##
**On a class of self-adjoint compact operators in Hilbert spaces and their relations with their finite-range truncations.**
*(English)*
Zbl 1437.47005

Summary: This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

### MSC:

47A58 | Linear operator approximation theory |

47B02 | Operators on Hilbert spaces (general) |

47J25 | Iterative procedures involving nonlinear operators |

47N70 | Applications of operator theory in systems, signals, circuits, and control theory |

### Keywords:

self-adjoint compact operators in Hilbert spaces; finite-range truncations; worst-case norm errors; iterated computations; boundedness properties; fixed points; iterated sequences; separable Hilbert spaces; numerable orthonormal bases
PDF
BibTeX
XML
Cite

\textit{M. De la Sen}, Abstr. Appl. Anal. 2013, Article ID 890657, 14 p. (2013; Zbl 1437.47005)

Full Text:
DOI

### References:

[1] | Franks, L. E., Signal Theory (1969), Englewood Cliffs, NJ, USA: Prentice Hall, Englewood Cliffs, NJ, USA · Zbl 0194.50303 |

[2] | Berezanskii, J. M., Expansion of eigenvectors of self-adjoint operators, Translation of Mathematical Monographs, 17 (1968) |

[3] | De la Sen, M., Some fixed point properties of self-maps constructed by switched sets of primary self-maps on normed linear spaces, Fixed Point Theory and Applications, 2010 (2010) · Zbl 1201.54030 |

[4] | De la Sen, M., Fixed and best proximity points of cyclic jointly accretive and contractive self-mappings, Journal of Applied Mathematics, 2012 (2012) · Zbl 1417.54023 |

[5] | De la Sen, M., About robust stability of Caputo linear fractional dynamic systems with time delays through fixed point theory, Fixed Point Theory and Applications, 2011 (2011) · Zbl 1219.34102 |

[6] | De la Sen, M., Total stability properties based on fixed point theory for a class of hybrid dynamic systems, Fixed Point Theory and Applications, 2009 (2009) · Zbl 1189.34023 |

[7] | Anh, P. N., A hybrid extragradient method for pseudomonotone equilibrium problems and fixed point problems, Bulletin of the Malaysian Mathematical Sciences Society, 36, 1, 107-116 (2013) · Zbl 1263.65066 |

[8] | Künzi, H.-P.; Olela Otafudu, O., q-hyperconvexity in quasipseudometric spaces and fixed point theorems, Journal of Function Spaces and Applications, 2012 (2012) · Zbl 1257.54040 |

[9] | Cojocaru, M. G.; Pia, S., Nonpivot and implicit projected dynamical systems on Hilbert spaces, Journal of Function Spaces and Applications, 2012 (2012) · Zbl 1284.37014 |

[10] | Chen, C. M.; Chang, T. H.; Juang, K. S., Common fixed point theorems for the stronger Meir-Keeler cone-type function in cone ball-metric spaces, Applied Mathematics Letters, 25, 4, 692-697 (2012) · Zbl 1244.54087 |

[11] | Chen, C.-M., Common fixed-point theorems in complete generalized metric spaces, Journal of Applied Mathematics, 2012 (2012) · Zbl 1244.54088 |

[12] | Kumam, P.; Katchang, P., Viscosity approximations with weak contraction for finding a common solution of fixed points and a general system of variational inequalities for two accretive operators, Journal of Computational Analysis and Applications, 14, 7, 1269-1287 (2012) · Zbl 1272.47080 |

[13] | Mursaleen, M.; Mohiuddine, S. A., Some new double sequence spaces of invariant means, Glasnik Matematički, 45, 65, 139-153 (2010) · Zbl 1195.46005 |

[14] | Duman, O.; Khan, M. K.; Orhan, C., A-statistical convergence of approximating operators, Mathematical Inequalities & Applications, 6, 4, 689-699 (2003) · Zbl 1086.41008 |

[15] | Gadjiev, A. D.; Orhan, C., Some approximation theorems via statistical convergence, The Rocky Mountain Journal of Mathematics, 32, 1, 129-138 (2002) · Zbl 1039.41018 |

[16] | Mohiuddine, S. A.; Alotaibi, A., Some spaces of double sequences obtained through invariant mean and related concepts, Abstract and Applied Analysis, 2013 (2013) · Zbl 1277.46003 |

[17] | Belen, C.; Mohiuddine, S. A., Generalized weighted statistical convergence and application, Applied Mathematics and Computation, 219, 18, 9821-9826 (2013) · Zbl 1308.40003 |

[18] | Ashyralyev, A.; Koksal, M. E., Stability of a second order of accuracy difference scheme for hyperbolic equation in a Hilbert space, Discrete Dynamics in Nature and Society, 2007 (2007) · Zbl 1156.65079 |

[19] | de la Sen, M., The reachability and observability of hybrid multirate sampling linear systems, Computers & Mathematics with Applications, 31, 1, 109-122 (1996) · Zbl 0854.93084 |

[20] | Ashyralyev, A.; Sharifov, Y. A., Optimal control problem for impulsive systems with integral boundary conditions, Proceedings of the 1st International Conference on Analysis and Applied Mathematics (ICAAM ’12) · Zbl 1286.49035 |

[21] | de la Sen, M., On some structures of stabilizing control laws for linear and time-invariant systems with bounded point delays and unmeasurable states, International Journal of Control, 59, 2, 529-541 (1994) · Zbl 0799.93048 |

[22] | Ortega, J. M., Numerical Analysis, xiii+201 (1972), New York, NY, USA: Academic Press, New York, NY, USA |

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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.