Scalar-type spectral operators and holomorphic semigroups |
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Authors: | Ralph de Laubenfels |
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Institution: | 1. Department of Mathematics, The University of Tulsa, 74104, Tulsa, OK
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Abstract: | We show that a linear operator (possibly unbounded), A, on a reflexive Banach space, X, is a scalar-type spectral operator, with non-negative spectrum, if and only if the following conditions hold. - A generates a uniformly bounded holomorphic semigroup {e?zA}Re(z)≥0.
- If \(F_N (s) \equiv \int_{ - N}^N {\tfrac{{\sin (sr)}}{r}} e^{irA} dr\) , then {‖FN‖} N=1 ∞ is uniformly bounded on 0,∞) and, for all x in X, the sequence {FN(s)x} N=1 ∞ converges pointwise on 0, ∞) to a vector-valued function of bounded variation.
The projection-valued measure, E, for A, may be constructed from the holomorphic semigroup {e?zA}Re(z)≥0 generated by A, as follows. $$\frac{1}{2}(E\{ s\} )x + (E0,s)) x = \mathop {\lim }\limits_{N \to \infty } \int_{ - N}^N {\frac{{\sin (sr)}}{r}} e^{irA} x\frac{{dr}}{\pi }$$ for any x in X. |
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