^2/\omega _q \} \). Then \(\mathcal {S}(\omega _p,\omega _q) \subset \mathcal {S}({\tilde{\omega }}_p,\omega _q)\) and \(c^2 \ge {\tilde{\omega }}_p \omega _q\). Replacing \(\omega _p\) with \({\tilde{\omega }}_p\) in the definition (43) of \(f_{n,\alpha ,(\omega _k)}\), we define the function \({\tilde{f}}_{n,\alpha ,(\omega _k)}\) by
where \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}({\tilde{\omega }}_p,\omega _q)\). Then, for all \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\),
where
Since \(r_1 \in \mathcal{L}\mathcal{M}\) (see Example 1.(a)), we have from [4, Proposition 2.2] that \(r_{\alpha } \in \mathcal {B}\).
Now we apply Lemmas 2 and 3 to \({\tilde{f}}_{n,\alpha ,(\omega _k)}\). The derivative of \({\tilde{f}}_{n,\alpha ,(\omega _k)}\) is given by
Let \(\xi >0\), \(\eta \in \mathbb {R}\), and \(s :=\eta ^2 \ge 0\). Since \(c^2 \ge {\tilde{\omega }}_p \omega _q\), Lemma 2 shows that
for all \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}({\tilde{\omega }}_p,\omega _q)\), where \(g_{n,\alpha ,{\tilde{\omega }}_p}\) and \(h_{n,\alpha ,{\tilde{\omega }}_p}\) are defined by (46) and (47) with \(\omega ={\tilde{\omega }}_p\), respectively. From Lemma 3, we see that there exist constants \({\tilde{M}}>0\) and \(n_0 \in \mathbb {N}\) such that
for all \(n \ge n_0\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}({\tilde{\omega }}_p,\omega _q)\). Since \(\mathcal {S}(\omega _p,\omega _q) \subset \mathcal {S}({\tilde{\omega }}_p,\omega _q) \) and since
for all \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\), the constant \(M :=2\Vert r_{\alpha }\Vert _{\mathcal {B}} {\tilde{M}} \) satisfies
for all \(n \ge n_0\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\). \(\square \)
Proof of Theorem 2
Since the \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) is exponentially stable, there exist constants \(K \ge 1\) and \(c >0\) such that
for all \(t \ge 0\). Define \(B :=A - c I\). Then \(-B\) is the generator of a bounded \(C_0\)-semigroup \((e^{-tB})_{t \ge 0}\), and \(\sup _{t \ge 0} \Vert e^{-tB}\Vert \le K\).
Define the function \(f_{n,\alpha ,(\omega _k)}\) by (43). Then,
The inequality (17) gives
Combining this with Proposition 4, we see that there exist constants \(M>0\) and \(n_0 \in \mathbb {N}\) such that the desired estimate (42) holds for all \(n \ge n_0\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\).
\(\square \)
We provide some remarks on the optimality of the estimates given in Proposition 4 and Theorem 2.
Remark 2
The estimate (44) for \(\Vert f_{n,\alpha ,(\omega _k)}\Vert _{\mathcal {B}_0}\) with \(\alpha >0\) cannot be improved in general even in the constant case \(\omega _k \equiv 1\). To see this, fix \(c>0\), and define
where \(n \in \mathbb {N}\) and \(\alpha >0\). Then, we have
Since
there exist \(M_1>0\) and \(n_1 \in \mathbb {N}\) such that
for all \(n \ge n_1\). The estimates (54) and (55) yield
for all \(n \ge n_1\).
Remark 3
One cannot in general improve the operator norm estimate (42) for \(\left\| \left( \prod _{k=1}^n V_{\omega _k}(A)\right) A^{-\alpha } \right\| \) with \(\alpha >0\) either. Let B be a self-adjoint operator on a Hilbert space H, and assume that
for some \(n_2 \in \mathbb {N}\). Fix \(c>0\) and define \(A :=iB + cI\). Then \(-A\) is the generator of an exponentially stable \(C_0\)-semigroup on H. For \(n \in \mathbb {N}\) and \(\alpha >0\), define the function \(f_{n,\alpha } \in \mathcal {B}\) by (53). Since
the spectral mapping theorem (see, e.g., [20, Theorem 2.7.8]) shows that
By the estimate (55),
for all \(n \ge \max \{n_1,n_2 \}\), where \(M_2 :=M_1/ \Vert ( I +A^{-1})^{-\alpha }\Vert \).
Remark 4
The argument in the proof of Proposition 4 gives a non-sharp constant \(M >0\) for the estimate (44), although the decay rate \(n^{-\alpha /2}\) cannot be improved in general. In the proof, we have replaced \(\omega _p\) by \(c^2/\omega _q\) when \(c^2/\omega _q < \omega _p\). Consequently, the constant M there may allow a unnecessary large variation of \((\omega _k)_{k\in \mathbb {N}}\) when c is small and \(\omega _q\) is large. Finding a sharp constant M for given constants \(\omega _p\) and \(\omega _q\) is left as a topic for further research.
3.2.2 Polynomially stable semigroups with normal generators
Using spectral properties of normal operators, one can also obtain an estimate of \(\Vert (\prod _{k=1}^n V_{\omega _k}(A) ) A^{-\alpha }\Vert \) for the normal generator \(-A\) of a polynomially stable \(C_0\)-semigroup. A similar result holds for multiplication operators on \(L^p\)-spaces and spaces of continuous functions that vanish at infinity.
Proposition 5
Consider the following two cases:
-
(a)
Let \((e^{-tA})_{t\ge 0}\) be a bounded \(C_0\)-semigroup on a Hilbert space H such that A is normal.
-
(b)
Let \(\Omega \) be a locally compact Hausdorff space, and let \(\mu \) be a \(\sigma \)-finite regular Borel measure on \(\Omega \). Assume that either
-
(i)
\(X :=L^p(\Omega ,\mu )\) for \(1\le p < \infty \) and \(\phi :\Omega \rightarrow \mathbb {C}\) is measurable with essential range in \(\mathbb {C}_+\); or that
-
(ii)
\(X :=C_0(\Omega )\) and \(\phi :\Omega \rightarrow \mathbb {C}\) is continuous with range in \(\mathbb {C}_+\).
Let A be the multiplication operator induced by \(\phi \) on X, i.e, \(Af = \phi f\) with domain \(D(A) :=\{f \in X:\phi f \in X\}\).
-
(i)
In both cases (a) and (b), assume that \(\Vert e^{-tA} (I+A)^{-1} \Vert = O(t^{-1/\beta })\) as \(t\rightarrow \infty \) for some \(\beta >0\). Then, for all \(\alpha >0\) and \(0< \omega _p \le \omega _q < \infty \), there exist constants \(M>0\) and \(n_0 \in \mathbb {N}\) such that
for all \(n \ge n_0\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\).
Proof
Step 1: Let \(n \in \mathbb {N}\), \(\alpha >0\), and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\). Define
for \(z \in \mathbb {C}_+\). The estimate \(\Vert e^{-tA} (I+A)^{-1} \Vert = O(t^{-1/\beta })\) as \(t\rightarrow \infty \) implies that \(\sigma (A)\) does not intersect with the imaginary axis \(i \mathbb {R}\); see [3, Theorem 1.1]. Hence \(\sigma (A) \subset \mathbb {C}_+\) and A has a bounded inverse in both cases (a) and (b). Moreover,
where we used the spectral mapping theorem (see, e.g., [20, Theorem 2.7.8]) for the case (a) and [12, Propositions I.4.2 and I.4.10] for the case (b); see also the proof of [23, Proposition 4.5], which provides a detailed derivation of (57) in the case \(\omega _k \equiv 1\).
By (57), it suffices to show that there exist constants \(M>0\) and \(n_0 \in \mathbb {N}\) such that
for all \(n \ge n_0\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\). We have from Proposition 1 that \(\Vert e^{-tA} A^{-\beta } \Vert = O(t^{-1})\) as \(t \rightarrow \infty \). Hence, in both cases (a) and (b), there exist \(\delta , C>0\) such that \(|\mathop {\textrm{Im}}\nolimits \lambda | \ge C (\mathop {\textrm{Re}}\nolimits \lambda )^{-1/\beta }\) for all \(\lambda \in \sigma (A)\) with \(\mathop {\textrm{Re}}\nolimits \lambda \le \delta \); see [2, Propositions 4.1 and 4.2]. Define
We divide the proof of the estimate (58) into two cases: \(\lambda \in \Omega _1\) and \(\lambda \in \Omega _2\). The following observation will be useful for later purposes: For all \(\lambda \in \sigma (A)\) and \(\omega >0\),
Step 2: First we assume that \(\lambda \in \Omega _1\). We have
for some constant \(C_0 \ge 1\) independent of \(\lambda \). Combining this estimate with \(\mathop {\textrm{Re}}\nolimits \lambda \ge C^{\beta } / |\mathop {\textrm{Im}}\nolimits \lambda |^{\beta }\), we obtain
for all \(\omega \in [\omega _p,\omega _q]\). Then (59) gives that for all \(\omega \in [\omega _p,\omega _q]\),
Hence,
for all \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\).
Let \(a,b,c>0\), and define the function \(g_n\) by
Then,
for all \(s > c^{1/b}\). Since \(g_n\) takes the maximum value at
we obtain
This and the estimate (60) show that there exist \(M_1>0\) and \(n_1 \in \mathbb {N}\) such that
for all \(n \ge n_1\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\).
Step 3: Next we assume that \(\lambda \in \Omega _2\). Since \(\mathop {\textrm{Re}}\nolimits \lambda > \delta \), we obtain
for all \(\omega \in [\omega _p,\omega _q]\). Together with (59), this implies that for all \(\omega \in [\omega _p,\omega _q]\),
From this estimate, we derive
for all \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\). Using the estimate (61) again, we see that there exist \(M_2>0\) and \(n_2\in \mathbb {N}\) such that
for all \(n \ge n_2\) and \((\omega _k)_{k \in \mathbb {N}} \in \mathcal {S}(\omega _p,\omega _q)\). Thus, the desired estimate (58) is obtained from (62) and (63). \(\square \)
4 Estimates for inverse generators
In this section, we are interested in the asymptotic behavior of \(\Vert e^{-tA^{-1}}A^{-\alpha }\Vert \) for \(\alpha >0\), which has been studied in the Banach space case [9, 26] and the Hilbert space case [24] as mentioned in Sect. 1. First we show that \(\Vert e^{-tA^{-1}}A^{-\alpha }\Vert \) is uniformly bounded for \(t \ge 0\) if \(-A\) is the generator of a bounded \(C_0\)-semigroup on a Hilbert space and has a bounded inverse. Next we estimate the decay rate of \(\Vert e^{-tA^{-1}}A^{-\alpha }\Vert \) for a polynomially stable \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) on a Hilbert space.
4.1 Bounded semigroups
We present an estimate of \(\Vert e^{-tA^{-1}}A^{-\alpha }\Vert \) for a bounded \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) such that the generator \(-A\) has a bounded inverse.
Theorem 3
Let \(-A\) be the generator of a bounded \(C_0\)-semigroup on a Hilbert space H such that \(0 \in \varrho (A)\). Then, for all \(\alpha >0\),
Let \(t >0\) and \(\alpha \ge 0\). To prove Theorem 3, we consider the function \(f_{t,\alpha }\) on \(\mathbb {C}_+\) defined by
One has \(f_{t,\alpha } \in \mathcal{L}\mathcal{M}\); see Example 1. In [5, Lemma 5.6], the estimate in the case \(\alpha = 0\),
has been obtained. The next proposition gives an estimate for \(\Vert f_{t,\alpha }\Vert _{\mathcal {B}_0}\) in the case \(\alpha >0\).
Proposition 6
Let \(t,\alpha >0\) and define the function \(f_{t,\alpha }\) by (65). Then,
The estimate (67) can be proved in a way similar to the estimate (66). We need the following auxiliary result.
Lemma 4
Let \(p >0\), and define
for \(t,\xi >0\) and \(s \ge 0\). Then
Proof
Define \(\zeta :=t \xi >0\) and
Then,
for all \(t,\xi >0\) and \(s \ge 0\). We have
There exists a unique positive solution \(\tau _1 = \tau _1(\zeta )\) of the equation
and
Since
we have
This yields
for all \(t,\xi >0\). This together with (70) implies that
for all \(t >0\). Thus, the estimate (69) holds. \(\square \)
Proof of Proposition 6
Let \(t,\alpha >0\) and let \(f_{t,\alpha }\) be defined as in (65). The derivative of \(f_{t,\alpha }\) is given by
Therefore,
for all \(\xi > 0\). To the term \(\phi _{t,1}\), we apply Lemma 4 with \(p =\alpha /2\) and \(s = \eta ^2\). Then we obtain
The term \(\phi _{t,2}\) satisfies
for all \(\xi >0\). We also have that
for all \(\xi >0\). Therefore,
Thus, the desired estimate (67) is obtained. \(\square \)
We are now ready to prove Theorem 3.
Proof of Theorem 3
Let \(t,\alpha >0\), and let the function \(f_{t,\alpha }\) be defined as in (65). As seen in Example 2, we have
Hence
By the estimate (17),
where \(K :=\sup _{t \ge 0} \Vert e^{-tA}\Vert \). Combining the estimates (72) and (73) with Proposition 6, we obtain the desired conclusion (64). \(\square \)
4.2 Polynomially stable semigroups
Here we consider a polynomially stable \(C_0\)-semigroup \((e^{-tA})_{t \ge 0}\) on a Hilbert space. The next result shows how a decay estimate for \((e^{-tA})_{t \ge 0}\) can be transferred to that for \((e^{-tA^{-1}})_{t \ge 0}\). As in Theorem 1, we use the notation \(\lfloor \xi \rfloor :=\max \{k \in \mathbb {Z}: k \le \xi \}\) for \(\xi \in \mathbb {R}\).
Theorem 4
Let \(-A\) be the generator of a polynomially stable \(C_0\)-semigroup with parameter \(\beta >0\) on a Hilbert space H. Then, for all \(\alpha >0\),
To prove Theorem 4, we need the following estimate for \(\Vert f_{t,2p}\Vert _{\mathcal {B}_0, q }\) with \(0< q < 1/2\), where \(f_{t,2p}\) is defined by (65) with \(\alpha = 2p\).
Proposition 7
Let \(p >0\) and \( q \in (0,1/2)\). Then the function \(f_{t,2p}\) defined by (65) with \(\alpha = 2p\) satisfies
Proof
Step 1: Fix \(p >0\) and \( q \in (0,1/2)\). The derivative \(f_{t,2p}'\) is given by
Since
for all \(z \in \mathbb {C}_+\), we have
where \(g_{t,p}\) is defined by (68) and
for \(\xi >0\) and \(s \ge 0\). We will prove in Steps 2 and 3 below that
and
The desired conclusion (75) follows immediately from the estimates (76)–(79).
Step 2: In this step, we will prove the estimate (78) for \(g_{t,p}\). From the estimates (70) and (71) in the proof of Lemma 4, we obtain
for all \(t,\xi >0\) and some constant \(c_1>0\) depending only on p.
Suppose that \(t > 1\). Since the function \(\psi _q\), defined as in (23), satisfies \(\psi _q(\xi ) = \xi ^q\) for all \(\xi \in (0,1)\), the estimate (80) gives
Hence,
as \(t \rightarrow \infty \). Recalling that \(\psi _q(\xi ) = 1\) for all \(\xi \ge 1\), we also have
which yields
as \(t \rightarrow \infty \). From (81) and (82), we obtain the estimate (78) for \(g_{t,p}\).
Step 3: It remains to prove the estimate (79) for \(h_{t,p}\). Define \(\zeta :=t\xi \) and
Then \(H_{\zeta }\) has similar properties to \(G_{\zeta }\) used in the proof of Lemma 4. In fact,
for all \(t, \xi >0\) and \(s \ge 0\). There exists a unique positive solution \(\tau _2 = \tau _2(\zeta )\) of \(H_{\zeta }'(\tau ) = 0\), and
Moreover, \( \tau _2 \ge c_2 \zeta \) for some constant \(c_2>0\) depending only on p. Therefore,
for all \(t,\xi >0\).
By definition, \(\psi _{ q } (\xi ) \le 1\) for all \(\xi >0\). Combining this with the estimates (83) and (84), we obtain
for all \(t >0\). Hence,
as \(t\rightarrow \infty \). On the other hand, we have from the definition (77) of \(h_{t,p}\) that
for all \(t, \xi >0\) and \(s \ge 0\). This gives
as \(t\rightarrow \infty \). From (85) and (86), we obtain the estimate (79) for \(h_{t,p}\). \(\square \)
Now we are in a position to give the proof of Theorem 4.
Proof of Theorem 4
Let \(p \in (0,1/2)\) and define \(f_{t,2p} \in \mathcal {B}\) by (65) with \(\alpha = 2p\) for \(t >0\). Using Example 3 in the case \(\alpha = 2p\), \(c= 1\), \(q = p\), and
we see that there exists a constant \(M>0\) such that
for all \(t >0\). The estimate (75) with \(p = q\) yields
as \(t \rightarrow \infty \). Therefore, the desired estimate (74) holds in the case \(k = \lfloor 2\alpha /(2+\beta ) \rfloor =0\). The case \(k \ge 1\) follows as in the proof of Theorem 1. \(\square \)
Remark 5
The estimate (75) with \(p = q \) for \(\Vert f_{t,2p}\Vert _{\mathcal {B}_0, q }\) has been used in the proof of Theorem 4. This is because the case \(p \not = q \) yields a less sharp estimate, which can be seen from the same argument as in Remark 1.
Data Availability
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References
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Wakaiki, M. Decay estimates for Cayley transforms and inverses of semigroup generators via the \(\mathcal {B}\)-calculus. J. Evol. Equ. 24, 48 (2024). https://doi.org/10.1007/s00028-024-00979-2
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DOI: https://doi.org/10.1007/s00028-024-00979-2