Linear topological properties of the Lumer-Smirnov class of the polydisc

• Autorzy: Marek Nawrocki
• Języki publikacji: EN

Abstrakty

EN

Linear topological properties of the Lumer-Smirnov class $LN_∗(𝕌^n)$ of the unit polydisc $𝕌^n$ are studied. The topological dual and the Fréchet envelope are described. It is proved that $LN_∗(𝕌^n)$ has a weak basis but it is nonseparable in its original topology. Moreover, it is shown that the Orlicz-Pettis theorem fails for $LN_∗(𝕌^n)$.

Kategorie tematyczne

• 32A35: H p -spaces, Nevanlinna spaces
• 46J15: Banach algebras of differentiable or analytic functions, H p -spaces
• 46E10: Topological linear spaces of continuous, differentiable or analytic functions

• Wydawca: Institute of Mathematics Polish Academy of Sciences
• Czasopismo: Studia Mathematica
• Rocznik: 1992
• Tom: 102
• Numer: 1
• Strony: 87-102

Daty

wydano

1992

otrzymano

1991-08-02

Twórcy

autor

Marek Nawrocki

• Institute of Mathematics, A. Mickiewicz University, Matejki 48/49, 60-769 Poznań, Poland

Bibliografia

• [1] L. Drewnowski, Un théorème sur les opérateurs de $ℓ_∞(Γ)$, C. R. Acad. Sci. Paris Sér. A 281 (1975), 967-969.
• [2] L. Drewnowski, Topological vector groups and the Nevanlinna class, preprint.
• [3] E. Dubinsky, Basic sequences in stable finite type power series spaces, Studia Math. 68 (1980), 117-130.
• [4] P. L. Duren, Theory of $H^p$ Spaces, Academic Press, New York 1970.
• [5] N. J. Kalton, Subseries convergence in topological groups and vector measures, Israel J. Math. 10 (1971), 402-412.
• [6] N. J. Kalton, Basic sequences in F-spaces and their applications, Proc. Edinburgh Math. Soc. 19 (1974), 151-167.
• [7] N. J. Kalton, Quotients of F-spaces, Glasgow Math. J. 19 (1978), 103-108.
• [8] N. J. Kalton, The Orlicz-Pettis theorem, in: Contemp. Math. 2, Amer. Math. Soc., 1980, 91-100.
• [9] N. J. Kalton, N. T. Peck and J. W. Roberts, An F-Space Sampler, London Math. Soc. Lecture Note Ser. 89, Cambridge Univ. Press, 1984.
• [10] G. Lumer, Espaces de Hardy en plusieurs variables complexes, C. R. Acad. Sci. Paris 273 (1971), 151-154.
• [11] M. Nawrocki, On the Orlicz-Pettis property in nonlocally convex F-spaces, Proc. Amer. Math. Soc. 101 (1987), 492-496.
• [12] M. Nawrocki, Multipliers, linear functionals and the Fréchet envelope of the Smirnov class $N_∗(𝕌^n)$, Trans. Amer. Math. Soc. 322 (1990), 493-506.
• [13] M. Nawrocki, The Fréchet envelopes of vector-valued Smirnov classes, Studia Math. 94 (1989), 61-75.
• [14] M. Nawrocki, Linear functionals on the Smirnov class of the unit ball in $ℂ^n$, Ann. Acad. Sci. Fenn. 14 (1989), 369-379.
• [15] M. Nawrocki, The Orlicz-Pettis theorem fails for Lumer's Hardy spaces $(LH)^p(𝔹)$, Proc. Amer. Math. Soc. 109 (1990), 957-963.
• [16] S. Rolewicz, Metric Linear Spaces, PWN, Warszawa, and Reidel, Dordrecht 1984.
• [17] L. A. Rubel, Internal-external factorization in Lumer's Hardy spaces, Adv. in Math. 50 (1983), 1-26.
• [18] W. Rudin, Function Theory in Polydiscs, Benjamin, New York 1969.
• [19] W. Rudin, Lumer's Hardy spaces, Michigan Math. J. 24 (1977), 1-5.
• [20] W. Rudin, Function Theory in the Unit Ball of $ℂ^n$, Grundlehren Math. Wiss. 241, Springer, 1980.
• [21] J. H. Shapiro, Mackey topologies, reproducing kernels, and diagonal maps on Hardy and Bergman spaces, Duke Math. J. 43 (1976), 187-202.
• [22] J. H. Shapiro, Some F-spaces of harmonic functions for which the Orlicz-Pettis theorem fails, Proc. London Math. Soc. 50 (1985), 299-313.
• [23] J. H. Shapiro, Linear topological properties of the harmonic Hardy spaces $h^p$ for 0 < p < 1, Illinois J. Math. 29 (1985), 311-339.
• [24] J. H. Shapiro and A. L. Shields, Unusual topological properties of the Nevanlinna Class, Amer. J. Math. 97 (1975), 915-936.
• [25] N. Yanagihara, The containing Fréchet space for the class $N^+$, Duke Math. J. 40 (1973), 93-103.
• [26] N. Yanagihara, Multipliers and linear functionals for the class $N^+$, Trans. Amer. Math. Soc. 180 (1973), 449-461.