Abstract
We introduce algebras which are inductive limits of Banach spaces and carry inequalities which are counterparts of the inequality for the norm in a Banach algebra, and show that the algebra of holomorphic functions on a compact set is such an algebra. We then define an associated Wiener algebra, and prove the corresponding version of the well-known Wiener theorem. Finally, we consider factorization theory in these algebras, and in particular, in the associated Wiener algebra.
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Alpay D., Salomon G.: Non-commutative stochastic distributions and applications to linear systems theory. Stoch. Process. Appl. 123, 2303–2322 (2013)
Alpay D., Salomon G.: Topological convolution algebras. J. Funct. Anal. 264(9), 2224–2244 (2013)
Bochner S., Phillips R.S.: Absolutely convergent Fourier expansions for non-commutative normed rings. Ann. Math. 43(2), 409–418 (1942)
Bourbaki, N.: Espaces vectoriels topologiques. Chapitres 1 à 5. Éléments de mathématique. [Elements of mathematics], new edn. Masson, Paris (1981)
Clancey K., Gohberg I.: Factorization of matrix functions and singular integral operators. In: Operator Theory: Advances and Applications, vol. 3. Birkhäuser, Basel (1981)
Dinculeanu, N.: Vector measures. In: Hochschulbücher für Mathematik, Band 64. VEB Deutscher Verlag der Wissenschaften, Berlin (1966)
Duren, P.L.: Theory of H p spaces. In: Pure and Applied Mathematics, vol. 38. Academic Press, New York (1970)
Gelfand I.M., Shilov G.E.: Generalized Functions, vol. 2. Academic Press, New York (1968)
Gohberg I., Goldberg S., Kaashoek M.A.: Classes of linear operators. Vol. II. In: Operator Theory: Advances and Applications, vol. 63. Birkhäuser, Basel (1993)
Holden, H., Øksendal, B., Ubøe, J., Zhang, T.: Stochastic partial differential equations. In: Probability and its Applications. Birkhäuser Boston Inc., Boston (1996)
Leonard I.E.: Banach sequence spaces. J. Math. Anal. Appl. 54(1), 245–265 (1976)
Morimoto, M.: An Introduction to Sato’s Hyperfunctions, 3rd edn. Wolters-Noordhoff Publishing, Groningen (1972). [Translated and revised from the 1976 Japanese original by the author. Translations of Mathematical Monographs, vol. 129. American Mathematical Society, Providence (1993)]
Naĭmark, M.A.: Normed Algebras, 3rd edn. Wolters-Noordhoff Publishing, Groningen (1972). (Translated from the second Russian edition by Leo F. Boron, Wolters-Noordhoff Series of Monographs and Textbooks on Pure and Applied Mathematics)
Schaefer H.H.: Topological Vector Spaces. Springer, New York (1986)
Wiener N.: Tauberian theorems. Ann. Math. 33, 1–100 (1932)
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D. Alpay thanks the Earl Katz family for endowing the chair which supported his research, and the Binational Science Foundation Grant No. 2010117.
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Alpay, D., Salomon, G. On Algebras Which are Inductive Limits of Banach Spaces. Integr. Equ. Oper. Theory 83, 211–229 (2015). https://doi.org/10.1007/s00020-015-2220-y
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DOI: https://doi.org/10.1007/s00020-015-2220-y