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Weighted $$(LB)$$-spaces of holomorphic functions: $$\nu H(G)=\nu_{0}H(G)$$ and completeness of $$\nu_{0}H(G)$$. (English) Zbl 1119.46028
In their interesting paper, the authors show that algebraic equalities between weighted spaces of holomorphic functions may have strong topological implications. Let $$\mathcal{V}:= \{v_n\mid n\in \mathbb{N}\}$$ be an inductive system of weights on an open set $$G\subset \mathbb{C}^N$$. Then $$\mathcal{V}$$ defines inductive limits $$\mathcal{V} H(G)$$ ($$\mathcal{V}_0H(G)$$) of holomorphic functions by means of $$O$$-growth conditions ($$o$$-growth conditions, respectively). Let $$H \overline{V}(G)$$ ($$H \overline{V}_0(G)$$) be the projective hulls of $$\mathcal{V} H(G)$$ ($$\mathcal{V}_0H(G)$$, respectively). Assume that, for each $$n\in \mathbb{N}$$, every discrete sequence in $$G$$ contains a subsequence which is interpolating in $$Hv_n(G)$$. Then the algebraic equality $$\mathcal{V} H(G)= \mathcal{V}_0H(G)$$ implies that $$\mathcal{V} H(G)$$ and $$\mathcal{V}_0H(G)$$ are $$(DFS)$$-spaces. Moreover, the algebraic equality $$H \overline{V}(G)=H \overline{V}_0(G)$$ implies that $$H \overline{V}(G)$$ is semi-Montel.
The interpolating assumption is discussed in detail, e.g., it is always satisfied if $$G\subset \mathbb{C}$$ is connected and no connected component of the complement of $$G$$ in the Riemann sphere consists of one point or if $$G\subset \mathbb{C}^N$$ is absolutely convex and bounded. If the biduality conditions are satisfied, i.e., if for each $$n\in \mathbb{N}$$ the closed unit ball $$B_n$$ of $$Hv_n(G)$$ is contained in the compact open closure of the unit ball $$C_n$$ of $$H(v_n)_0(G)$$, then the algebraic equality $$H \overline{V}(G)=H \overline{V}_0(G)$$ is equivalent to $$H \overline{V}_0(G)$$ being semireflexive. Finally, let $$\mathbb{D}\subset \mathbb{C}$$ be the unit disc and assume that $$\mathcal{V}_0 H(\mathbb{D})= H \overline{V}_0(\mathbb{D})$$ algebraically and topologically. Then the inductive limit $$\mathcal{V}_0 H(\mathbb{D})$$ is boundedly retractive.

##### MSC:
 46E10 Topological linear spaces of continuous, differentiable or analytic functions 30H05 Spaces of bounded analytic functions of one complex variable 46A13 Spaces defined by inductive or projective limits (LB, LF, etc.) 46M40 Inductive and projective limits in functional analysis
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##### References:
 [1] Bierstedt, K.D., An introduction to locally convex inductive limits, (), 35-133 · Zbl 0786.46001 [2] Bierstedt, K.D., A survey of some results and open problems in weighted inductive limits and projective description for spaces of holomorphic functions, Bull. soc. roy. sci. liège, 70, 167-182, (2001) · Zbl 1030.46002 [3] Bierstedt, K.D.; Bonet, J., Stefan Heinrich’s density condition for Fréchet spaces and the characterization of the distinguished Köthe echelon spaces, Math. nachr., 135, 149-180, (1988) · Zbl 0688.46001 [4] Bierstedt, K.D.; Bonet, J., Biduality in Fréchet and (LB)-spaces, (), 113-133 · Zbl 0804.46007 [5] Bierstedt, K.D.; Bonet, J., Projective description of weighted (LF)-spaces of holomorphic functions on the disc, Proc. Edinburgh math. soc., 46, 435-450, (2003) · Zbl 1060.46018 [6] Bierstedt, K.D.; Meise, R., Distinguished echelon spaces and the projective description of weighted inductive limits of type $$\mathcal{V}_d \mathcal{C}(X)$$, (), 169-226 [7] Bierstedt, K.D.; Meise, R., Weighted inductive limits and their projective descriptions, DOğA tr. J. math., 10, 1, 54-82, (1986) · Zbl 0970.46541 [8] Bierstedt, K.D.; Summers, W.H., Biduals of weighted Banach spaces of analytic functions, J. austral. math. soc. ser. A, 54, 70-79, (1993) · Zbl 0801.46021 [9] Bierstedt, K.D.; Bonet, J.; Galbis, A., Weighted spaces of holomorphic functions on balanced domains, Michigan math. J., 40, 271-297, (1993) · Zbl 0803.46023 [10] Bierstedt, K.D.; Bonet, J.; Taskinen, J., Associated weights and spaces of holomorphic functions, Studia math., 127, 137-168, (1998) · Zbl 0934.46027 [11] Bierstedt, K.D.; Meise, R.; Summers, W.H., A projective description of weighted inductive limits, Trans. amer. math. soc., 272, 107-160, (1982) · Zbl 0599.46026 [12] Boiti, C.; Nacinovich, M., The overdetermined Cauchy problem, Ann. inst. Fourier, 47, 155-199, (1997) · Zbl 0865.35091 [13] Bonet, J.; Taskinen, J., The subspace problem for weighted inductive limits of spaces of holomorphic functions, Michigan math. J., 42, 259-268, (1995) · Zbl 0841.46014 [14] Bonet, J.; Wolf, E., A note on weighted Banach spaces of holomorphic functions, Arch. math. (basel), 81, 650-654, (2003) · Zbl 1047.46018 [15] Bonet, J.; Domański, P.; Lindström, M., Essential norm and weak compactness of composition operators on weighted Banach spaces of analytic functions, Canad. math. bull., 42, 139-148, (1999) · Zbl 0939.47020 [16] Bonet, J.; Engliš, M.; Taskinen, J., Weighted $$L^\infty$$-estimates for Bergman projections, Studia math., 171, 67-92, (2005) · Zbl 1082.47027 [17] Bonet, J.; Friz, M.; Jordá, E., Composition operators between weighted inductive limits of spaces of holomorphic functions, Publ. math. debrecen, 67, 333-348, (2005) · Zbl 1097.46013 [18] Bonet, J.; Meise, R.; Melikhov, S.N., Projective representations of spaces of quasianalytic functions, Studia math., 164, 91-102, (2004) · Zbl 1062.46022 [19] Floret, K., Folgenretraktive sequenzen lokalkonvexer Räume, J. reine angew. math., 259, 65-85, (1973) · Zbl 0251.46003 [20] S. Holtmanns, Operator representation and biduals of weighted function spaces, dissertation, Paderborn, 2000 · Zbl 1011.46035 [21] Jarchow, H., Locally convex spaces, Math. leitfäden, (1981), Teubner Stuttgart · Zbl 0466.46001 [22] Jasiczak, M., On locally convex extension of $$H^\infty$$ in the unit ball and continuity of the Bergman projection, Studia math., 156, 261-275, (2003) · Zbl 1026.32007 [23] Marco, N.; Massaneda, X.; Ortega-Cerdà, J., Interpolating and sampling sequences for entire functions, Geom. funct. anal., 13, 862-914, (2003) · Zbl 1097.30041 [24] Mattila, P.; Saksman, E.; Taskinen, J., Weighted spaces of harmonic and holomorphic functions: sequence space representations and projective descriptions, Proc. Edinburgh math. soc., 40, 41-62, (1997) · Zbl 0898.46022 [25] Meise, R.; Vogt, D., Introduction to functional analysis, Grad. texts in math., vol. 2, (1997), Clarendon [26] Robertson, A.P.; Robertson, W., Topological vector spaces, Cambridge tracts in math. math. phys., vol. 53, (1973), Cambridge Univ. Press Cambridge · Zbl 0251.46002 [27] Summers, W.H., Dual spaces of weighted spaces, Trans. amer. math. soc., 151, 323-333, (1970) · Zbl 0203.12401 [28] Taskinen, J., On the continuity of the Bergman and szegö projections, Houston J. math., 30, 171-190, (2004) · Zbl 1064.46019 [29] Wojtaszczyk, P., Banach spaces for analysts, (1991), Cambridge Univ. Press Cambridge · Zbl 0724.46012
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