Polynomial gaps below linear growth for Kreiss bounded semigroups and operators

Abstract.

We prove that every Kreiss bounded $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ on a Hilbert space satisfies

$$\|T_{t}\|\leq C(1+t)^{1-\varepsilon_{K}},\qquad t\geq 0,$$

where $\varepsilon_{K}>0$ depends explicitly only on the Kreiss constant. The same conclusion is obtained for positive Kreiss bounded $C_{0}$-semigroups on $L^{p}$-spaces, $1<p<\infty$, and in discrete time for Kreiss bounded operators on Hilbert spaces and positive Kreiss bounded operators on $L^{p}$-spaces. Finally, we obtain a non-quantitative polynomial gap for individually eventually positive Kreiss bounded $C_{0}$-semigroups on $L^{p}$-spaces.

1. Introduction

A classical problem in operator theory and in the asymptotic analysis of evolution equations is to determine to what extent first-order resolvent estimates control the growth of the corresponding semigroup or of the powers of an operator. In this paper, we consider the continuous and discrete Kreiss conditions, both on Hilbert spaces and, under positivity assumptions, on $L^{p}$-spaces.

Background and motivation

On Hilbert spaces, it was first established that Kreiss boundedness of a $C_{0}$-semigroup $(T_{t})_{t\geq 0}$ implies the linear bound $\|T_{t}\|=O(t)$ (see [6, 7]). This estimate was subsequently improved in [1] to

$$\|T_{t}\|=O\left(\frac{t}{\sqrt{\log(t+1)}}\right).$$

Related estimates for uniformly eventually positive Kreiss bounded semigroups on Banach lattices were obtained in [3]. In particular, on $L^{p}$-spaces, $1<p<\infty$, the logarithmic improvement

$$\|T_{t}\|=O\left(\frac{t}{(\log(t+1))^{\max\{1/p,1/p^{\prime}\}}}\right)$$

was proved, whereas on $(AL)$- and $(AM)$-spaces a genuine polynomial improvement

$$\|T_{t}\|=O(t^{1-\varepsilon})$$

for some $\varepsilon>0$ was already obtained.

In discrete time, and in contrast to the continuous-time setting, Kreiss boundedness of an operator $T$ on any Banach space $X$ always implies the linear bound $\|T^{n}\|=O(n).$ In the Hilbert space setting, this estimate was improved by Cohen, Cuny, Eisner and Lin [5] to

$$\|T^{n}\|=O\!\left(\frac{n}{\sqrt{\log(n+1)}}\right).$$

There is, however, a strong obstruction to any uniform polynomial improvement. Indeed, Eisner and Zwart constructed, for every $\gamma\in(0,1)$, a Kreiss bounded $C_{0}$-semigroup on a Hilbert space such that

$$\left\|T_{t}\right\|\gtrsim t^{\gamma}\qquad(t\to\infty);$$

see [6, Example 4.4]. Thus, there is no $\varepsilon>0$ such that

$$\left\|T_{t}\right\|=O(t^{1-\varepsilon})$$

for every Kreiss bounded $C_{0}$-semigroup on a Hilbert space. The same phenomenon occurs in discrete time: Bonilla and Müller showed that no universal polynomial gap below linear growth can hold, even for uniformly Kreiss bounded operators; see [4].

The purpose of this paper is to show that a genuine polynomial gap nevertheless exists once the exponent is allowed to depend on the Kreiss constant. We prove this both in continuous and discrete time, and we show that the same phenomenon holds for positive operators and semigroups on $L^{p}$-spaces. We finally extend the continuous-time result, in a non-quantitative form, to individually eventually positive semigroups on $L^{p}$-spaces.

The four quantitative results are obtained through the same self-improvement mechanism. A Kreiss-type resolvent estimate first yields a triangular estimate. Applied to a suitably normalized reversed orbit, this gives a reciprocal-weight inequality. Combined with Hölder’s inequality, it produces a fixed gain when the corresponding orbit quantity is evaluated at twice the scale. Iterating this gain and using an upper orbit estimate yields the polynomial improvement. In the Hilbertian setting, the triangular estimate follows from a Fourier–Plancherel multiplier argument, whereas in the positive $L^{p}$ setting , it follows from a positive convolution principle due to Weis and its discrete counterpart. The individually eventually positive case is then obtained by combining the same self-improvement argument with a non-quantitative uniformisation principle.

Notation and main results

Throughout the paper, $H$ denotes a complex Hilbert space and $(\Omega,\Sigma,\mu)$ a measure space. For $1<p<\infty$, we write $L^{p}(\Omega)=L^{p}(\Omega,\Sigma,\mu)$ (or simply $L^{p}$ when convenient). Its positive cone is

$$L^{p}(\Omega,\mu)_{+}=\{f\in L^{p}(\Omega,\mu;\mathbb{R}):f\geq 0\text{ a.e.}\}.$$

Moreover, we denote by $p^{\prime}$ the conjugate exponent of $p$.

For $f\in L^{2}(\mathbb{R};H)$, we define its Fourier transform $\widehat{f}\in L^{2}(\mathbb{R};H)$ (initially on $L^{1}\cap L^{2}$) by

$$\widehat{f}(\beta):=\int_{\mathbb{R}}e^{-i\beta s}f(s)\,ds,\qquad\beta\in\mathbb{R}.$$

Similarly, for a sequence $f=(f_{k})_{k\in\mathbb{Z}}\in\ell^{2}(\mathbb{Z};H)$, its Fourier transform $\widehat{f}\in L^{2}(\mathbb{T};H)$ is defined by

$$\widehat{f}(\gamma):=\sum_{k\in\mathbb{Z}}f_{k}\gamma^{-k},\qquad\gamma\in\mathbb{T},$$

where $\mathbb{T}=\{z\in\mathbb{C}:|z|=1\}$ carries the normalized Haar measure.

If $B$ is a closed operator, we write

$$R(\lambda,B):=(\lambda I-B)^{-1},\qquad\lambda\in\rho(B),$$

for its resolvent. We write $B(X)$ for the algebra of bounded linear operators on a Banach space $X$.

In continuous time, $(T_{t})_{t\geq 0}$ will always denote a $C_{0}$-semigroup with generator $-A$. Its Kreiss constant and Abel constant are respectively defined by

$$C_{K}(T):=\sup_{\begin{subarray}{c}r>0\\ \beta\in\mathbb{R}\end{subarray}}r\,\left\|R(r+i\beta,-A)\right\|\qquad\text{and}\qquad C_{A}(T):=\sup_{r>0}r\,\left\|R(r,-A)\right\|.$$

The semigroup $(T_{t})_{t\geq 0}$ is said to be Kreiss bounded when $C_{K}(T)<\infty$ and Abel bounded when $C_{A}(T)<\infty$. Notice that $C_{A}(T)\leq C_{K}(T)$, so that Kreiss boundedness implies Abel boundedness. Moreover, when $(T_{t})_{t\geq 0}$ acts positively on an $L^{p}$-space, the reverse implication holds (see [3, Proposition 2.6.]).

In discrete time, for $T\in B(X)$, we set

$$C_{K}(T):=\sup_{|\lambda|>1}(|\lambda|-1)\left\|R(\lambda,T)\right\|\qquad\text{and}\qquad C_{A}(T):=\sup_{\rho>1}(\rho-1)\left\|R(\rho,T)\right\|.$$

We use the same notation $C_{K}(T)$ and $C_{A}(T)$ whether $T$ represents a continuous semigroup or a discrete-time operator, as there is no risk of confusion between the two. The operator $T$ is said to be Kreiss bounded if $C_{K}(T)<\infty$ and Abel bounded if $C_{A}(T)<\infty$. As before, one always has $C_{A}(T)\leq C_{K}(T)$. Furthermore, if $T$ is positive on an $L^{p}$-space, then Abel boundedness implies Kreiss boundedness (see [5, Proposition 5.13.]).

For $r>1$ and $C>0$, it will be convenient to introduce the exponent

(1) $$\varepsilon_{r}(C):=\frac{1}{r}\log_{2}\!\left(1+(2eC)^{-r}\right)>0.$$

With this notation, our main results take a particularly simple form.

Theorem 1.1 (Continuous time).

Let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup.

1. If $(T_{t})_{t\geq 0}$ acts on a complex Hilbert space $H$ and is Kreiss bounded, then there exists $C>0$ such that

   (2) $$\left\|T_{t}\right\|\leq C(1+t)^{1-\varepsilon_{2}(C_{K}(T))},\qquad t\geq 0.$$

2. Let $1<p<\infty$. If $(T_{t})_{t\geq 0}$ is a positive $C_{0}$-semigroup on $L^{p}(\Omega)$ with $C_{A}(T)<\infty$, then there exists $C>0$ such that

   $$\left\|T_{t}\right\|\leq C(1+t)^{1-\varepsilon_{q}(C_{A}(T))},\qquad t\geq 0,$$

   where $q:=p\wedge p^{\prime}$.

The discrete analogue is as follows.

Theorem 1.2 (Discrete time).

Let $T$ be a bounded operator.

1. If $T\in B(H)$ is Kreiss bounded on a complex Hilbert space $H$, then there exists $C>0$ such that

   $$\left\|T^{N}\right\|\leq C(N+1)^{1-\varepsilon_{2}(C_{K}(T))},\qquad N\geq 0.$$

2. Let $1<p<\infty$. If $T$ is positive on $L^{p}(\Omega)$ with $C_{A}(T)<\infty$, then there exists $C>0$ such that

   $$\left\|T^{N}\right\|\leq C(N+1)^{1-\varepsilon_{q}(C_{A}(T))},\qquad N\geq 0.$$

Organization of the paper

Section 2 collects the continuous and discrete preliminary estimates used throughout the paper. In particular, we record the triangular estimates and the upper orbit estimates in both the Hilbertian and positive $L^{p}$ settings. Section 3 contains the continuous-time self-improvement argument and the proof of Theorem 1.1. Section 4 develops the discrete counterpart and proves Theorem 1.2. Section 5 extends the continuous-time result, in a non-quantitative form, to individually eventually positive semigroups on $L^{p}$-spaces. Finally, Appendix A gathers the Fourier multiplier arguments underlying the triangular estimates in the Hilbert space setting.

2. Preliminaries

We begin by stating a positive convolution principle that will be useful. It is a special case of the positive convolution theorem of Weis; see [9, Theorem 2]. The discrete version follows from the same argument.

Proposition 2.1.

Let $1\leq p<\infty$.

1. If $u\mapsto K(u)$ is a strongly measurable family of positive operators on $L^{p}(\Omega)$ and $\int_{\mathbb{R}}K(u)\,du$ exists as a bounded operator, then the convolution

   $$(\mathcal{S}f)(s):=\int_{\mathbb{R}}K(u)f(s-u)\,du$$

   satisfies

   $$\left\|\mathcal{S}\right\|_{B(L^{p}(\mathbb{R};L^{p}(\Omega)))}\leq\left\|\int_{\mathbb{R}}K(u)\,du\right\|_{B(L^{p})}.$$

2. If $(K_{n})_{n\in\mathbb{Z}}$ is a finitely supported family of positive operators on $L^{p}(\Omega)$, then

   $$(Sf)_{j}:=\sum_{n\in\mathbb{Z}}K_{n}f_{j-n}$$

   satisfies

   $$\left\|S\right\|_{B(\ell^{p}(\mathbb{Z};L^{p}(\Omega)))}\leq\left\|\sum_{n\in\mathbb{Z}}K_{n}\right\|_{B(L^{p})}.$$

2.1. Continuous time

Let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup with generator $-A$. For $m>0$, set

$$I_{m}:=[0,m],\qquad J_{m}:=[m,2m],$$

and define

$$(\mathcal{K}_{m}f)(s):=\int_{J_{m}}T_{v-s}f(v)\,dv,\qquad s\in I_{m}.$$

We now prove the two triangular estimates which will be used in Section 3.

Proposition 2.2.

Let $m\geq 1$.

1. If $(T_{t})_{t\geq 0}$ is Kreiss bounded on a Hilbert space $H$, then

   (3) $$\left\|\mathcal{K}_{m}\right\|_{B(L^{2}(J_{m};H),L^{2}(I_{m};H))}\leq 2eC_{K}(T)m.$$

2. If $(T_{t})_{t\geq 0}$ is positive and Abel bounded on $L^{p}(\Omega)$, then

   $$\left\|\mathcal{K}_{m}\right\|_{B(L^{p}(J_{m};L^{p}),L^{p}(I_{m};L^{p}))}\leq 2eC_{A}(T)m.$$

Proof.

We first prove (i). Set $r:=\frac{1}{2m}$. By Proposition A.1(i),

(4) $$\left\|\mathcal{C}_{r}\right\|_{B(L^{2}(\mathbb{R};H))}\leq\frac{C_{K}(T)}{r}=2mC_{K}(T).$$

Define

$$(\mathcal{K}_{m}^{(r)}f)(s):=\int_{J_{m}}e^{-r(v-s)}T_{v-s}f(v)\,dv.$$

If $\mathcal{E}_{m}:L^{2}(J_{m};H)\to L^{2}(\mathbb{R};H)$ denotes extension by zero and $\mathcal{P}_{m}:L^{2}(\mathbb{R};H)\to L^{2}(I_{m};H)$ restriction to $I_{m}$, then

(5) $$\mathcal{K}_{m}^{(r)}=\mathcal{P}_{m}\mathcal{C}_{r}\mathcal{E}_{m}.$$

Indeed let $f\in L^{2}(J_{m};H)$. For $s\in I_{m}$, we have

$$(\mathcal{C}_{r}\mathcal{E}_{m}f)(s) =\int_{0}^{\infty}e^{-ru}T_{u}(\mathcal{E}_{m}f)(s+u)\,du$$

$$=\int_{\{u\geq 0:\,s+u\in J_{m}\}}e^{-ru}T_{u}f(s+u)\,du.$$

Since $s\in I_{m}$, the change of variables $v=s+u$ yields

$$(\mathcal{C}_{r}\mathcal{E}_{m}f)(s)=\int_{J_{m}}e^{-r(v-s)}T_{v-s}f(v)\,dv=(\mathcal{K}_{m}^{(r)}f)(s),$$

which gives (5). Thus, by (4), $\left\|\mathcal{K}_{m}^{(r)}\right\|\leq 2mC_{K}(T)$.

Define the multiplication operators $(D_{I}f)(s):=e^{-rs}f(s)$, $s\in I_{m}$, and $(D_{J}f)(v):=e^{-rv}f(v)$, $v\in J_{m}$. We have $\mathcal{K}_{m}=D_{I}\mathcal{K}_{m}^{(r)}D_{J}^{-1}$ and $\|D_{I}\|\,\|D_{J}^{-1}\|\leq e^{r\cdot 2m}=e.$ Therefore

$$\left\|\mathcal{K}_{m}\right\|\leq 2meC_{K}(T),$$

which proves (i).

We now prove (ii). Since the semigroup is positive and Kreiss bounded on $L^{p}$ its growth bound is nonpositive and the Laplace representation

$$R(r,-A)x=\int_{0}^{\infty}e^{-ru}T_{u}x\,du$$

holds for every $r>0$ and $x\in L^{p}(\Omega)$.

Taking $r=\frac{1}{2m}$, we have for every $x\in L^{p}_{+}(\Omega)$,

$$0\leq\int_{0}^{2m}T_{u}x\,du\leq e\int_{0}^{2m}e^{-u/(2m)}T_{u}x\,du\leq eR\left(\frac{1}{2m},-A\right)x.$$

Hence

$$\left\|\int_{0}^{2m}T_{u}\,du\right\|\leq 2eC_{A}(T)m.$$

Define the operator $(\mathcal{V}_{2m}f)(s):=\int_{0}^{2m}T_{u}f(s+u)\,du$. Applying Proposition 2.1 (i) to the positive kernel $u\longmapsto\mathbbm{1}_{[-2m,0]}(u)T_{-u}$ yields

(6) $$\left\|\mathcal{V}_{2m}\right\|_{B(L^{p}(\mathbb{R};L^{p}))}\leq 2eC_{A}(T)m.$$

Extend $f\in L^{p}(J_{m};L^{p})$ by zero outside $J_{m}$. For $s\in I_{m}$,

(7) $$(\mathcal{V}_{2m}f)(s)=\int_{0}^{2m}T_{u}f(s+u)\,du=\int_{J_{m}}T_{v-s}f(v)\,dv=(\mathcal{K}_{m}f)(s).$$

Thus, by (6),

$$\left\|\mathcal{K}_{m}\right\|\leq 2eC_{A}(T)m.$$

This proves (ii). ∎

The second ingredient is an upper estimate for the orbit. In the Hilbertian case we use the following result from [1], while in the positive case we use [3, Proposition 3.2]. Although this result is stated there for $\sigma$-finite measure spaces, the $\sigma$-finiteness assumption can be removed by the standard reduction described in Section 5.

Proposition 2.3.

1. If $(T_{t})_{t\geq 0}$ is Kreiss bounded on a Hilbert space $H$, then there exists $M_{2}>0$ such that

   $$\left(\int_{0}^{t}\left\|T_{s}x\right\|^{2}\,ds\right)^{1/2}\leq M_{2}t\left\|x\right\|,\qquad t\geq 1,$$

   for every $x\in H$.

2. If $(T_{t})_{t\geq 0}$ is positive and Kreiss bounded on $L^{p}(\Omega)$, then there exists $M_{p}>0$ such that

   $$\left(\int_{0}^{t}\left\|T_{s}x\right\|_{p}^{p}\,ds\right)^{1/p}\leq M_{p}t\left\|x\right\|_{p},\qquad t\geq 1,$$

   for every $x\in L^{p}(\Omega)$.

Since $1<p<\infty$, the adjoint semigroup $(T_{t}^{*})_{t\geq 0}$ is a positive $C_{0}$-semigroup on $L^{p^{\prime}}(\Omega)$. Moreover, since $R(r,-A^{*})=R(r,-A)^{*}$ for $r>0$,

(8) $$C_{A}(T^{*})=C_{A}(T).$$

2.2. Discrete time

Let $T\in B(X)$. For $m\in\mathbb{N}^{*}$, set

$$I_{m}:=\{0,\ldots,m-1\},\qquad J_{m}:=\{m,\ldots,2m-1\},$$

and define

$$(K_{m}f)_{i}:=\sum_{j\in J_{m}}T^{j-i}f_{j},\qquad i\in I_{m}.$$

Proposition 2.4.

Let $m\geq 1$.

1. If $T$ is Kreiss bounded on a Hilbert space $H$, then

   $$\left\|K_{m}\right\|_{B(\ell^{2}(J_{m};H),\ell^{2}(I_{m};H))}\leq 2eC_{K}(T)m.$$

2. If $T$ is positive and Kreiss bounded on $L^{p}(\Omega)$, then

   $$\left\|K_{m}\right\|_{B(\ell^{p}(J_{m};L^{p}),\ell^{p}(I_{m};L^{p}))}\leq 2eC_{A}(T)m.$$

Proof.

We begin with (i). Set $\rho:=1-\frac{1}{2m}$. By Proposition A.1(ii),

$$\left\|C_{\rho}\right\|_{B(\ell^{2}(\mathbb{Z};H))}\leq\frac{C_{K}(T)}{1-\rho}=2mC_{K}(T).$$

Define $(K_{m}^{(\rho)}f)_{i}:=\sum_{j\in J_{m}}\rho^{j-i}T^{j-i}f_{j}$, $i\in I_{m}$. If $E_{m}$ denotes extension by zero from $J_{m}$ to $\mathbb{Z}$ and $P_{m}$ restriction to $I_{m}$, then $K_{m}^{(\rho)}=P_{m}C_{\rho}E_{m}$. Therefore

$$\left\|K_{m}^{(\rho)}\right\|\leq\frac{C_{K}(T)}{1-\rho}=2mC_{K}(T).$$

Define $(D_{I}f)_{i}:=\rho^{i}f_{i}$ and $(D_{J}f)_{j}:=\rho^{j}f_{j}$. Then $K_{m}=D_{I}K_{m}^{(\rho)}D_{J}^{-1}$. Moreover,

$$\left\|D_{I}\right\|\left\|D_{J}^{-1}\right\|\leq\rho^{-(2m-1)}=\left(1-\frac{1}{2m}\right)^{-(2m-1)}\leq e.$$

Thus

$$\left\|K_{m}\right\|\leq 2eC_{K}(T)m.$$

We now prove (ii). For $m\geq 1$, define

$$(V_{2m}f)_{i}:=\sum_{n=0}^{2m-1}T^{n}f_{i+n},\qquad f\in\ell^{p}(\mathbb{Z};L^{p}(\Omega)),$$

and again set $\rho:=1-\frac{1}{2m}$. Since $T$ is positive and Abel bounded, for every $x\in L^{p}_{+}(\Omega)$,

$$0\leq\sum_{n=0}^{2m-1}T^{n}x \leq\rho^{-(2m-1)}\sum_{n=0}^{2m-1}\rho^{n}T^{n}x$$

$$\leq e(I-\rho T)^{-1}x.$$

Hence

$$\left\|\sum_{n=0}^{2m-1}T^{n}\right\|\leq\frac{eC_{A}(T)}{1-\rho}=2eC_{A}(T)m.$$

By the discrete positive convolution principle,

$$\left\|V_{2m}\right\|_{B(\ell^{p}(\mathbb{Z};L^{p}))}\leq 2eC_{A}(T)m.$$

Extend $f\in\ell^{p}(J_{m};L^{p})$ by zero outside $J_{m}$. For $i\in I_{m}$, setting $j=i+n$ gives

$$(V_{2m}f)_{i}=\sum_{n=0}^{2m-1}T^{n}f_{i+n}=\sum_{j=i}^{i+2m-1}T^{j-i}f_{j}=\sum_{j\in J_{m}}T^{j-i}f_{j}=(K_{m}f)_{i}.$$

We conclude that

$$\left\|K_{m}\right\|_{B(\ell^{p}(J_{m};L^{p}),\ell^{p}(I_{m};L^{p}))}\leq 2eC_{A}(T)m.$$

This proves (ii). ∎

We next record the corresponding upper orbit estimates.

Proposition 2.5.

1. If $T$ is Kreiss bounded on a Hilbert space $H$, then there exists $M_{2}>0$ such that

   $$\left(\sum_{k=0}^{N}\left\|T^{k}x\right\|^{2}\right)^{1/2}\leq M_{2}(N+1)\left\|x\right\|,\qquad N\geq 0$$

   for every $x\in H$; see [5].

2. If $T$ is positive and Kreiss bounded on $L^{p}(\Omega)$, then there exists $M_{p}>0$ such that

   (9) $$\left(\sum_{k=0}^{N}\left\|T^{k}x\right\|_{p}^{p}\right)^{1/p}\leq M_{p}(N+1)\left\|x\right\|_{p},\qquad N\geq 1.$$

for every $x\in L^{p}(\Omega)$.

Proof.

We only provide the proof of (ii), since it seems to be absent from the literature. Let $N\geq 1$, $\rho:=1-\frac{1}{N+1}$, and $x\in L^{p}(\Omega)$. By positivity, $|T^{k}x|\leq T^{k}|x|$ for all $k\geq 0$. Since $\rho^{-N}=\left(1+\frac{1}{N}\right)^{N}\leq e$, the continuous embedding $\ell^{1}\hookrightarrow\ell^{p}$ yields

$$\left(\sum_{k=0}^{N}|T^{k}x|^{p}\right)^{1/p}\leq\sum_{k=0}^{N}T^{k}|x|\leq\rho^{-N}\sum_{k=0}^{N}\rho^{k}T^{k}|x|\leq e\rho^{-1}R(\rho^{-1},T)|x|.$$

Taking the $L^{p}$-norm and using the definition of $C_{A}(T)$, we get

$$\left(\sum_{k=0}^{N}\left\|T^{k}x\right\|_{p}^{p}\right)^{1/p} \leq e\rho^{-1}\left\|R(\rho^{-1},T)\right\|\left\|x\right\|_{p}$$

$$\leq e\,\frac{C_{A}(T)}{1-\rho}\left\|x\right\|_{p}$$

$$=eC_{A}(T)(N+1)\left\|x\right\|_{p},$$

which proves (9). ∎

Finally, if $T$ is positive on $L^{p}(\Omega)$, then $T^{*}$ is positive on $L^{p^{\prime}}(\Omega)$. Moreover, for every $\lambda>1$, $R(\lambda,T^{*})=R(\lambda,T)^{*}.$ Hence

(10) $$C_{A}(T^{*})=C_{A}(T).$$

3. The continuous-time self-improvement

We now isolate the reciprocal-weight argument. Once the triangular and upper orbit estimates are available, the remaining self-improvement argument is the same in the Hilbertian and positive $L^{p}$ settings.

In this section, $I_{m}$ and $J_{m}$ refer to the intervals $[0,m]$ and $[m,2m]$, respectively.

Lemma 3.1.

Let $1<r<\infty$ and let $(T_{t})_{t\geq 0}$ be a $C_{0}$-semigroup on a Banach space $X$. Assume that there exist $C_{0},C_{1}>0$ such that, for every $m\geq 1$,

(11) $$\left\|\mathcal{K}_{m}\right\|_{B(L^{r}(J_{m};X),L^{r}(I_{m};X))}\leq 2eC_{0}m,$$

and, for every $t\geq 1$ and $x\in X$,

(12) $$\left(\int_{0}^{t}\left\|T_{s}x\right\|^{r}\,ds\right)^{1/r}\leq C_{1}t\left\|x\right\|.$$

Then there exists $C>0$ such that

(13) $$\left\|T_{t}\right\|\leq C(1+t)^{1-\varepsilon_{r}(C_{0})},\qquad t\geq 0.$$

Proof.

Fix $t\geq 2$ and $x\in X$. If $T_{t}x=0$, there is nothing to prove, so we assume that $T_{t}x\neq 0$. For $0\leq s\leq t$, set

$$z_{s}:=T_{t-s}x,\qquad w(s):=\left\|z_{s}\right\|^{r},$$

and, for $0<m\leq t$, define

$$W(m):=\int_{0}^{m}w(s)\,ds.$$

The function $w$ is continuous and strictly positive on $[0,t]$. Indeed, if $w(s)=0$ for some $s\in[0,t]$, then $z_{s}=0$ and the semigroup property would give

$$T_{t}x=T_{s}T_{t-s}x=T_{s}z_{s}=0,$$

which contradicts our assumption.

Let $m\geq 1$ be such that $2m\leq t$. Since $w$ is continuous and strictly positive on the compact interval $J_{m}=[m,2m]$, the function

$$f(v):=w(v)^{-1/(r-1)}z_{v},\qquad v\in J_{m},$$

belongs to $L^{r}(J_{m};X)$. Put

$$A_{m}:=\int_{m}^{2m}w(v)^{-1/(r-1)}\,dv.$$

Since $T_{v-s}z_{v}=z_{s}$ for $s\in I_{m}$ and $v\in J_{m}$, we have $(\mathcal{K}_{m}f)(s)=A_{m}z_{s}$.

Thus

$$\left\|\mathcal{K}_{m}f\right\|_{L^{r}(I_{m};X)}^{r}=A_{m}^{r}W(m)\quad\text{and}\quad\left\|f\right\|_{L^{r}(J_{m};X)}^{r}=A_{m}.$$

Applying (11) yields

$$W(m)^{1/r}A_{m}^{1/r^{\prime}}\leq 2eC_{0}m,\qquad r^{\prime}=r/(r-1).$$

Comparing this with Hölder’s inequality

$$m=\int_{m}^{2m}\frac{w(s)^{1/r}}{w(s)^{1/r}}\,ds \leq\left(\int_{m}^{2m}w(s)\,ds\right)^{1/r}\left(\int_{m}^{2m}w(s)^{-1/(r-1)}\,ds\right)^{1/r^{\prime}}$$

$$=\bigl(W(2m)-W(m)\bigr)^{1/r}A_{m}^{1/r^{\prime}},$$

we obtain

(14) $$W(2m)\geq\left(1+(2eC_{0})^{-r}\right)W(m).$$

Setting $\theta_{r}:=(2eC_{0})^{-r}$ and $\delta_{r}:=\log_{2}(1+\theta_{r})=r\varepsilon_{r}(C_{0})$, iterating (14) gives

$$W(2^{\ell})\geq(1+\theta_{r})^{\ell}W(1)=2^{\ell\delta_{r}}W(1)$$

for every integer $\ell\geq 0$ such that $2^{\ell}\leq t$.

To relate $W(1)$ to $\left\|T_{t}x\right\|$, let $D:=\sup_{0\leq s\leq 1}\left\|T_{s}\right\|<\infty$. Since $\left\|T_{t-s}x\right\|\geq D^{-1}\left\|T_{t}x\right\|$ for $s\in[0,1]$, we have

$$W(1)\geq D^{-r}\left\|T_{t}x\right\|^{r}.$$

Choosing $\ell\geq 0$ such that $M:=2^{\ell}\leq t<2M$, we obtain

$$M^{\delta_{r}}D^{-r}\left\|T_{t}x\right\|^{r}\leq W(M)=\int_{t-M}^{t}\left\|T_{u}x\right\|^{r}\,du\leq C_{1}^{r}t^{r}\left\|x\right\|^{r},$$

where we used estimate (12). Since $M>t/2$, it follows that for $t\geq 2$

$$\left\|T_{t}x\right\|\leq 2^{\delta_{r}/r}DC_{1}t^{1-\delta_{r}/r}\left\|x\right\|=2^{\varepsilon_{r}(C_{0})}DC_{1}t^{1-\varepsilon_{r}(C_{0})}\left\|x\right\|.$$

Extending this bound to $t\geq 0$ by local boundedness concludes the proof of (13). ∎

We are now in a position to prove Theorem 1.1

Proof of Theorem 1.1.

For (i), apply Lemma 3.1 with

$$r=2,\qquad C_{0}=C_{K}(T),\quad C_{1}=M_{2}$$

using Propositions 2.2(i) and 2.3(i). This gives (2).

For (ii), apply the same lemma with

$$r=p,\qquad C_{0}=C_{A}(T),\quad C_{1}=M_{p}$$

using Propositions 2.2(ii) and 2.3(ii). We obtain

$$\left\|T_{t}\right\|\leq C(1+t)^{1-\varepsilon_{p}(C_{A}(T))}$$

Applying the same argument to $(T_{t}^{*})_{t\geq 0}$, which is positive and Abel bounded with $C_{A}(T^{*})=C_{A}(T)$ by (8), gives the estimate with $p^{\prime}$ in place of $p$. For every $a\in(0,1)$, the map

$$r\longmapsto\frac{1}{r}\log(1+a^{r}),\qquad r>0,$$

is decreasing. Hence the better exponent is $\varepsilon_{q}(C_{A}(T))$, where we recall $q=p\wedge p^{\prime}$. ∎

4. The discrete self-improvement

The discrete-time proof is the exact analogue of the preceding one, with integrals replaced by sums.

Lemma 4.1.

Let $1<r<\infty$ and let $T\in B(X)$. Assume that there exist $C_{0},C_{1}>0$ such that, for every $m\geq 1$,

$$\left\|K_{m}\right\|_{B(\ell^{r}(J_{m};X),\ell^{r}(I_{m};X))}\leq 2eC_{0}m,$$

and, for every $N\geq 0$ and $x\in X$,

$$\left(\sum_{k=0}^{N}\left\|T^{k}x\right\|^{r}\right)^{1/r}\leq C_{1}(N+1)\left\|x\right\|.$$

Then

$$\left\|T^{N}\right\|\leq 2^{\varepsilon_{r}(C_{0})}C_{1}(N+1)^{1-\varepsilon_{r}(C_{0})},\qquad N\geq 0.$$

Proof.

Fix $N\in\mathbb{N}$ and $x\in X$. If $T^{N}x=0$, there is nothing to prove. Set

$$z_{j}:=T^{N-j}x,\qquad w_{j}:=\left\|z_{j}\right\|^{r},\qquad W_{m}:=\sum_{j=0}^{m-1}w_{j}.$$

All $w_{j}$ are strictly positive. Let $2m\leq N+1$ and define for $j\in J_{m}$

$$f_{j}:=w_{j}^{-1/(r-1)}z_{j},\qquad\text{and}\qquad A_{m}:=\sum_{j=m}^{2m-1}w_{j}^{-1/(r-1)}.$$

Since $T^{j-i}z_{j}=z_{i}$ for $i\in I_{m}$ and $j\in J_{m}$,

$$(K_{m}f)_{i}=A_{m}z_{i},\qquad\left\|f\right\|_{\ell^{r}(J_{m};X)}^{r}=A_{m}.$$

Hence

$$W_{m}^{1/r}A_{m}^{1/r^{\prime}}\leq 2eC_{0}m.$$

Hölder’s inequality yields

$$m\leq(W_{2m}-W_{m})^{1/r}A_{m}^{1/r^{\prime}},$$

and therefore

$$W_{2m}\geq\left(1+(2eC_{0})^{-r}\right)W_{m}.$$

Let $\delta_{r}:=r\varepsilon_{r}(C_{0}).$ Iteration gives, whenever $2^{\ell}\leq N+1$,

$$W_{2^{\ell}}\geq 2^{\ell\delta_{r}}W_{1}=2^{\ell\delta_{r}}\left\|T^{N}x\right\|^{r}.$$

Choose $M=2^{\ell}$ with $M\leq N+1<2M$. Then

$$M^{\delta_{r}}\left\|T^{N}x\right\|^{r}\leq W_{M}\leq\sum_{k=0}^{N}\left\|T^{k}x\right\|^{r}\leq C_{1}^{r}(N+1)^{r}\left\|x\right\|^{r}.$$

Since $M>(N+1)/2$, we obtain

$$\left\|T^{N}x\right\|\leq 2^{\delta_{r}/r}C_{1}(N+1)^{1-\delta_{r}/r}\left\|x\right\|,$$

which is the desired estimate. ∎

Proof of Theorem 1.2.

For (i), apply Lemma 4.1 with

$$r=2,\qquad C_{0}=C_{K}(T),\quad C_{1}=M_{2}$$

using Propositions 2.4(i) and 2.5(i).

For (ii), apply the same lemma with

$$r=p,\qquad C_{0}=C_{A}(T),\quad C_{1}=M_{p}$$

using Propositions 2.4(ii) and 2.5(ii). Apply the result once more to $T^{*}$ and use (10); as in continuous time, the better of the $p$ and $p^{\prime}$ exponents is $\varepsilon_{q}(C_{A}(T))$. ∎

5. Individually eventually positive $C_{0}$-semigroups on $L^{p}$-spaces

We conclude with an extension of the positive $L^{p}$ result to individually eventually positive semigroups. We deliberately treat this case separately from the positive case considered above. Indeed, for positive semigroups the positive convolution principle gives the required triangular estimate directly, with an explicit constant, and leads to a shorter and quantitative proof. Under individual eventual positivity there is no common positivity time, and a different argument is needed. The proof below relies on the operator-range uniformisation principle used in [2] and is therefore also of independent interest.

If $E$ is a complex Banach lattice, we denote by $E_{\mathbb{R}}$ its real part and by $E_{+}$ its positive cone. For $f\in E_{\mathbb{R}}$, we write $f\geq 0$ if $f\in E_{+}$, and $|f|$ denotes the lattice modulus of $f$. The same notation $|\cdot|$ is used for the modulus in the complexification of $E$.

For $g\in E_{+}$, the principal ideal generated by $g$ is

$$E_{g}:=\{f\in E:\ |f|\leq cg\text{ for some }c\geq 0\}.$$

Equipped with the gauge norm

$$\left\|f\right\|_{g}:=\inf\{c\geq 0:\ |f|\leq cg\},\qquad f\in E_{g},$$

the space $E_{g}$ is itself a Banach lattice.

We say that $(T_{t})_{t\geq 0}$ is individually eventually positive if, for every $f\in E_{+}$, there exists $t_{f}\geq 0$ such that

$$T_{t}f\geq 0,\qquad t\geq t_{f}.$$

We shall use the Cesàro constant

$$C_{\mathrm{Ces}}(T):=\sup_{t\geq 1}\frac{1}{t}\left\|\int_{0}^{t}T_{s}\,ds\right\|.$$

We recall that according to [3, Proposition 2.6], for individually eventually positive $C_{0}$-semigroups, Cesàro boundedness and Kreiss boundedness are equivalent, that is $C_{\mathrm{Ces}}(T)<\infty$.

Theorem 5.1.

Let $1<p<\infty$ and let $(T_{t})_{t\geq 0}$ be an individually eventually positive, Kreiss bounded $C_{0}$-semigroup on $L^{p}(\Omega)$. Then there exist $C\geq 1$ and $\varepsilon\in(0,1)$ such that

$$\left\|T_{t}\right\|\leq C(1+t)^{1-\varepsilon},\qquad t\geq 0.$$

We will use the following.

Proposition 5.2.

Let $(T_{t})_{t\geq 0}$ be an individually eventually positive semigroup on a complex Banach lattice $E$. Let $Y$ be a real Banach space, let $g\in E_{+}$, and let

$$J:Y\longrightarrow E_{\mathbb{R}}$$

be bounded with $JY\subseteq E_{g}$. Then there exist $\tau\geq 0$ and $c\geq 0$ such that, for every $t\geq\tau$ and $y\in Y$,

$$|T_{t}Jy|\leq c\left\|y\right\|_{Y}T_{t}g.$$

In particular, $T_{t}g\geq 0$ for every $t\geq\tau$.

Proof.

This is [2, Corollary 2.2]. ∎

From now on, for $1<p<\infty$, we denote $E=L^{p}(\Omega)$ so that $E_{\mathbb{R}}=L^{p}(\Omega,\mathbb{R})$ and $E_{+}=L^{p}(\Omega,\mathbb{R}_{+})$.

We shall repeatedly use the following standard observation. If $U:\mathbb{R}\to L^{p}(\Omega)$ is strongly measurable, then its essential range is contained in a separable subspace of $L^{p}(\Omega)$ and hence, up to null sets, in $L^{p}(\Omega_{0})$ for some $\sigma$-finite measurable subset $\Omega_{0}\subseteq\Omega$. Thus the usual Fubini-Tonelli arguments may be applied after this reduction. We shall use this observation without further comment; see also the proof of [8, Lemma 1].

Lemma 5.3.

Let $U\in L^{p}(\mathbb{R};E)$ and $H\in E_{+}$ such that

$$\left|\int_{\mathbb{R}}h(s)U(s)\,ds\right|\leq H$$

for every $h\in L^{p^{\prime}}(\mathbb{R};\mathbb{R})$ with $\left\|h\right\|_{p^{\prime}}\leq 1$. Then

(15) $$\left(\int_{\mathbb{R}}|U(s)|^{p}\,ds\right)^{1/p}\leq 2H.$$

Proof.

After the $\sigma$-finite reduction described above, apply the real-valued argument of [8, Lemma 1] to $\operatorname{Re}U$ and $\operatorname{Im}U$, and use $|U|\leq|\operatorname{Re}U|+|\operatorname{Im}U|$. ∎

The specific ingredients needed to apply the self-improvement argument from Section 3 are contained in the next proposition.

Proposition 5.4.

Assume that $(T_{t})_{t\geq 0}$ is individually eventually positive and Cesàro bounded on $E=L^{p}(\Omega)$. Then there exist $\Gamma,\Lambda<\infty$ such that the following assertions hold.

1. For $L>0$, define $\mathcal{V}_{L}$ by $(\mathcal{V}_{L}F)(s):=\int_{0}^{L}T_{u}F(s+u)\,du$ for $F\in L^{p}(\mathbb{R};E).$ Then for every $L\geq 1$,

   (16) $$\left\|\mathcal{V}_{L}\right\|_{B(L^{p}(\mathbb{R};E))}\leq\Gamma L.$$

2. For every $R\geq 1$ and $f\in E$,

   (17) $$\left(\int_{0}^{R}\left\|T_{t}f\right\|_{p}^{p}\,dt\right)^{1/p}\leq\Lambda R\left\|f\right\|_{p}.$$

Proof.

For $a\geq 0$, set $D_{a}:=\int_{0}^{a}\left\|T_{u}\right\|\,du$. Fix $F\in L^{p}(\mathbb{R};E_{\mathbb{R}})$ and set

$$g_{F}:=\left(\int_{\mathbb{R}}|F(v)|^{p}\,dv\right)^{1/p}\in E_{+}.$$

By Fubini-Tonelli, $\left\|g_{F}\right\|_{p}=\left\|F\right\|_{L^{p}(\mathbb{R};E)}$.

Define $J_{F}:L^{p^{\prime}}(\mathbb{R};\mathbb{R})\longrightarrow E_{\mathbb{R}}$ by $J_{F}h:=\int_{\mathbb{R}}h(v)F(v)\,dv$. Hölder’s inequality gives $|J_{F}h|\leq\left\|h\right\|_{p^{\prime}}g_{F}$, so $J_{F}L^{p^{\prime}}(\mathbb{R};\mathbb{R})\subseteq E_{g_{F}}$. Proposition 5.2 yields $\tau_{F},c_{F}\geq 0$ such that for every $u\geq\tau_{F}$,

(18) $$|T_{u}J_{F}h|\leq c_{F}\left\|h\right\|_{p^{\prime}}T_{u}g_{F}.$$

For $0\leq a\leq L$, define $(\mathcal{V}_{a,L}F)(s):=\int_{a}^{L}T_{u}F(s+u)\,du$. Let $L\geq\tau_{F}$ and $h\in L^{p^{\prime}}(\mathbb{R};\mathbb{R})$. Fubini’s theorem and (18) give

$$\left|\int_{\mathbb{R}}h(s)(\mathcal{V}_{\tau_{F},L}F)(s)\,ds\right|\leq c_{F}\left\|h\right\|_{p^{\prime}}\int_{\tau_{F}}^{L}T_{u}g_{F}\,du.$$

Now Lemma 5.3 yields

$$\left\|\mathcal{V}_{\tau_{F},L}F\right\|_{L^{p}(\mathbb{R};E)}\leq 2c_{F}\left\|\int_{\tau_{F}}^{L}T_{u}g_{F}\,du\right\|_{p}.$$

Consequently, for $L\geq\tau_{F}$,

$$\left\|\mathcal{V}_{L}F\right\|_{L^{p}(\mathbb{R};E)}\leq\Bigl(2c_{F}C_{\mathrm{Ces}}(T)L+(2c_{F}+1)D_{\tau_{F}}\Bigr)\left\|F\right\|_{L^{p}(\mathbb{R};E)}.$$

Since for $1\leq L<\tau_{F}$, $\left\|\mathcal{V}_{L}F\right\|_{L^{p}(\mathbb{R};E)}\leq D_{\tau_{F}}\left\|F\right\|_{L^{p}(\mathbb{R};E)}$, it follows that

$$\sup_{L\geq 1}\frac{1}{L}\left\|\mathcal{V}_{L}F\right\|_{L^{p}(\mathbb{R};E)}<\infty$$

for every real $F\in L^{p}(\mathbb{R};E)$, and hence for every complex $F$ by decomposition into real and imaginary parts. Banach-Steinhaus yields (16).

We now prove (ii). Fix $f\in E_{+}$. Choose $a\geq 0$ such that for every $t\geq a$, $T_{t}f\geq 0$ and define

$$F:=\left(\int_{a}^{a+1}|T_{r}f|^{p}\,dr\right)^{1/p}\in E_{+}.$$

Let $Y:=L^{p^{\prime}}([a,a+1];\mathbb{R})$ and $J_{f}h:=\int_{a}^{a+1}h(r)T_{r}f\,dr.$ Hölder’s inequality gives

$$|J_{f}h|\leq\left\|h\right\|_{p^{\prime}}F.$$

Proposition 5.2 therefore yields $\tau_{F},c_{F}\geq 0$ such that for every $u\geq\tau_{F}$

(19) $$|T_{u}J_{f}h|\leq c_{F}\left\|h\right\|_{p^{\prime}}T_{u}F.$$

Set $b:=a+\tau_{F}+1.$ Then $T_{t}f\geq 0$ for every $t\geq b$. Fix $R>b$, and let

$$0\leq h\in L^{p^{\prime}}([b,R]),\qquad\left\|h\right\|_{p^{\prime}}\leq 1,$$

extended by zero outside $[b,R]$. For $u\geq\tau_{F}$, define $h_{u}(r):=h(u+r)$ for $r\in[a,a+1].$ Then $\left\|h_{u}\right\|_{p^{\prime}}\leq 1$. Using Fubini’s theorem and the semigroup property gives

$$\int_{b}^{R}h(t)T_{t}f\,dt =\int_{b}^{R}h(t)T_{t}f\left(\int_{t-a-1}^{t-a}du\right)dt$$

$$=\int_{b-a-1}^{R-a}\int_{u+a}^{u+a+1}h(t)T_{t}f\,dt\,du$$

$$=\int_{\tau_{F}}^{R-a}T_{u}\left(\int_{a}^{a+1}h_{u}(r)T_{r}f\,dr\right)du$$

$$=\int_{\tau_{F}}^{R-a}T_{u}J_{f}(h_{u})\,du$$

Since the vector on the left-hand side is positive, (19) implies

(20) $$0\leq\int_{b}^{R}h(t)T_{t}f\,dt\leq c_{F}\int_{\tau_{F}}^{R-a}T_{u}F\,du.$$

Hence [8, Lemma 1] gives

$$\left(\int_{b}^{R}(T_{t}f)^{p}\,dt\right)^{1/p}=\sup_{\begin{subarray}{c}0\leq h\in L^{p^{\prime}}([b,R])\\ \left\|h\right\|_{p^{\prime}}\leq 1\end{subarray}}\int_{b}^{R}h(t)T_{t}f\,dt.$$

It follows from (20) that

$$\left(\int_{b}^{R}(T_{t}f)^{p}\,dt\right)^{1/p}\leq c_{F}\int_{\tau_{F}}^{R-a}T_{u}F\,du.$$

Taking $L^{p}$-norms and using Cesàro boundedness, we obtain

$$\left(\int_{b}^{R}\left\|T_{t}f\right\|_{L^{p}}^{p}\,dt\right)^{1/p} \leq c_{F}\left\|\int_{\tau_{F}}^{R-a}T_{u}F\,du\right\|_{L^{p}}$$

$$\leq c_{F}\left(\left\|\int_{0}^{R-a}T_{u}\,du\right\|+D_{\tau_{F}}\right)\left\|F\right\|_{L^{p}}$$

$$\leq c_{F}\bigl(C_{\mathrm{Ces}}(T)R+D_{\tau_{F}}\bigr)\left\|F\right\|_{L^{p}}.$$

The integral over the fixed interval $[0,b]$ is controlled by local boundedness. Therefore

$$\sup_{R\geq 1}\frac{1}{R}\left(\int_{0}^{R}\left\|T_{t}f\right\|_{p}^{p}\,dt\right)^{1/p}<\infty$$

for every $f\in E_{+}$ and so for every $f\in E$. Banach-Steinhaus therefore yields (17). ∎

We are now ready for proving Theorem 5.1.

Proof of Theorem 5.1.

Increase $\Gamma$, if necessary, so that $\Gamma\geq 1$. For $m\geq 1$, we recall that $I_{m}=[0,m]$,  $J_{m}=[m,2m]$ and for $f\in L^{p}(J_{m},L^{p})$,

$$(\mathcal{K}_{m}f)(s)=\int_{J_{m}}T_{v-s}f(v)\,dv,\qquad s\in I_{m}.$$

From (7) and (16) we deduce

$$\left\|\mathcal{K}_{m}\right\|_{B(L^{p}(J_{m};E),L^{p}(I_{m};E))}\leq 2\Gamma m.$$

Together with (17), this is precisely the setting of Lemma 3.1, with

$$r=p,\qquad C_{0}=\frac{\Gamma}{e},\qquad C_{1}=\Lambda.$$

We therefore obtain

$$\left\|T_{t}\right\|\leq C(1+t)^{1-\varepsilon},\qquad t\geq 0,$$

where

$$\varepsilon=\frac{1}{p}\log_{2}\left(1+(2\Gamma)^{-p}\right)>0.$$

Since $\Gamma\geq 1$, we have $\varepsilon<1$. This proves the assertion under Cesàro boundedness. The Kreiss bounded case follows immediately from [3, Proposition 2.6]. ∎

Remark 5.5.

The preceding argument is non-quantitative. Indeed, the constant $\Gamma$ in (16) is obtained by combining Proposition 5.2 from [2] with Banach-Steinhaus, and the proof does not provide a bound for $\Gamma$ in terms of $p$ and $C_{K}(T)$ (or $C_{\mathrm{Ces}}(T)$) alone. This contrasts with the positive case treated above, where the positive convolution principle yields an explicit triangular estimate and therefore an explicit exponent depending on the Abel constant. Moreover, the argument above directly gives the exponent associated with $p$. If the adjoint semigroup on $L^{p^{\prime}}(\Omega)$ is also individually eventually positive (which is not automatic), the same proof can be applied to $(T_{t}^{*})_{t\geq 0}$, and one may then take the better of the two resulting exponents.

Remark 5.6.

The preceding argument also has a discrete counterpart. Recall that an operator $T\in B(L^{p}(\Omega))$ is called individually eventually positive if, for every $f\in L^{p}(\Omega)_{+}$, there exists $N_{f}\in\mathbb{N}$ such that

$$T^{n}f\geq 0,\qquad n\geq N_{f}.$$

If $1<p<\infty$ and $T$ is individually eventually positive and Kreiss bounded on $L^{p}(\Omega)$, then there exist $C\geq 1$ and $\varepsilon>0$ such that

$$\left\|T^{n}\right\|\leq C(n+1)^{1-\varepsilon},\qquad n\geq 0.$$

We do not pursue the details here. In view of the arguments developed above, the proof is obtained in the same way as in continuous time, with integrals replaced by sums and using the discrete self-improvement Lemma 4.1. As in the semigroup case, the result is non-quantitative: the argument does not provide an exponent $\varepsilon$ depending only on the Kreiss constant (or the Cesàro constant).

Appendix A Proofs of the triangular estimates

We collect here the Fourier multiplier arguments used in the Hilbertian parts of Propositions 2.2 and 2.4.

Proposition A.1.

1. Let $(T_{t})_{t\geq 0}$ be a Kreiss bounded $C_{0}$-semigroup on a complex Hilbert space $H$ with generator $-A$. For $r>0$, the operator

   $$(\mathcal{C}_{r}f)(s):=\int_{0}^{\infty}e^{-ru}T_{u}f(s+u)\,du\qquad(s\in\mathbb{R}),$$

   defined initially on $L^{1}(\mathbb{R};H)\cap L^{2}(\mathbb{R};H)$, extends uniquely to $\mathcal{C}_{r}\in B\bigl(L^{2}(\mathbb{R};H)\bigr)$ with

   (21) $$\left\|\mathcal{C}_{r}\right\|_{B(L^{2}(\mathbb{R};H))}\leq\frac{C_{K}(T)}{r}.$$

   Moreover, for every $f\in L^{2}(\mathbb{R};H)$ and almost every $\beta\in\mathbb{R}$,

   (22) $$\widehat{\mathcal{C}_{r}f}(\beta)=R(r-i\beta,-A)\widehat{f}(\beta).$$

2. Let $T\in B(H)$ be Kreiss bounded and $0<\rho<1$. The operator

   $$(C_{\rho}f)_{i}:=\sum_{n=0}^{\infty}\rho^{n}T^{n}f_{i+n}\qquad(i\in\mathbb{Z}),$$

   defined initially on $\ell^{1}(\mathbb{Z};H)\cap\ell^{2}(\mathbb{Z};H)$, extends uniquely to $C_{\rho}\in B\bigl(\ell^{2}(\mathbb{Z};H)\bigr)$ with

   (23) $$\left\|C_{\rho}\right\|_{B(\ell^{2}(\mathbb{Z};H))}\leq\frac{C_{K}(T)}{1-\rho}.$$

   Moreover, for every $f\in\ell^{2}(\mathbb{Z};H)$ and almost every $\gamma\in\mathbb{T}$ (with normalized Haar measure),

   (24) $$\widehat{C_{\rho}f}(\gamma)=(I-\rho\gamma T)^{-1}\widehat{f}(\gamma).$$

Proof.

For (i), since $(T_{t})_{t\geq 0}$ is Kreiss bounded on $H$, for each $r>0$

$$\int_{0}^{\infty}e^{-ru}\left\|T_{u}\right\|\,du<\infty.$$

For $f\in L^{1}(\mathbb{R};H)\cap L^{2}(\mathbb{R};H)$, Minkowski’s inequality shows that $\mathcal{C}_{r}f\in L^{2}(\mathbb{R};H)$. Moreover,

$$\int_{\mathbb{R}}\int_{0}^{\infty}e^{-ru}\left\|T_{u}f(s+u)\right\|\,du\,ds\leq\left\|f\right\|_{L^{1}(\mathbb{R};H)}\int_{0}^{\infty}e^{-ru}\left\|T_{u}\right\|\,du<\infty,$$

so Fubini’s theorem and the change of variables $v=s+u$ give

$$\widehat{\mathcal{C}_{r}f}(\beta) =\int_{0}^{\infty}e^{-ru}T_{u}\left(\int_{\mathbb{R}}e^{-i\beta s}f(s+u)\,ds\right)du$$

$$=\left(\int_{0}^{\infty}e^{-(r-i\beta)u}T_{u}\,du\right)\widehat{f}(\beta)$$

$$=R(r-i\beta,-A)\widehat{f}(\beta).$$

By the Fourier-Plancherel theorem and the Kreiss condition,

$$\left\|\mathcal{C}_{r}f\right\|_{L^{2}(\mathbb{R};H)}\leq\frac{C_{K}(T)}{r}\left\|f\right\|_{L^{2}(\mathbb{R};H)}.$$

Density gives (21) and (22) on all of $L^{2}(\mathbb{R};H)$.

For (ii), since $T$ is Kreiss bounded, for every $0<\rho<1$,

$$\sum_{n=0}^{\infty}\rho^{n}\left\|T^{n}\right\|<\infty.$$

For $f\in\ell^{1}(\mathbb{Z};H)\cap\ell^{2}(\mathbb{Z};H)$,

$$\sum_{i\in\mathbb{Z}}\sum_{n=0}^{\infty}\rho^{n}\left\|T^{n}f_{i+n}\right\|\leq\left\|f\right\|_{\ell^{1}(\mathbb{Z};H)}\sum_{n=0}^{\infty}\rho^{n}\left\|T^{n}\right\|<\infty.$$

Thus the sums may be interchanged and then

$$\widehat{C_{\rho}f}(\gamma) =\sum_{n=0}^{\infty}\rho^{n}T^{n}\sum_{i\in\mathbb{Z}}f_{i+n}\gamma^{-i}=\sum_{n=0}^{\infty}(\rho\gamma T)^{n}\widehat{f}(\gamma)=(I-\rho\gamma T)^{-1}\widehat{f}(\gamma).$$

If $\lambda=(\rho\gamma)^{-1}$, then $(I-\rho\gamma T)^{-1}=\lambda R(\lambda,T)$ and therefore

$$\left\|(I-\rho\gamma T)^{-1}\right\|\leq\frac{C_{K}(T)}{1-\rho}.$$

The Parseval theorem yields

$$\left\|C_{\rho}f\right\|_{\ell^{2}(\mathbb{Z};H)}\leq\frac{C_{K}(T)}{1-\rho}\left\|f\right\|_{\ell^{2}(\mathbb{Z};H)}.$$

Since $\ell^{1}(\mathbb{Z};H)\cap\ell^{2}(\mathbb{Z};H)$ is dense in $\ell^{2}(\mathbb{Z};H)$, this proves (23) and the multiplier identity (24) extends by density. ∎

Declaration on the use of generative AI

The author used ChatGPT (GPT-5.6 Sol, OpenAI) during the preparation of this manuscript for language editing, presentation, and exploratory feedback. All mathematical content was independently verified by the author, who takes full responsibility for the manuscript.