Publication: Solution properties of some classes of operator equations in hilbert spaces
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Issued Date
2010-06-01
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ISSN
17937183
17935571
17935571
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2-s2.0-84857543343
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Mahidol University
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SCOPUS
Bibliographic Citation
Asian-European Journal of Mathematics. Vol.3, No.2 (2010), 263-273
Suggested Citation
Phakdi Charoensawan, Vu Quoc Phong, Nguyen Van Sanh Solution properties of some classes of operator equations in hilbert spaces. Asian-European Journal of Mathematics. Vol.3, No.2 (2010), 263-273. doi:10.1142/S1793557110000180 Retrieved from: https://repository.li.mahidol.ac.th/handle/123456789/29330
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Title
Solution properties of some classes of operator equations in hilbert spaces
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Abstract
We study properties of solutions of the operator equation ℒu = f, u, f ⋯ H,(*), where ℒ a closable linear operator on a Hilbert space H, such that there exists a self-adjoint operator D on H, with the resolution of identity E(·), which commutes with ℒ. We are interested in the question of regular admissibility of the subspace H(Λ): =E(Λ )H, i.e. when for every f ⋯ H(Λ) there exists a unique (mild) solution u in H (Λ) of this equation. We introduce the notion of equation spectrum Σ associated with Eq. (*), and prove that if Λ ⊂ is a compact subset such that Λ ∩ Σ = ∅, then H (Λ) is regularly admissible. If Λ ⊂ is an arbitrary Borel subset such that Λ ∩ Σ = ∅, then, in general, H(Λ) needs not be regularly admissible, but we derive necessary and sufficient conditions, in terms of some inequalities, for the regular admissibility of H(Λ). Our results are generalizations of the well-known spectral mapping theorem of Gearhart-Herbst-Howland-Prüss [4], [5], [6], [9], as well as of the recent results of Cioranescu-Lizama [3], Schüler [10] and Vu [11], [12]. © 2010 World Scientific Publishing Company.