数理解析研究所講究録
第 1570 巻 2007 年 33-35

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Boundedness of -Ces\‘aro means $(\gamma>0)$ ofoperators$\gamma$

Ryotaro Sato (Okayama University)
E-mail: satoryot@math.okayama-u.ac.jp

In this talk I would like to report some recent results on the boundedness properties of -Ces\‘aro means of operators, where $\gamma>0$ . The results are taken from joint$\gamma$works with Jeng-Chung Chen, Yuan-Chuan Li, and Sen-Yen Shaw (cf. [1], [3]).

1. The discrete case. Let$T$: $Xarrow X$ be a bounded linear operator on aBanach space $X$ . The Ces\‘aro means of order$\gamma\geq 0$$\gamma$(or -Ces\‘aro means) of $T$ , where$\gamma$, are defined by$C_{n}^{\gamma}=C_{n}^{\gamma}(T)$where$\sigma_{n}^{\beta}=(\begin{array}{l}\beta+nn\end{array})$$:= \frac{1}{\sigma_{n}^{\gamma}}\sum_{k=0}^{n}\sigma_{n-k}^{\gamma-1}T^{k}$for $n\geq 1$ , and$\sigma_{0}^{\beta}=1$,$n=0,1,2,$$\ldots$,(see Zygmund [5, Chapter 3]). Amongthem are the $f_{0}n_{oW}ing$ two particular means: $C_{n}^{0}=C_{n}^{0}(T)=T^{n}$ , and $C_{n}^{1}=C_{n}^{1}(T)=$.
The Abel means of are the operators $A_{r}=A_{r}(T)$$(n+1)^{-1} \sum_{k=0}^{n}T^{k}$$T$

for $0<r<1/r(T)$ , where$r(T)= \lim_{narrow\infty}\Vert T^{n}\Vert^{1/n}$

Clearly, $r(T)\leq 1$ if and only if$A_{r}$$:=(1-r)\Sigma_{k=0}^{\infty}r^{k}T^{k}$, defined

denotes the spectral radius of .
$T$

exists for all $0<r<1$ . (Moreover, in this cse,

we have $A_{r}=(1-r)(I-rT)^{-1}$ for each $0<r<1.$ ) The following is well-known(cf. [5]):If $0<\gamma<\beta<\infty$ , then(1);$\sup_{0<r<1}\Vert A_{r}\Vert\leq\sup_{\mathfrak{n}\geq 0}\Vert C_{n}^{\beta}\Vert\leq\sup_{n\geq 0}\Vert C_{n}^{\gamma}\Vert\leq\sup_{n\geq 0}\Vert C_{n}^{0}\Vert(=\sup_{n\geq 0}\Vert T^{n}\Vert)$

in particular, if $T$ is a positive linear operator on a Banach lattice $X$ , then(2)$\sup_{0<r<1}\Vert A_{r}||<\infty\Leftrightarrow\sup_{n\geq 0}\Vert C_{n}^{1}\Vert<\infty$(cf. Emilion [2]).

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In connection with these relations, two questions come up naturally:(A) If $T$ is positive, then does the implication$<\infty$“holds for a cert ain constant(B) If$T$$\gamma_{f}$$\sup_{0<r<1}$

li il$A_{r}$$< \infty\Rightarrow\sup_{n\geq 1}\Vert C_{n}^{\gamma}\Vert$with $0<\gamma<1$ ?

is not assumed to be positive, then does the implication$\infty\Rightarrow\sup_{n\geq 1}\Vert C_{n}^{\gamma}\Vert<\infty$

hold for a certain constant , with”$\gamma$$\sup_{0<r<1}\Vert A_{r}\Vert<$$\gamma\geq 1$?

Our answers are as follows.
Theorem 1. For any
$T$

on an

$L_{1}$$\gamma_{f}$-space such thatwith $0<\gamma<1$ , there erists a positive linear operator$\sup_{n\geq 1}||C_{n}^{\beta}\Vert<\infty$

for all$\beta>\gamma$

Theorem 2. There erists a positive linear operator$\sup_{n\geq 1}\Vert C_{n}^{\beta}\Vert<\infty$

for all$\beta>0$, but$\sup_{n\geq 1}\Vert T^{n}\Vert=\infty$

$T$

, but

$\sup_{n\geq 1}\Vert C_{\mathfrak{n}}^{\gamma}\Vert=\infty$on an$L_{1}$.

-space such that.

Theorem 3. Let dim $X=\infty$ . Then the following hold:(i) For any integer $k\geq 1$ , there exists a bounded linear operator

that$\sup_{n\geq 1}\Vert C_{n}^{k}\Vert<\infty$, but$\sup_{n\geq 1}\Vert C_{n}^{\beta}||=\infty$

for all , with $0\leq\beta<k$ .

(ii) There evists a bounded linear operator$\sup_{0<r<1}||A_{r}\Vert<\infty$

2.

, but

$\sup_{n\geq 1}\Vert C_{n}^{\beta}\Vert=\infty$

The continuous case. Let$\beta$$T$

for all$T(\cdot)$

be aon $X$ , with $r(T)=1$ , such that$\beta\geq 0$$C_{0}$.

-semigroup of bounded linear

operators on a Banach space $X$ . The -th Ces\‘aro means of$\gamma$

defined as$C_{0}^{\gamma}=C_{0}^{\gamma}(T(\cdot))$$T(\cdot)$$\lambda>0$$\gamma\geq 0$$:=\{\begin{array}{ll}T(t) if \gamma=0,\gamma t^{-\gamma}\int_{0}^{t}(t-u)^{\gamma-1}T(u)du if \gamma>0.\end{array}$$:= \lambda\int_{0}^{\infty}e^{-\lambda u}T(u)du=\lim_{tarrow}\lambda\int_{0}^{t}e^{-\lambda u}T(u)du$,

if the limit exists. As in the discrete case, we have (cf. [4]):If $0<\gamma<\beta<\infty,$(3), whereare the operators$A_{\lambda}=A_{\lambda}(T(\cdot))$

defined for$T(\cdot)$$:=T(0)$ and, for $t>0$ ,$C_{t}^{\gamma}=C_{t}^{\gamma}(T(\cdot))$

The Abel means ofon $X$ such$T$$\cdot then$$\sup_{0<\lambda<\infty}||A_{\lambda}\Vert\leq\sup_{t>}||C_{t}^{\beta}\Vert\leq\sup_{t>}\Vert C_{t}^{\gamma}\Vert\leq\sup_{t>}\Vert C_{t}^{0}||(=\sup_{t>}||T(t)\Vert)$;, are35
inparticular, if(4)$T(\cdot)$is a positive$C_{0}$-semigroup on a Banach lattice $X$ , then(cf. [2]).

$\sup_{0<\lambda<\infty}\Vert A_{\lambda}\Vert<\infty\Leftrightarrow\sup_{t>0}\Vert C_{t}^{1}\Vert<\infty$

The following are the continuous case results:

Theorem 1’. For any , with $0<\gamma<1$ , there exists a positive$\gamma$$T(\cdot)$on an$L_{1}$-space such that$\sup_{t>0}\Vert C_{t}^{\beta}\Vert<\infty$

Theorem 2’. There eansts a positivethat$\sup_{t>0}\Vert C_{t}^{\beta}\Vert<\infty$

for all$\beta>0$$C_{0}$

for all$\beta>\gamma$-semigroup, but$T(\cdot)$$C_{0}$-semigroup$\sup_{t>0}\Vert\sigma_{t}\Vert=$

on an$L_{1}$科科.

-space such

, but $\sup_{t>0}\Vert T(t)\Vert=\infty$ .

Theorem 3’. Let dim $X=\infty$ . Then the following hold:(i) For any integer $k>1$ , there enists a$\sup_{t>0}\Vert C_{t}^{k}\Vert<\infty$, but(ii) There exzsts a$\sup_{t>0}||C_{t}^{\beta}||=\infty$$\sup_{t>0}\Vert C_{t}^{\beta}\Vert=\infty$$C_{0}$

for all-semigroup$\beta\geq 0$$T(\cdot)$$C_{0}$-semigroupfor all$\beta$$T(\cdot)$

on$X$

such that, with $0\leq\beta<k$ .

on $X$ such that$\sup_{0<\lambda<\infty}\Vert A_{\lambda}||<\infty$, but.
References[1] J.-C. Chen, R. Sato, and S.-Y. Shaw, Growth orders of Ces\‘aro and Abelmeans of functions in Banach spaces, preprint.
[2] R. Emilion, Mean-bounded operators and mean ergodic $th\infty rems$ , J. Rrnct.

Anal. 61 (1985), 1-14.
[3] Y.-C. Li, R. Sato, and S.-Y. Shaw, Boundedness and growth orders of meansof discrete and continuous semigroups, preprint.
[4] R. Sato, On ergodic averages and the range of a closed operator, TaiwaneseJ. Math. 10 (2006), 1193-1223.
[5] A. Zygmund, $ffigonomet_{7\dot{\eta}}c$ Series. Vol. I, Cmbridge University Press, Cmbridge, 1959.