Abstract
The aim of this paper is to consider operator-valued functions that can be approximated in the strong and weak operator topology by simple functions. We relate these notions with the classical formulations of measurability and provide conditions for their coincidence. A number of examples and counterexamples are exhibited.
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References
Ahmed N.: A note on Radon–Nikodym theorem for operator valued measures and its applications. Commun. Korean Math. Soc. 28(2), 285–295 (2013)
Berkson E.: Marcinkiewicz multipliers of higher variation and summability of operator-valued Fourier series. Stud. Math. 222(2), 123–155 (2014)
Bahrami F., Bayati Eshkaftaki A., Manjegani S.M.: Operator-valued Bochner integrable functions and Jensen’s inequality. Georgian Math. J. 20(4), 625–640 (2013)
Bruzual R., Dominguez M.: Representation of operator valued measurable indefinite functions. Acta Cient. Venezolana. 52(3), 223–225 (2001)
Blasco, O., van Neerven, J.: Spaces of operator-valued functions measurable with respect to the strong operator topology, Vector measures, integration and related topics. Oper. Theory Adv. Appl. Birkhuser. 201, 65–78 (2010)
Badrikian A., Johnson G.W., Yoo I.: The composition of operator-valued measurable functions is measurable. Proc. Am. Math. Soc. 123(6), 1815–1820 (1995)
Diestel, J., Uhl, J.J.: Vector Measures. American Mathematical Society, Providence (1977)
Dunford N.: On one parameter groups of linear transformations. Ann. Math. 39(2), 569–573 (1938)
Dunford, N., Schwartz, J.: Linear Operators, Part I: General Theory. Interscience, New York (1958)
Edwards, R.E.: Functional Analysis. Theory and Applications. Dover Pub. Inc, New York (1995)
Fourie, J.H.: On operator-valued measurable functions. Vector measures, integration and related topics. Oper. Theory Adv. Appl., 201, Birkhuser, Basel, pp. 205–214 (2010)
Halmos, P.R.: Measure Theory. Springer, New York (1974)
Hille, E., Phillips, R.S.: Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc, Providence, RI (1957)
Johnson G.W.: The product of strong operator measurable functions is strong operator measurable. Proc. Am. Math. Soc. 117, 1097–1104 (1993)
Michael, E.: On a theorem of Rudin and Klee. Proc. Am. Math. Soc. 12, 921 (1961)
Pettis B.J.: On integration in vector spaces. Am. Math. Soc. 44, 277–304 (1938)
Reed, M., Simon, B.: Methods of Modern Mathematical Physics, Vol. I. Functional Analysis, Revised Enlarged ed., Academic Press, New York (1980)
Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)
Rudin M.E., Klee V.L. Jr: A note on certain function spaces. Arch. Math. (Basel) 7, 469–470 (1957)
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Blasco, O., García-Bayona, I. Remarks on Measurability of Operator-Valued Functions. Mediterr. J. Math. 13, 5147–5162 (2016). https://doi.org/10.1007/s00009-016-0798-1
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DOI: https://doi.org/10.1007/s00009-016-0798-1