Displaying similar documents to “A resolvent condition implying power boundedness”

On the growth of the resolvent operators for power bounded operators

Olavi Nevanlinna (1997)

Banach Center Publications

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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...

Vasilescu-Martinelli formula for operators in Banach spaces

V. Kordula, V. Müller (1995)

Studia Mathematica

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We prove a formula for the Taylor functional calculus for functions analytic in a neighbourhood of the splitting spectrum of an n-tuple of commuting Banach space operators. This generalizes the formula of Vasilescu for Hilbert space operators and is closely related to a recent result of D. W. Albrecht.

Characterizing spectra of closed operators through existence of slowly growing solutions of their Cauchy problems

Sen Huang (1995)

Studia Mathematica

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Let A be a closed linear operator in a Banach space E. In the study of the nth-order abstract Cauchy problem u ( n ) ( t ) = A u ( t ) , t ∈ ℝ, one is led to considering the linear Volterra equation (AVE) u ( t ) = p ( t ) + A ʃ 0 t a ( t - s ) u ( s ) d s , t ∈ ℝ, where a ( · ) L l o c 1 ( ) and p(·) is a vector-valued polynomial of the form p ( t ) = j = 0 n 1 / ( j ! ) x j t j for some elements x j E . We describe the spectral properties of the operator A through the existence of slowly growing solutions of the (AVE). The main tool is the notion of Carleman spectrum of a vector-valued function. Moreover, an extension...

Compact AC-operators

Ian Doust, Byron Walden (1996)

Studia Mathematica

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We prove that compact AC-operators have a representation as a combination of disjoint projections which mirrors that for compact normal operators. We also show that unlike arbitrary AC-operators, compact AC-operators admit a unique splitting into real and imaginary parts, and that these parts must necessarily be compact.

Some spectral inequalities involving generalized scalar operators

B. Aupetit, D. Drissi (1994)

Studia Mathematica

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In 1971, Allan Sinclair proved that for a hermitian element h of a Banach algebra and λ complex we have ∥λ + h∥ = r(λ + h), where r denotes the spectral radius. Using Levin's subordination theory for entire functions of exponential type, we extend this result locally to a much larger class of generalized spectral operators. This fundamental result improves many earlier results due to Gelfand, Hille, Colojoară-Foiaş, Vidav, Dowson, Dowson-Gillespie-Spain, Crabb-Spain, I. & V. Istrăţescu,...