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Uniform Abel–Kreiss Boundedness and the Extremal Behaviour of the Volterra Operator
Author(s) -
MontesRodríguez Alfonso,
SánchezÁlvarez Juan,
Zemánek Jaroslav
Publication year - 2005
Publication title -
proceedings of the london mathematical society
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 1.899
H-Index - 65
eISSN - 1460-244X
pISSN - 0024-6115
DOI - 10.1112/s002461150501539x
Subject(s) - mathematics , bounded function , bounded operator , banach space , norm (philosophy) , resolvent , uniform boundedness , laguerre polynomials , operator (biology) , finite rank operator , pure mathematics , discrete mathematics , combinatorics , mathematical analysis , biochemistry , chemistry , repressor , transcription factor , gene , political science , law
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 ⩽ p ⩽ ∞, are obtained. As a consequence, I − V acting on L p [0, 1], with 1 ⩽ p ⩽ ∞, 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 I − V is uniformly Kreiss bounded on the spaces L p [0,1], for 1 ⩽ p ⩽ ∞. A bounded linear operator T on a Banach space is said to be Kreiss bounded if there is a constant C > 0 such that ∥ ( T − λ ) − 1 ∥ ⩽ C ( | λ | − 1 ) − 1for ∣ λ ∣ > 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 finite‐dimensional Banach spaces, Kreiss' Matrix Theorem asserts that Kreiss boundedness is equivalent to T being power bounded. Thus, in the infinite‐dimensional 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. 2000 Mathematics Subject Classification 47B38, 47G10.