Imbedding of power series spaces and spaces of analytic functions. (English) Zbl 0724.46009
Let E denote a nuclear Fréchet space, which satisfies the conditions (DN) and (\(\Omega\)). It is proved that E contains a complemented copy of the power series space \(\Lambda_{\infty}(\epsilon)\) provided the diametral dimensions of E and \(\Lambda_{\infty}(\epsilon)\) are equal and \(\epsilon\) is stable. Several interesting applications of these results to spaces of analytic functions are given. For instance it is shown that the space \({\mathcal O}({\mathbb{C}}^ d)\) of entire functions is isomorphic to a complemented subspace of \({\mathcal O}(D)\), where D is a complete Reinhardt domain, if the characteristic function of D is almost everywhere infinite. In the one dimensional case, the spaces \({\mathcal O}(G)\) and \({\mathcal O}({\mathbb{C}})\) are isomorphic, if and only if their diametral dimensions are equal.
Reviewer: F.Haslinger (Wien)
MSC:
| 46A45 | Sequence spaces (including Köthe sequence spaces) |
| 46E10 | Topological linear spaces of continuous, differentiable or analytic functions |
| 46A11 | Spaces determined by compactness or summability properties (nuclear spaces, Schwartz spaces, Montel spaces, etc.) |
| 46A04 | Locally convex Fréchet spaces and (DF)-spaces |
| 32A07 | Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010) |
References:
| [1] | Aytuna, A.: Analitik fonksiyon uzaylarinin lineer topolojik yapilan üzerine. (Turkish) Habilitation thesis, 1981 |
| [2] | Aytuna, A.: On the linear topological structures of spaces of analytic functions. Doĝa Tr. J. Math.10, 46–49 (1986) · Zbl 0970.32500 |
| [3] | Aytuna, A., Terzioĝlu, T.: On certain subspaces of a nuclear power series space of finite type. Studia Math.69, 79–86 (1980) · Zbl 0467.46009 |
| [4] | Aytuna, A., Krone, J., Terzioĝlu, T.: Complemented infinite type power series subspaces of nuclear Fréchet spaces. Math. Ann.283, 193–202 (1989) · Zbl 0643.46001 · doi:10.1007/BF01446430 |
| [5] | Crone, L., Dubinsky, E., Robinson, W.B.: Regular bases in products of power series spaces. J. Func. Anal.24, 211–222 (1977) · Zbl 0345.46006 · doi:10.1016/0022-1236(77)90053-2 |
| [6] | Djakov, P.B.: On isomorphisms of some spaces of holomorphic functions (Russian). Func. anal. and appl. (Trans. of 6. Winter School in Progobych 1973); ed. B. Mityagin, 129–135, (1975) |
| [7] | Dubinsky, E.: The structure of nuclear Fréchet spaces. Lecture Notes in Math. 720 (1979) · Zbl 0403.46005 |
| [8] | Edwards, R.E.: Functional analysis, theory and applications. Holt, Rinehart and Winston Inc. 1965 · Zbl 0182.16101 |
| [9] | Fisher, S.D., Micchelli, C.A.: Then-th width of sets of analytic functions. Duke Math. J.47, 789–801 (1980) · Zbl 0451.30032 · doi:10.1215/S0012-7094-80-04746-8 |
| [10] | Fisher, S.D.: Function theory on planar domains: A second course in complex analysis. John Wiley and Sons, 1983 · Zbl 0511.30022 |
| [11] | Gamelin, T.W.: Uniform algebras. Prentice-Hall, 1969 · Zbl 0213.40401 |
| [12] | Gunning, R., Rossi, H.: Analytic functions of several complex variables. Englewood Cliffs, 1965 · Zbl 0141.08601 |
| [13] | Helms, L.L.: Introduction to potential theory. John Wiley and Sons, 1969 · Zbl 0188.17203 |
| [14] | Jarchow, H.: Locally convex spaces, Teubner-Verlag, 1981 · Zbl 0466.46001 |
| [15] | Köthe, G.: Topologische lineare Räume. Springer-Verlag, 1960 · Zbl 0093.11901 |
| [16] | Mityagin, B.S., Henkin, G.M.: Linear problems of complex analysis. Russian Math. Surveys26, 99–164 (1971) · Zbl 0245.46027 · doi:10.1070/RM1971v026n04ABEH003983 |
| [17] | Nurlu, Z.: Embedding {\(\Lambda\)}{\(\alpha\)}) into {\(\Lambda\)}1({\(\alpha\)}) and some consequences. Math. Balkanica. New Series1, 14–24 (1987) · Zbl 0645.46010 |
| [18] | Pietsch, A.: Nuclear locally convex spaces. Springer-Verlag 1972 · Zbl 0308.47024 |
| [19] | Rolewicz, S.: On spaces of holomorphic functions. Studia Math.21, 135–160 (1962) · Zbl 0123.30304 |
| [20] | Terzioĝlu, T.: Die diametrale Dimension von lokalkonvexen Räumen. Collect. Math.20, 49–99 (1969) · Zbl 0175.41602 |
| [21] | Terzioĝlu, T.: On the diametral dimension of some classes ofF-spaces. J. Karadeniz Univ. Ser. Math.-Physics18, 1–13 (1985) · Zbl 0609.46001 |
| [22] | Vogt, D.: Vektorwertige Distributionen als Randverteilungen holomorpher Funktionen. Manuscripta Math.17, 267–290 (1975) · Zbl 0349.46040 · doi:10.1007/BF01170313 |
| [23] | Vogt, D.: Charakterisierung der Unterräume vons. Math. Z.155, 109–117 (1977) · Zbl 0337.46015 · doi:10.1007/BF01214210 |
| [24] | Vogt, D.: Charakterisierung der Unterräume eines nuklearen stabilen Potenzreihenräumes von endlichem Typ. Studia Math.71, 251–270 (1982) · Zbl 0539.46009 |
| [25] | Vogt, D.: Eine Charakterisierung der Potenzreihenräume von endlichem Typ und ihre Folgerungen. Manuscripta Math.37, 269–301 (1982) · Zbl 0512.46003 · doi:10.1007/BF01166224 |
| [26] | Vogt, D.: Some results on continuous linear maps between Fréchet spaces. Functional Analysis: surveys and recent results III. K.D. Bierstedt and B. Fuchssteiner (eds.), 349–381, North-Holland 1984 |
| [27] | Vogt, D.: On the functorsExt 1 (E, F) for Fréchet spaces. Studia Math.85, 163–197 (1987) |
| [28] | Vogt, D., und Wagner, M.J.: Charakterisierung der Quotientenräume vons und eine Vermutung von Martineau. Studia Math.67, 225–240 (1979) · Zbl 0464.46010 |
| [29] | Vogt, D., und Wagner, M.J.: Charakterisierung der Unterräume und Quotientenräume der nuklearen stabilen Potenzreihenräume von unendlichem Typ. ibid70, 65–80 (1981) · Zbl 0402.46008 |
| [30] | Zaharjuta, V.P.: Spaces of functions of one variable, analytic in open sets and on compacta. Math. USSR Sbornik11, 75–88 (1970) · Zbl 0216.15602 · doi:10.1070/SM1970v011n01ABEH002063 |
| [31] | Zaharjuta, V.P.: Generalized Mityagin invariants and a continuum of pairwise nonisomorphic spaces of analytic functions. Func. Anal. and Appl.11, 182–188 (1978) · Zbl 0423.46015 · doi:10.1007/BF01079463 |
| [32] | Zaharjuta, V.P.: Isomorphism of spaces of analytic functions. Sov. Math. Dokl.22, 631–634 (1980) · Zbl 0467.32009 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
