\[ s _ { m _ { n } n } = \frac { \sum _ { j \ge n + 1 } t _ { m _ { n } j } } { \sum _ { j \ge 1 } j t _ { m _ { n } j } } = \frac { p + 1 } { p n + 1 } \left( \frac { p - 1 } { p } \right) ^ { p } \alpha _ { n } \, , \]

where \( \alpha _ { n } \rightarrow 1 ^ { + } \). The conclusion follows from Theorem 4.1.

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For the special choice \( w _ { n } = n ^ { r } \),\( n \geq \) 1, with \( r \ge \) 0 fixed, Corollary 4.6 yields the implication

\[ ( 4 . 5 ) \qquad \qquad \operatorname * { s u p } _ { n \geq 1 } n ^ { - r } \| M _ { n } ^ { ( p ) } ( \lambda T ) \| < \infty \Longrightarrow \| T ^ { n } \| = o ( n ^ { r + 1 } ) , \quad n \to \infty \, , \]

whenever \( p \geq \) 1 and \( \sigma ( T ) \cap \mathbb { T } \) has arclength measure zero. It turns out that such ”regular” growth restrictions for the Ces\`aro means can be reformulated in terms of resolvent estimates. This has been first proved by Strikwerda and Wade [SW] in the case when \( p \) = 2 and \( r \) = 0. The Ces\`aro means of order one are related to the partial sums of the Taylor expansion at infinity of the resolvent function of T. The uniform boundedness of these partial sums in the exterior of the unit disc is called the uniform Kreiss boundedness condition (see [MSZ] and[GH] for the boundedness only on the real line) and is equivalent to the uniform boundedness of \( \{ M _ { n } ( \lambda T ) \} \) on the unit circle. These last two conditions are more restrictive. The result below provides an extension of the theorems wej ust mentioned.

Theorem 4.7. Given \( T \in { \mathcal { B } } ( { \mathcal { X } } ) \) we have:

(i) If \( p \geq \) 2 and \( r \ge \) 0 then

\[ \operatorname* { s u p } _ { \stackrel { n \geq 1 } { \lambda \in \mathbb { T } } } n ^ { - r } \| M _ { n } ^ { ( p ) } ( \lambda T ) \| < \infty \iff \operatorname* { s u p } _ { | \lambda | > 1 } \frac { ( | \lambda | - 1 ) ^ { r + 1 } } { | \lambda | ^ { r } } \| ( T - \lambda I ) ^ { - 1 } \| < \infty \, . \]

(ii) If \( p \) = 1 and \( r \ge \) 0 then

\[ \operatorname* { s u p } _ { \frac { n \geq 1 } { \lambda \in \mathbb { T } } } n ^ { - r } \| M _ { n } ( \lambda T ) \| < \infty \iff \operatorname* { s u p } _ { \frac { n \in \mathbb { N } } { | \lambda | > 1 } } \frac { ( | \lambda | - 1 ) ^ { r + 1 } } { | \lambda | ^ { r } } \left\| \sum _ { k = 0 } ^ { n } \lambda ^ { - k - 1 } T ^ { k } \right\| < \infty . \]

Proof. (=\( \Rightarrow \)) The argument uses the ideas in [SW]. We start with the identity proved in [SW],formula (6.3), for 0 \( < \rho < \) 1 and \( \lambda \in \mathbb { T } \), namely

\[ ( I - \rho \lambda T ) ^ { - 1 } = ( 1 - \rho ) ^ { p } \sum _ { n = 0 } ^ { \infty } \binom { n + p } { p } M _ { n } ^ { ( p ) } ( \lambda T ) \rho ^ { n } \, . \]

For \( p \geq \) 2 the resolvent estimate follows directly from

\[ \sum _ { n = 0 } ^ { \infty } \binom { n + p } { p } n ^ { r } \rho ^ { n } \leq C ( 1 - \rho ) ^ { - p - r - 1 } \, . \]

By a straightforward argument we see that the above equality holds for all \( \rho \in \mathbb { D } \) and by comparing the coefficients of \( \rho ^ { k } \) we obtain for \( p \) = 1,

\[ \sum _ { k = 0 } ^ { n } \rho ^ { k } \lambda ^ { k } T ^ { k } = ( 1 - \rho ) \sum _ { k = 0 } ^ { n - 1 } ( k + 1 ) M _ { k } ( \lambda T ) \rho ^ { k } + ( n + 1 ) M _ { n } ( \lambda T ) \rho ^ { n } \, . \]

In this case the desired estimate follows from the inequalities

\[ \sum _ { n = 0 } ^ { n - 1 } ( k + 1 ) ^ { r + 1 } \rho ^ { k } \leq C ( 1 - \rho ) ^ { - r - 2 } \, , \quad ( 1 - \rho ) ^ { r + 1 } \rho ^ { n } \leq C ^ { \prime } ( n + 1 ) ^ { - r - 1 } \, , \]

for 0 \( < \rho < \) 1.

(\( \Leftarrow \)=) We begin with (i) and assume first that \( p \) = 2. Use (2.6) to write for 0 \( < \rho < \) 1,\( \lambda \in \mathbb { T } \),

\[ \frac { ( n + 1 ) ( n + 2 ) } { 2 } M _ { n } ^ { ( 2 ) } ( \lambda T ) = \int _ { 0 } ^ { 2 \pi } ( I - \rho \lambda e ^ { i t } T ) ^ { - 1 } \sum _ { j = 0 } ^ { n } ( n - j + 1 ) e ^ { - i j t } \rho ^ { - j } \frac { \mathrm { d } t } { 2 \pi } \, . \]

so that

\[ \begin{array} { r l } & { \left\| \frac { ( n + 1 ) ( n + 2 ) } { 2 } M _ { n } ^ { ( 2 ) } ( \lambda T ) \right\| \le \frac { C } { ( 1 - \rho ) ^ { r + 1 } } \rho ^ { - n } \int _ { 0 } ^ { 2 \pi } \frac { 1 } { | 1 - \rho e ^ { - i t } | ^ { 2 } } \frac { \mathrm { d } t } { 2 \pi } } \\ & { \qquad \qquad \qquad \qquad = \frac { C } { ( 1 - \rho ) ^ { r + 1 } ( 1 - \rho ^ { 2 } ) } \rho ^ { - n } } \\ & { \qquad \qquad \qquad \qquad \le \frac { C } { ( 1 - \rho ) ^ { r + 2 } } \rho ^ { - n } \, . } \end{array} \]

If we now choose \( \textstyle \rho = 1 - { \frac { 1 } { n } } \) we obtain the estimate for \( M _ { n } ^ { ( 2 ) } ( T ) \) in (i) for all \( r \ge \) 0. For \( p > \) 2, this follows immediately from the first equality in (2.6).

Let us turn to the case when \( p \) = 1. For \( n \geq \) 1, \( \mu \in \mathbb { T } \) and \( \lambda \in \mathbb { C } \) with \|λ\| > 1 we have by the Abel summation formula

\[ \begin{array} { c } { { \displaystyle M _ { n } ( \mu T ) = \frac { 1 } { n + 1 } \sum _ { k = 0 } ^ { n } ( \mu T ) ^ { k } = \frac { 1 } { n + 1 } \sum _ { k = 0 } ^ { n } \lambda ^ { k + 1 } \lambda ^ { - k - 1 } ( \mu T ) ^ { k } } } \\ { { = \frac { 1 } { n + 1 } \left( \lambda ^ { n + 1 } \sum _ { k = 0 } ^ { n } \lambda ^ { - k - 1 } ( \mu T ) ^ { k } + ( 1 - \lambda ) \sum _ { k = 0 } ^ { n - 1 } \lambda ^ { k + 1 } \sum _ { j = 0 } ^ { k } \lambda ^ { - j - 1 } ( \mu T ) ^ { j } \right) } } \\ { { = \frac { 1 } { n + 1 } \left( \lambda ^ { n + 1 } \overline { { { \mu } } } \sum _ { k = 0 } ^ { n } ( \lambda \overline { { { \mu } } } ) ^ { - k - 1 } T ^ { k } + ( 1 - \lambda ) \sum _ { k = 0 } ^ { n - 1 } \lambda ^ { k + 1 } \overline { { { \mu } } } \sum _ { j = 0 } ^ { k } ( \lambda \overline { { { \mu } } } ) ^ { - j - 1 } T ^ { j } \right) . } } \end{array} \]

From the assumption

\[ \operatorname* { s u p } _ { \stackrel { n \in \mathbb { N } } { | \lambda | > 1 } } \frac { ( | \lambda | - 1 ) ^ { r + 1 } } { | \lambda | ^ { r } } \left\| \sum _ { k = 0 } ^ { n } \lambda ^ { - k - 1 } T ^ { k } \right\| < \infty \, , \]

we obtain

\[ \| M _ { n } ( \mu T ) \| \leq C \frac { | \lambda | ^ { r + 1 } } { ( | \lambda | - 1 ) ^ { r + 1 } ( n + 1 ) } \left( | \lambda | ^ { n } + \frac { | 1 - \lambda | ( | \lambda | ^ { n } - 1 ) } { | \lambda | - 1 } \right) \, , \]

and if \( \textstyle \lambda = 1 + { \frac { 1 } { n } } \),

\[ \| M _ { n } ( \mu T ) \| \le 2 ^ { r } ( 2 e - 1 ) C n ^ { r } \]

for \( n \geq \) 1 and \( \mu \in \mathbb { T } \), which concludes the proof.

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