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Aleman, Alexandru and Suciu, Laurian 2016. On Ergodic Operator Means in Banach Spaces. Integral Equations and Operator Theory, Vol. 85, Issue. 2, p. 259.
Cohen, Guy and Lin, Michael 2016. Remarks on rates of convergence of powers of contractions. Journal of Mathematical Analysis and Applications, Vol. 436, Issue. 2, p. 1196.
Suciu, Laurian 2016. Estimations of the Operator Resolvent by Higher Order Cesàro Means. Results in Mathematics, Vol. 69, Issue. 34, p. 457.
Gomilko, Alexander and Zemánek, Jaroslav 2013. On the Strong Kreiss Resolvent Condition. Complex Analysis and Operator Theory, Vol. 7, Issue. 2, p. 421.
Kozitsky, Yuri Shoikhet, David and Zemánek, Jaroslav 2013. Power convergence of Abel averages. Archiv der Mathematik, Vol. 100, Issue. 6, p. 539.
Léka, Zoltán 2013. On Orbits of Functions of the Volterra Operator. Complex Analysis and Operator Theory, Vol. 7, Issue. 4, p. 1321.
Reich, Simeon Shoikhet, David and Zemánek, Jaroslav 2013. Ergodicity, numerical range, and fixed points of holomorphic mappings. Journal d'Analyse Mathématique, Vol. 119, Issue. 1, p. 275.
Paulauskas, Vygantas 2012. A generalization of sectorial and quasisectorial operators. Journal of Functional Analysis, Vol. 262, Issue. 5, p. 2074.
Gomilko, Oleksandr 2011. Optimal estimates for the semigroup generated by the classical Volterra operator on L p spaces. Semigroup Forum, Vol. 83, Issue. 2, p. 343.
Léka, Zoltán 2010. A note on the powers of Cesàro bounded operators. Czechoslovak Mathematical Journal, Vol. 60, Issue. 4, p. 1091.
Dungey, Nick 2009. Asymptotic type for sectorial operators and an integral of fractional powers. Journal of Functional Analysis, Vol. 256, Issue. 5, p. 1387.
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Bermudo, Sergio MontesRodríguez, Alfonso and Shkarin, Stanislav 2008. Orbits of operators commuting with the Volterra operator. Journal de Mathématiques Pures et Appliquées, Vol. 89, Issue. 2, p. 145.
Gomilko, A. M. and Zemánek, J. 2008. On the uniform Kreiss resolvent condition. Functional Analysis and Its Applications, Vol. 42, Issue. 3, p. 230.
Гомилко, Александр Mихайлович Gomilko, Aleksandr Mikhailovich Земанек, Я and Zemanek, Ya 2008. О равномерном резольвентном условии Крейсса. Функциональный анализ и его приложения, Vol. 42, Issue. 3, p. 81.
Let $V$ denote the classical Volterra operator. In this work, sharp estimates of the norm of $(I  V)^n$ acting on $L^p [0, 1]$, for $1 \leq p \leq \infty$, are obtained. As a consequence, $I  V$ acting on $L^p [0, 1]$, with $1 \leq p \leq \infty$, is power bounded if and only if $p = 2$. Thus the Volterra operator characterizes when $L^p [0, 1]$ is a Hilbert space. By means of sharp estimates of the $L^1$norm of the $n$th partial sums of the generating function of the Laguerre polynomials on the unit circle, it is also proved that <formula form="inline" disc="math" id="frm012"><formtex notation="AMSTeX">$I  V$ is uniformly Kreiss bounded on the spaces $L^p [0,1]$, for $1 \leq p \leq \infty$.
A bounded linear operator $T$ on a Banach space is said to be Kreiss bounded if there is a constant $C > 0$ such that $\Vert (T  \lambda)^{1} \Vert \leq C(  \lambda   1)^{1}$ for $ \lambda  > 1$. If the same upper estimate holds for each of the partial sums of the resolvent, then $T$ is said to be uniformly Kreiss bounded. This is, for instance, true for power bounded operators. For finitedimensional Banach spaces, Kreiss' Matrix Theorem asserts that Kreiss boundedness is equivalent to $T$ being power bounded. Thus, in the infinitedimensional setting, even a much stronger property than Kreiss boundedness still does not imply power boundedness. It is also shown that, for general operators, uniform Abel boundedness characterizes Cesàro boundedness and, as a consequence, uniform Kreiss boundedness is characterized in terms of a Cesàro type boundedness of order 1.
Let $V$ denote the classical Volterra operator. In this work, sharp estimates of the norm of $(I  V)^n$ acting on $L^p [0, 1]$, for $1 \leq p \leq \infty$, are obtained. As a consequence, $I  V$ acting on $L^p [0, 1]$, with $1 \leq p \leq \infty$, is power bounded if and only if $p = 2$. Thus the Volterra operator characterizes when $L^p [0, 1]$ is a Hilbert space. By means of sharp estimates of the $L^1$norm of the $n$th partial sums of the generating function of the Laguerre polynomials on the unit circle, it is also proved that <formula form="inline" disc="math" id="frm012"><formtex notation="AMSTeX">$I  V$ is uniformly Kreiss bounded on the spaces $L^p [0,1]$, for $1 \leq p \leq \infty$.
A bounded linear operator $T$ on a Banach space is said to be Kreiss bounded if there is a constant $C > 0$ such that $\Vert (T  \lambda)^{1} \Vert \leq C(  \lambda   1)^{1}$ for $ \lambda  > 1$. If the same upper estimate holds for each of the partial sums of the resolvent, then $T$ is said to be uniformly Kreiss bounded. This is, for instance, true for power bounded operators. For finitedimensional Banach spaces, Kreiss' Matrix Theorem asserts that Kreiss boundedness is equivalent to $T$ being power bounded. Thus, in the infinitedimensional setting, even a much stronger property than Kreiss boundedness still does not imply power boundedness. It is also shown that, for general operators, uniform Abel boundedness characterizes Cesàro boundedness and, as a consequence, uniform Kreiss boundedness is characterized in terms of a Cesàro type boundedness of order 1.
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