Sequences of the Projection-valued Measures and Functional Calculi
DOI:
https://doi.org/10.37134/jsml.vol11.2.5.2023Keywords:
functional calculus, projection-valued measure, projection operator, measurable calculusAbstract
Let a pair be functional calculus, where is a homomorphism from the space of the measurable functions on into the space of all linear bounded operators on a reflexive Banach space . We define a norm of functional calculus by , the convergence of the sequence of functional calculi is a convergence relative to this norm. We study the correspondence between sequences of spectral decompositions, well-bounded operators defined on the reflexive Banach space , and their correspondence with the theory of functional calculus for such operators. In this article, we establish that if a sequence of the projection-valued measures strongly converges to then the sequence of the functional calculi converges to the functional calculus Results of the article can be employed in the modern extensions of the quantum theory and theory of quantum information.
Downloads
References
Alfeus M., Schlogl E. (2018). On numerical methods for spread options. Quantitative Finance Research Centre. http://www.qfrc.uts.edu.au.
Arendt W, Vogt H, Voigt J. (2019). Form Methods for Evolution Equations. Lecture Notes of the 18th International Internet seminar, version: 6 March.
Batty C, Gomilko A, Tomilov Y. (2015). Product formulas in functional calculi for sectorial operators. Mathematische Zeitschrift, 279, 479-507.
Budde C, Landsman K. (2016). A bounded transform approach to self-adjoint operators: functional calculus and affiliated von Neumann algebras. Annals of Functional Analysis, 7, 411-420.
Colombo F, Gentili, G, Sabadini I, Struppa DC. (2010). Non-commutative functional calculus: Unbounded operators. Journal of Geometry and Physics, 60(2), 251-259.
Colombo F, Sabadini I. (2009). On Some Properties of the Quaternionic Functional Calculus. Journal of Geometric Analysis, 19, 601.
DeLaubenfels R. (1995). Automatic extensions of functional calculi. Studia Mathematica, 114, 237-259.
Dungey N. (2009). Asymptotic type for sectorial operators and an integral of fractional powers. Journal of Functional Analysis, 256, 1387-1407.
Eisner T, Farkas B, Haase M, Nagel R. (2015). Operator theoretic aspects of ergodic theory. Vol. 272 of Graduate Texts in Mathematics. Springer, Cham.
Florentino C, Nozad A, Zamora A. (2020). Serre polynomials of S Ln - and PG Ln-character varieties of free groups, Journal of Geometry and Physics, 161, 104008.
Haase M. (2014). Functional analysis. An Elementary Introduction. Vol. 156 of Graduate Studies in Mathematics. American Mathematical Society, Providence, RI.
Mhlanga FJ. (2015). Calculations of Greeks for jump-diffusion Processes. Mediterranean Journal of Mathematics, 12, 1141-1160.
Ringrose J. (1960). On well-bounded operators. Journal of the Australian Mathematical Society, 1(3), 334-343.
Schmudgen K. (2012). Unbounded Self-adjoint Operators on Hilbert Space. Vol. 265 of Graduate Texts in Mathematics. Springer, Dordrecht.
Smart DR. (1959). Eigenfunction expansions in Lp and C, Illinois Journal of Mathematics 3, 82-97.
Van Belle J, Vanduffelm S, Yao J. (2019). Closed-form approximations for spread options in Levy markets. Applied Stochastic Models in Business and Industry, 35, 732-746.
Yaremenko MI. (2012). Calderon-Zygmund operators and singular integrals. Applied Mathematics & Information Sciences, 15, 97-107.
Yilmaz B. (2018). Computation of option Greeks under hybrid stochastic volatility models via Malliavin calculus. Modern Stochastics: Theory and Applications, 5(2), 145-165.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2023 Mykola Ivanovich Yaremenko
This work is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License.
