New counterexamples on Ritt operators, sectorial operators and R-boundedness

TL;DR

This paper constructs counterexamples of Ritt and sectorial operators that are multipliers but lack R-boundedness, highlighting limitations in the theory of operator boundedness on Banach spaces.

Contribution

It demonstrates that non-$R$-Schauder decompositions can produce operators with non-$R$-bounded powers or semigroups, revealing new limitations in operator theory.

Findings

Existence of Ritt operators that are multipliers but not $R$-bounded

Existence of sectorial operators with non-$R$-bounded semigroups

Counterexamples challenge assumptions about $R$-boundedness in Banach spaces

Abstract

Let $\mathcal D$ be a Schauder decomposition on some Banach space $X$. We prove that if $\mathcal D$ is not $R$-Schauder, then there exists a Ritt operator $T\in B(X)$ which is a multiplier with respect to $\mathcal D$, such that the set $\{T^n\, :\, n\geq 0\}$ is not $R$-bounded. Likewise we prove that there exists a bounded sectorial operator $A$ of type $0$ on $X$ which is a multiplier with respect to $\mathcal D$, such that the set $\{e^{-tA}\, : \, t\geq 0\}$ is not $R$-bounded.

Equations

$$\Bigl{\|}\sum_{j=1}^{k}\varepsilon_{j}\otimes x_{j}\Bigr{\|}_{R,X}=\int_{\Omega}\Bigl{\|}\sum_{j=1}^{k}\varepsilon_{j}(u)x_{j}\Bigr{\|}_{X}\,d\mathbb{P}(u).$$

$$\Bigl{\|}\sum_{j=1}^{k}\varepsilon_{j}\otimes T_{j}(x_{j})\Bigr{\|}_{R,X}\,\leq\,K\,\Bigl{\|}\sum_{j=1}^{k}\varepsilon_{j}\otimes x_{j}\Bigr{\|}_{R,X}.$$

$$\bigl{\{}\lambda R(\lambda,A)\,:\,\lambda\in\mathbb{C}\setminus\overline{\Sigma_{\theta}}\,\bigr{\}}$$

$$\lambda R(\lambda ,A^{-1})=I_X-{\lambda }^{-1}R({\lambda }^{-1},A).$$

$$\bigl{\{}e^{-tA}\,:\,t\geq 0\bigr{\}}\qquad\hbox{and}\qquad\bigl{\{}tAe^{-tA}\,:\,t\geq 0\bigr{\}}$$

$$\bigl{\{}T^{n}\,:\,n\geq 0\bigr{\}}\qquad\hbox{and}\qquad\bigl{\{}nT^{n}(I_{X}-T)\,:\,n\geq 1\bigr{\}}$$

$$\{P_N:N\ge 1\}$$

$$\mathrm{var}(c)=|{\ell }_c|+\sum _{n=1}^{\infty }|c_n-c_{n+1}|.$$

$$M_c(x)={\ell }_{c}x+\sum _{N=1}^{\infty }(c_N-c_{N+1})P_N(x),x\in X.$$

$$\Vert M_c\Vert \le \mathrm{var}(c)\underset{N\ge 1}{\sup }\Vert P_N\Vert .$$

$$A(x)=\sum _{n=1}^{\infty }{a}_n{p}_n(x).$$

$$\sigma(A)=\overline{\bigl{\{}a_{n}\,:\,n\geq 1\bigr{\}}}$$

$$K_{\theta}=\sup\bigl{\{}|\lambda|{\rm var}(c(\lambda))\,:\,\lambda\in\mathbb{C}\setminus\overline{\Sigma_{\theta}}\bigr{\}}\,<\infty,$$

$$T(x)=\sum _{n=1}^{\infty }{c}_n{p}_n(x),x\in X.$$

$$\bigr{[}\lambda R(\lambda,A)\bigr{]}(x)\,=\,\sum_{N=1}^{\infty}\,\lambda\bigl{(}c(\lambda)_{N}-c(\lambda)_{N+1}\bigr{)}\,P_{N}(x),\qquad x\in X,$$

$$\bigl{\{}\lambda R(\lambda,A)\,:\,\lambda\in\mathbb{C}\setminus\overline{\Sigma_{\theta}}\bigr{\}}\,\subset K_{\theta}\,\cdotp\overline{{\rm aco}}^{so}(F),$$

$$\bigl{\{}e^{-tA^{-1}}\,:\,t\geq 0\bigr{\}}$$

$$\bigl{\{}T^{n}\,:\,n\geq 0\bigr{\}}$$

$$\bigl{(}M_{c}-Q_{N}\bigr{)}(x)\,=\,\sum_{n=1}^{N}c_{n}p_{n}(x)\,+\,\sum_{n=N+1}^{\infty}(c_{n}-1)p_{n}(x).$$

$$\sum _{n=1}^N{c}_n{p}_n(x)=c_N{P}_N(x)+\sum _{n=1}^{N-1}(c_n-c_{n+1})P_n(x).$$

$$\sum _{n=N+1}^{\infty }(c_n-1)p_n(x)$$

$$=(c_{N+1}-1)Q_N(x)+\sum _{n=N+1}^{\infty }(c_{n+1}-c_n)Q_n(x).$$

$$\bigl{\|}M_{c}-Q_{N}\bigr{\|}\,\leq(1+K)\biggl{(}\sum_{n=1}^{N-1}|c_{n+1}-c_{n}|\,+|c_{N}|+|1-c_{N+1}|+\,\sum_{n=N+1}^{\infty}|c_{n+1}-c_{n}|\biggr{)}.$$

$$c_n=e^{-t{a}_n^{-1}},n\ge 1.$$

$$|c_N|=c_N=e^{-t{a}_N^{-1}},|1-c_{N+1}|=1-c_{N+1}=1-e^{-t{a}_{N+1}^{-1}},$$

$$\sum _{n=1}^{N-1}|c_{n+1}-c_n|=c_N-c_1\le c_N=e^{-t{a}_N^{-1}},$$

$$\sum _{n=N+1}^{\infty }|c_{n+1}-c_n|=1-c_{N+1}=1-e^{-t{a}_{N+1}^{-1}},$$

$$\bigl{\|}e^{-tA^{-1}}-Q_{N}\bigr{\|}\leq 2(1+K)\bigl{(}e^{-ta_{N}^{-1}}+(1-e^{-ta_{N+1}^{-1}})\bigr{)}.$$

$$a_n=(n!{)}^3,n\ge 1.$$

$$t_N=N(N!{)}^3,N\ge 1.$$

$$\bigl{\|}e^{-t_{N}A^{-1}}-Q_{N}\bigr{\|}\leq 2(1+K)\bigl{(}e^{-N}+(1-e^{-\frac{N}{(N+1)^{3}}})\bigr{)}\leq 2(1+K)\bigl{(}e^{-N}+N^{-2}\bigr{)}$$

Full text

2000 Mathematics Subject Classification: 47A99, 46B15.

$R$-boundedness plays a prominent role in the study of sectorial operators and Ritt operators. Namely the notions of $R$-sectorial operators and $R$-Ritt operators have been instrumental in the development of $H^{\infty}$-functional calculus, square function estimates and applications to maximal regularity and to many other aspects of the harmonic analysis of semigroups (in either the continuous or the discrete case).

The existence of sectorial operators which are not $R$-sectorial was discovered by Kalton and Lancien in their paper solving the $L^{p}$-maximal regularity problem [6]. The existence of Ritt operators which are not $R$-Ritt was established a bit later by Portal [14]. More recently, Fackler [4] extended the work of Kalton-Lancien in various directions. In contrast with [6], which focused on existence results, [4] supplied explicit constructions of sectorial operators which are not $R$-sectorial. Further it is easy to derive from the latter paper explicit constructions of Ritt operators which are not $R$-Ritt. In [4, 6, 14], sectorial operators which are not $R$-sectorial (resp. Ritt operators which are not $R$-Ritt) are defined as multipliers with respect to Schauder decompositions having various “bad” properties. In particular, these Schauder decompositions cannot be $R$-Schauder (see Lemma 0.2).

The aim of this note is two-fold. First we show that given any Schauder decomposition ${\mathcal{D}}$ which is not $R$-Schauder, one can define a sectorial operator $A$ which is a multiplier with respect to ${\mathcal{D}}$ and which is not $R$-sectorial (resp. a Ritt operator $T$ which is a multiplier with respect to ${\mathcal{D}}$ and which is not $R$-Ritt). Second we strengthen these negative results in both cases by showing that $A$ can be chosen bounded and such that $\{e^{-tA}\,:\,t\geq 0\}$ is not $R$-bounded, whereas $T$ is taken such that $\{T^{n}:\,n\geq 0\}$ is not $R$-bounded. (See Remark 0.6 for more comments.)

In addition to the above mentioned papers, we refer the reader to [3, 8, 12, 16] for relevant information on $R$-sectorial and $R$-Ritt operators. We also mention [9] which contains examples of Ritt operators which are not $R$-Ritt. They are of a different nature to those in [14].

We now introduce the relevant definitions and constructions to be used in this paper. Throughout we let $X$ be a complex Banach space and we let $B(X)$ denote the Banach algebra of all bounded operators on $X$. We let $I_{X}$ denote the identity operator on $X$.

Let $(\varepsilon_{j})_{j\geq 1}$ be an independent sequence of Rademacher variables on some probability space $(\Omega,d\mathbb{P})$. Given any $x_{1},\ldots,x_{k}$ in $X$, we set

[TABLE]

Then we say that a subset $F\subset B(X)$ is $R$-bounded provided that there exists a constant $K\geq 0$ such that for any $k\geq 1$, for any $T_{1},\ldots,T_{k}$ in $F$ and for any $x_{1},\ldots,x_{k}$ in $X$,

[TABLE]

We refer the reader to e.g. [5, Chap. 8] for basic information on $R$-boundedness.

For any $\omega\in(0,\pi)$, we let $\Sigma_{\omega}=\{\lambda\in\mathbb{C}^{*}\,:\,|{\rm Arg}(\lambda)|<\omega\}$. Let $A$ be a densely defined closed operator $A\colon D(A)\to X$, with domain $D(A)\subset X$. Let $\sigma(A)$ denote the spectrum of $A$ and let $R(\lambda,A)=(\lambda I_{X}-A)^{-1}$ denote the resolvent operator for $\lambda\notin\sigma(A)$. We say that $A$ is sectorial of type $\omega\in(0,\pi)$ if $\sigma(A)\subset\overline{\Sigma_{\omega}}$ and for any $\theta\in(\omega,\pi)$, the set

[TABLE]

is bounded. We further say that $A$ is sectorial of type [math] if it is sectorial of type $\omega$ for any $\omega\in(0,\pi)$.

Note that if $A$ is sectorial of type $\omega$ and $A$ is invertible, then $A^{-1}\in B(X)$ is sectorial of type $\omega$ as well. This readily follows from the fact that for any $\lambda\notin\overline{\Sigma_{\omega}}$, we have $\lambda^{-1}\notin\overline{\Sigma_{\omega}}$ and

[TABLE]

We recall that $A$ is sectorial of type $<\frac{\pi}{2}$ if and only if $-A$ generates a bounded analytic semigroup. In this case, the latter is denoted by $(e^{-tA})_{t\geq 0}$.

Next we say that $A$ is $R$-sectorial of $R$-type $\omega\in(0,\pi)$ if $A$ is sectorial of type $\omega$ and for any $\theta\in(\omega,\pi)$, the set (0.1) is $R$-bounded.

The following lemma is a straightforward consequence of (0.2).

Let $A$ be a sectorial operator of type $<\frac{\pi}{2}$. We recall that by [16], $A$ is $R$-sectorial of $R$-type $<\frac{\pi}{2}$ if and only if the two sets

[TABLE]

are $R$-bounded.

Let $T\in B(X)$. We say that $T$ is a Ritt operator if the two sets

[TABLE]

are bounded. We further say that $T$ is $R$-Ritt if these two sets are $R$-bounded.

These notions are closely related to sectoriality. Indeed let $\mathbb{D}=\{\lambda\in\mathbb{C}\,:\,|\lambda|<1\}$ be the open unit disc. Then $T$ is a Ritt operator if and only if $\sigma(T)\subset\mathbb{D}\cup\{1\}$ and $I_{X}-T$ is sectorial of type $<\frac{\pi}{2}$. Further in this case, $T$ is $R$-Ritt if and only if $I_{X}-T$ is $R$-sectorial of $R$-type $<\frac{\pi}{2}$. We refer the reader to [3, 12] and the references therein for these results and various informations on Ritt operators and their applications.

We recall from [13, Section 1.g] that a Schauder decomposition on $X$ is a sequence \mbox{{\mathcal{D}}}=\{X_{n}\,:\,n\geq 1\} of closed subspaces of $X$ such that for any $x\in X$, there exists a unique sequence $(x_{n})_{n\geq 1}$ of $X$ such that $x_{n}\in X_{n}$ for any $n\geq 1$ and $x=\sum_{n=1}^{\infty}x_{n}.$ For any $n\geq 1$, we let $p_{n}\in B(X)$ be the projection defined for $x$ as above by $p_{n}(x)=x_{n}$. For any integer $N\geq 1$, consider their sum $P_{N}=\sum_{n=1}^{N}p_{n}\,$. This is a projection and the set

[TABLE]

is bounded.

We say that ${\mathcal{D}}$ is an $R$-Schauder decomposition if this set is actually $R$-bounded. Then a Schauder basis is called $R$-Schauder if its associated Schauder decomposition is $R$-Schauder.

Let $c=(c_{n})_{n\geq 1}$ be a sequence of complex numbers. Assume that the sum $\sum_{n=1}^{\infty}|c_{n}-c_{n+1}|\,$ is finite (in which case we say that the sequence has a bounded variation). Then $c$ has a limit. Let $\ell_{c}$ denote this limit and set

[TABLE]

For any $x\in X$, the series $\sum_{n}c_{n}p_{n}(x)\,$ converges. This follows from an Abel transformation argument, using the boundedness of $\{P_{N}\,:\,N\geq 1\}$. Let $M_{c}\colon X\to X$ be defined by $M_{c}(x)=\sum_{n=1}^{\infty}c_{n}p_{n}(x)$, then we actually have

[TABLE]

This implies that

[TABLE]

Let $(a_{n})_{n\geq 1}$ be a nondecreasing sequence of $(0,\infty)$. Then we may define an operator $A\colon D(A)\to X$ as follows. We let $D(A)$ be the space of all $x\in X$ such that the series $\sum_{n}a_{n}p_{n}(x)\,$ converges and for any $x\in D(A)$, we set

[TABLE]

Such operators were first introduced in [2, 15]. It is well-known that

[TABLE]

and that $A$ is a sectorial operator of type [math] (see [6, 10]). More precisely, for any $\lambda\in\mathbb{C}\setminus\mathbb{R}_{+}$, $R(\lambda,A)$ is the operator $M_{c(\lambda)}$ associated with the sequence c(\lambda)=\bigl{(}(\lambda-a_{n})^{-1}\bigr{)}_{n\geq 1} and for any $\theta\in(0,\pi)$, we have

[TABLE]

see e.g. [10, Section 2]. This estimate and (0.7) show that $A$ is sectorial of type [math].

We note for further use that by (0.9), the above operator $A$ is invertible.

In the sequel, any sectorial operator $A$ of this form will be called a ${\mathcal{D}}$-multiplier.

Likewise let $c=(c_{n})_{n\geq 1}$ be a nondecreasing sequence of $(0,1)$. Then $c$ has a bounded variation, which allows the definition of $T=M_{c}\in B(X)$ given by

[TABLE]

It turns out that $T$ is a Ritt operator on $X$. Indeed, let $A$ be the sectorial operator (0.8) associated with the sequence $(a_{n})_{n\geq 1}$ defined by $a_{n}=(1-c_{n})^{-1}$. Then $I_{X}-T=A^{-1}$ is sectorial of type [math] and $\sigma(T)\subset[0,1]$, which ensures that $T$ is a Ritt operator.

In the sequel, any Ritt operator $T$ of this form will be called a ${\mathcal{D}}$-multiplier.

The following is well-known to specialists.

• (a)

Any sectorial operator $A$ on $X$ which is a ${\mathcal{D}}$-multiplier is $R$-sectorial or $R$-type [math].

• (b)

Any Ritt operator $T\in B(X)$ which is a ${\mathcal{D}}$-multiplier is $R$-Ritt.

[TABLE]

with c(\lambda)=\bigl{(}(\lambda-a_{n})^{-1}\bigr{)}_{n\geq 1}. This implies that

[TABLE]

where $K_{\theta}$ is given by (0.10) and $\overline{{\rm aco}}^{so}(F)$ stands for the the closure of the absolute convex hull of $F$ in the strong operator topology of $B(X)$. Since $F$ is $R$-bounded, $\overline{{\rm aco}}^{so}(F)$ is $R$-bounded as well, see e.g. [5, Subsection 8.1.e]. Then the set (0.1) is $R$-bounded, which shows (a).

Let $T$ be given by (0.11). It follows from the above discussion that $T=I_{X}-A^{-1}$ for some sectorial operator $A$ on $X$ which is a ${\mathcal{D}}$-multiplier. By (a) and Lemma 0.1, $A^{-1}$ is $R$-sectorial of type $<\frac{\pi}{2}$. This entails that $T$ is $R$-Ritt. ∎

Our main result is the following.

• (a)

There exists a sectorial operator $A$ on $X$ which is a ${\mathcal{D}}$-multiplier, such that the set

[TABLE]

is not $R$-bounded.

• (b)

There exists a Ritt operator $T\in B(X)$ which is a ${\mathcal{D}}$-multiplier, such that the set

[TABLE]

is not $R$-bounded.

Let $c=(c_{n})_{n\geq 1}$ be a complex sequence with a bounded variation and let $N\geq 1$ be a fixed integer. For any $x\in X$, $Q_{N}(x)=\sum_{n=N+1}^{\infty}p_{n}(x)$ hence

[TABLE]

On the one hand, we have

[TABLE]

On the other hand,

[TABLE]

Let $K=\sup_{N\geq 1}\|P_{N}\|$. If follows from these identities that

[TABLE]

Let $(a_{n})_{n\geq 1}$ be a nondecreasing sequence of $(0,\infty)$, with $\lim_{n}a_{n}=\infty$, and let $A$ be the associated sectorial operator defined by (0.8). Let $t>0$ and apply the above with

[TABLE]

Then $c_{n}\in(0,1)$ for any $n\geq 1$, the sequence $(c_{n})_{n\geq 1}$ is nondecreasing (hence has a bounded variation) and $M_{c}=e^{-tA^{-1}}$. Further $\lim_{n}c_{n}=1$. Consequently we have

[TABLE]

[TABLE]

[TABLE]

and hence

[TABLE]

We apply the above with

[TABLE]

Next we consider the sequence $(t_{N})_{N\geq 1}$ of positive integers given by we set

[TABLE]

Then by (0.12), we have

[TABLE]

for any $N\geq 1$. This estimate implies that

[TABLE]

Let $C$ be the above sum. Then for any $x_{1},\ldots,x_{k}$ in $X$, we have

[TABLE]

Hence

[TABLE]

By assumption, the set $\{P_{N}\,:\,N\geq 1\}$ is not $R$-bounded, hence $\{Q_{N}\,:\,N\geq 1\}$ is nor $R$-bounded. The above estimate therefore shows that the set \bigl{\{}e^{-tA^{-1}}\,:\,t\geq 0\bigr{\}} cannot be $R$-bounded. This proves (a).

To prove (b), we consider $T=e^{-A^{-1}}$. Then $T$ is the Ritt operator defined by $(\ref{T})$ for the sequence $c_{n}=e^{-a_{n}^{-1}}$. For any $N\geq 1$, $t_{N}$ is an integer and $T^{t_{N}}=e^{-t_{N}A^{-1}}$. Hence the above argument shows that $\{T^{n}\,:\,n\geq 1\}$ is not $R$-bounded, which proves (b). ∎

Theorem 0.3 provides a converse to Lemma 0.2, as follows.

Combining the above fact with Theorem 0.3 (b) and Lemma 0.2, we deduce both ‘if and only if’ results. ∎

It follows from [4, 6] that if $X$ has an unconditional basis and $X$ is not isomorphic to a Hilbert space, then $X$ has a Schauder basis which is not $R$-Schauder. The above theorem therefore applies to all these spaces.

Further the arguments in [6, Theorem 3.7 $\&$ Corollary 3.8] show that we actually have the following.

• (a)

There exists a bounded sectorial operator $A$ of type [math] on $X$ such that $\{e^{-tA}\,:\,t\geq 0\}$ is not $R$-bounded.

• (b)

There exists a Ritt operator $T\in B(X)$ such that the set $\{T^{n}\,:\,n\geq 0\}$ is not $R$-bounded.

(1) It follows from [14] that there exists a Ritt operator $T\in B(X)$ such that $T$ is not $R$-Ritt. Recall that by definition, $T$ is not $R$-Ritt if and only if one of the two sets in (0.4) is not $R$-bounded. Part (b) of Corollary 0.5 strengthens [14] by providing a Ritt operator $T$ on $X$ for which we know that the first of the two sets in (0.4) is not $R$-bounded. This is an important step in the understanding of the class of power bounded operators $T$ such that $\{T^{n}\,:\,n\geq 0\}$ is $R$-bounded. This class will be investigated in a future paper (in preparation). We refer to [11] for the study of invertible operators $T\in B(X)$ such that $\{T^{n}\,:\,n\in\mathbb{Z}\}$ is $R$-bounded.

(2) The existence of a sectorial operators $A$ of type [math] on $X$ such that $\{e^{-tA}\,:\,t\geq 0\}$ is not $R$-bounded follows from [14]. Part (a) of Corollary 0.5 shows that this can be achieved with a bounded $A$. We refer to [1] for various results on bounded $C_{0}$-semigroups $(T_{t})_{t\geq 0}$ on Banach space such thatthe set $\{T_{t}\,:\,t\geq 0\}$ is/is not $R$-bounded.

Acknowledgements. The authors were supported by the French “Investissements d’Avenir” program, project ISITE-BFC (contract ANR-15-IDEX-03).