4,196 research outputs found
Quantified asymptotic behaviour of Banach space operators and applications to iterative projection methods
We present an extension of our earlier work [Ritt operators and convergence
in the method of alternating projections, J. Approx. Theory, 205:133-148, 2016]
by proving a general asymptotic result for orbits of an operator acting on a
reflexive Banach space. This result is obtained under a condition involving the
growth of the resolvent, and we also discuss conditions involving the location
and the geometry of the numerical range of the operator. We then apply the
general results to some classes of iterative projection methods in
approximation theory, such as the Douglas-Rachford splitting method and, under
suitable geometric conditions either on the ambient Banach space or on the
projection operators, the method of alternating projections
Compressions of Resolvents and Maximal Radius of Regularity
Suppose that is left-invertible in for all , where is an open subset of the complex plane. Then an
operator-valued function is a left resolvent of in if
and only if has an extension , the resolvent of which is a
dilation of of a particular form. Generalized resolvents exist on
every open set , with included in the regular domain of . This
implies a formula for the maximal radius of regularity of in terms of the
spectral radius of its generalized inverses. A solution to an open problem
raised by J. Zem\'anek is obtained.Comment: 15 pages, to appear in Trans. Amer. Math. So
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