Range transformations on a Banach function algebra
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- by Osamu Hatori
- Trans. Amer. Math. Soc. 297 (1986), 629-643
- DOI: https://doi.org/10.1090/S0002-9947-1986-0854089-3
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Abstract:
We study the range transformations $\operatorname {Op} ({A_{D,}}\operatorname {Re} B)$ and $\operatorname {Op} ({A_D},B)$ for Banach function algebras $A$ and $B$. As a special instance, the harmonicity of functions in $\operatorname {Op} ({A_D},\operatorname {Re} A)$ for a nontrivial function algebra $A$ is established and is compared with previous investigations of $\operatorname {Op} ({A_D},A)$ and $\operatorname {Op} ({(\operatorname {Re} A)_I},(\operatorname {Re} A))$ for an interval $I$. In $\S 2$ we present some results on $\operatorname {Op} ({A_D},B)$ and use them to show that functions in ${\operatorname {Op} ^C}({A_D},B)$ are analytic for certain Banach function algebras.References
- J. M. Bachar, Jr., Composition mappings between function spaces, Ph.D. Thesis, UCLA, 1970.
- John M. Bachar Jr., Some results on range transformations between function spaces, Proceedings of the conference on Banach algebras and several complex variables (New Haven, Conn., 1983) Contemp. Math., vol. 32, Amer. Math. Soc., Providence, RI, 1984, pp. 35–62. MR 769496, DOI 10.1090/conm/032/769496 W. G. Bade and P. C. Curtis, Jr., Banach algebras on $F$-spaces, Function Algebras, Proc. Internat. Sympos. on Function Algebras, Tulane University, (F. T. Birtel, ed.), Scott, Foresman & Co., Glenview, Ill., 1966, pp. 90-92.
- B. T. Batikjan and E. A. Gorin, Ultraseparating algebras of continuous functions, Vestnik Moskov. Univ. Ser. I Mat. Meh. 31 (1976), no. 2, 15–20 (Russian, with English summary). MR 0412812
- A. Bernard, Espace des parties réelles des éléments d’une algèbre de Banach de fonctions, J. Functional Analysis 10 (1972), 387–409 (French, with English summary). MR 0343037, DOI 10.1016/0022-1236(72)90036-5
- Alain Bernard and Alain Dufresnoy, Calcul symbolique sur la frontière de Šilov de certaines algèbres de fonctions holomorphes, Ann. Inst. Fourier (Grenoble) 25 (1975), no. 2, xi, 33–43 (French). MR 388108
- Frank T. Birtel, Products of maximal function algebras, Function Algebras (Proc. Internat. Sympos. on Function Algebras, Tulane Univ., 1965) Scott-Foresman, Chicago, Ill., 1966, pp. 65–69. MR 0198277
- Errett Bishop, A generalization of the Stone-Weierstrass theorem, Pacific J. Math. 11 (1961), 777–783. MR 133676
- R. B. Burckel, Characterizations of $C(X)$ among its subalgebras, Lecture Notes in Pure and Applied Mathematics, Vol. 6, Marcel Dekker, Inc., New York, 1972. MR 0442687
- Osamu Hatori, Functions which operate on the real part of a function algebra, Proc. Amer. Math. Soc. 83 (1981), no. 3, 565–568. MR 627693, DOI 10.1090/S0002-9939-1981-0627693-6
- Osamu Hatori, Functions which operate by composition on the real part of a Banach function algebra, Tokyo J. Math. 6 (1983), no. 2, 423–429. MR 732095, DOI 10.3836/tjm/1270213882
- Osamu Hatori, Functional calculus for certain Banach function algebras, J. Math. Soc. Japan 38 (1986), no. 1, 103–112. MR 816226, DOI 10.2969/jmsj/03810103 —, Remarks on the functional calculus on a Banach function algebra (preprint).
- Karel de Leeuw and Yitzhak Katznelson, Functions that operate on non-self-adjoint algebras, J. Analyse Math. 11 (1963), 207–219. MR 158282, DOI 10.1007/BF02789985
- S. J. Sidney, Functions which operate on the real part of a uniform algebra, Pacific J. Math. 80 (1979), no. 1, 265–272. MR 534716 W. Spraglin, Partial interpolation and the operational calculus in Banach algebras, Ph. D. Thesis, UCLA, 1966.
- John Wermer, The space of real parts of a function algebra, Pacific J. Math. 13 (1963), 1423–1426. MR 156223
Bibliographic Information
- © Copyright 1986 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 297 (1986), 629-643
- MSC: Primary 46J10; Secondary 46H30
- DOI: https://doi.org/10.1090/S0002-9947-1986-0854089-3
- MathSciNet review: 854089