Kinoshita, Takehiko; Watanabe, Yoshitaka; Nakao, Mitsuhiro T. Some lower bound estimates for resolvents of a compact operator on an infinite-dimensional Hilbert space. (English) Zbl 1498.47014 J. Comput. Appl. Math. 369, Article ID 112561, 8 p. (2020). Authors’ abstract: This paper presents some lower estimates for resolvents of a compact operator \(A\) on an infinite-dimensional Hilbert space. Usually, upper estimates for resolvents are known under suitable conditions but lower estimates are not known. These estimates enable us to calculate suitability results for approximation of a resolvent. Reviewer: Costică Moroşanu (Iaşi) MSC: 47A10 Spectrum, resolvent 47B02 Operators on Hilbert spaces (general) 47B07 Linear operators defined by compactness properties 35B45 A priori estimates in context of PDEs 35A35 Theoretical approximation in context of PDEs 47G10 Integral operators 65N30 Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs 35J25 Boundary value problems for second-order elliptic equations 35J61 Semilinear elliptic equations Keywords:Hilbert space; compact operator; resolvent; lower bound estimate; lower estimates for resolvents; upper estimates for resolvents Software:INTLAB PDFBibTeX XMLCite \textit{T. Kinoshita} et al., J. Comput. Appl. Math. 369, Article ID 112561, 8 p. (2020; Zbl 1498.47014) Full Text: DOI References: [1] Nakao, M. T.; Hashimoto, K.; Watanabe, Y., A numerical method to verify the invertibility of linear elliptic operators with applications to nonlinear problems, Computing, 75, 1-14 (2005) · Zbl 1151.35337 [2] Kinoshita, T.; Watanabe, Y.; Nakao, M. T., Some remarks on the rigorous estimation of inverse linear elliptic operators, LNCS, 9553, 225-235 (2016) · Zbl 1354.65232 [3] Nakao, M. T.; Watanabe, Y.; Kinoshita, T.; Kimura, T.; Yamamoto, N., Some considerations of the invertibility verifications for linear elliptic operators, Japan J. Ind. Appl. Math., 32, 19-32 (2015) · Zbl 1320.35164 [4] Oishi, S., Numerical verification of existence and inclusion of solutions for nonlinear operator equations, J. Comput. Appl. Math., 60, 1-2, 171-185 (1995) · Zbl 0871.65045 [5] Plum, M., Eigenvalue inclusions for second-order ordinary differential operators by a numerical homotopy method, J. Appl. Math. Phys. (ZAMP), 41, 205-226 (1990) · Zbl 0707.65060 [6] Watanabe, Y.; Kinoshita, T.; Nakao, M. T., A posteriori estimates of inverse operators for boundary value problems in linear elliptic partial differential equations, Math. Comp., 82, 1543-1557 (2013) · Zbl 1280.65119 [7] Rump, S. M., INTLAB - INTerval LABoratory, (Csendes, T., Developments, Reliable Computing (1999)), 77-104, http://www.ti3.tu-harburg.de/rump/ · Zbl 0949.65046 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.