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Resolvent conditions and powers of operators

100%
Nevanlinna O.
Studia Mathematica
|
2001
|
tom 145
|
nr 2
113-134
EN
We discuss the relation between the growth of the resolvent near the unit circle and bounds for the powers of the operator. Resolvent conditions like those of Ritt and Kreiss are combined with growth conditions measuring the resolvent as a meromorphic function.
2
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On the growth of the resolvent operators for power bounded operators

100%
Nevanlinna O.
Banach Center Publications
|
1997
|
tom 38
|
nr 1
247-264
EN
Outline. In this paper I discuss some quantitative aspects related to power bounded operators T and to the decay of $T^{n}(T-1)$. For background I refer to two recent surveys J. Zemánek [1994], C. J. K. Batty [1994]. Here I try to complement these two surveys in two different directions. First, if the decay of $T^{n}(T-1)$ is as fast as O(1/n) then quite strong conclusions can be made. The situation can be thought of as a discrete version of analytic semigroups; I try to motivate this in Section 1 by demonstrating the similarity and lack of it between power boundedness of T and uniform boundedness of $e^{t(cT-1)}$ where c is a constant of modulus 1 and t > 0. Section 2 then contains the main result in this direction. I became interested in studying the quantitative aspects of the decay of $T^{n}(T-1)$ since it can be used as a simple model for what happens in the early phase of an iterative method (O. Nevanlinna [1993]). Secondly, the so called Kreiss matrix theorem relates bounds for the powers to bounds for the resolvent. The estimate is proportional to the dimension of the space and thus has as such no generalization to operators. However, qualitatively such a result holds in Banach spaces e.g. for Riesz operators: if the resolvent satisfies the resolvent condition, then the operator is power bounded operator (but without an estimate). I introduce in Section 3 a growth function for bounded operators. This allows one to obtain a result of the form: if the resolvent condition holds and if the growth function is finite at 1, then the powers are bounded and can be estimated. In Section 4 in addition to the Kreiss matrix theorem, two other applications of the growth function are given.
3
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Real linear matrix analysis

63%
Huhtanen M., Nevanlinna O.
Banach Center Publications
|
2007
|
tom 75
|
nr 1
171-189
4
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Approximating real linear operators

63%
Huhtanen M., Nevanlinna O.
Studia Mathematica
|
2007
|
tom 179
|
nr 1
7-25
EN
A framework to extend the singular value decomposition of a matrix to a real linear operator $ℳ : ℂⁿ → ℂ^{p}$ is suggested. To this end real linear operators called operets are introduced, to have an appropriate generalization of rank-one matrices. Then, adopting the interpretation of the singular value decomposition of a matrix as providing its nearest small rank approximations, ℳ is approximated with a sum of operets.
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