MINISTRY OF EDUCATION AND TRAINING QUY NHON UNIVERSITY DUONG THANH VY CONVERGENCE OF TAUBERIAN TYPE FOR HOLOMORPHIC FUNCTIONS AND HOLOMORPHIC MAPPINGS Speciality: Mathematical Analysis Code: 9460102 SUMMARY OF DOCTORAL THESIS IN MATHEMATICS BINH DINH - 2019 This thesis has been completed at: Quy Nhon University Scienti c Advisor: Assoc Prof Dr Thai Thuan Quang Referee 1: Prof Dr Dang Duc Trong Referee 2: Prof Dr Sci Nguyen Quang Dieu Referee 3: Dr Dao Van Duong The thesis shall be defended before the University level Thesis Assessment Council at Quy Nhon University at : : : : : : The thesis can be found in: - The National Library of Viet Nam - Learning Resources Center Quy Nhon University DECLARATION OF AUTHORSHIP This thesis has been completed at Quy Nhon University, under the supervison of Assoc Prof Dr Thai Thuan Quang I hereby assure that this research project is mine All results have been formally approved by co-authors and have not been released by anyone else before Author Duong Thanh Vy ACKNOWLEDGMENTS First and foremost, I am extremely grateful to my thesis supervisor Assoc Prof Thai Thuan Quang for his skilled guidance, valuable suggestions and fruitful discussions I would like to express my deepest gratitude to my supervisor and his family I would like to deeply thank Prof Nguyen Van Khue, Prof Le Mau Hai (Hanoi National University of Education) and Prof Sean Dineen (Dublin University, Ireland) for their valuable comments, fruitful discussions to complete some results in Chapters and of this thesis I would like to deeply thank the Department of Mathematics of Quy Nhon University This is where I have spent a lot of years studying and teaching mathematics, have bene ted from the lectures and have received a lot of support and encouragement I would like to take advantage of this occasion to express my gratitude to all of the lecturers of the Department of Mathematics I would like to give my special thanks to the Board of Management of Quy Nhon University and Postgraduate Training O ce for their help and administrative assistance My sincerest thanks go to Dr Le Quang Thuan, Dr Lam Thi Thanh Tam, and Assoc Prof Dr Luong Dang Ky for their valuable comments, fruitful discussions and constant encouragement Last but not least, I want to thank my whole family and friends for their endless love, their enthusiastic support and their patience during these years and this thesis is dedicated to them Author Duong Thanh Vy LIST OF SYMBOLS AvpDq : Subspace of HvpDq with compact closed unit ball AvpD; F q AG;vpDq : : AG;vpD; F q acx pDq BpEq cspF q EB E1 E1 : : : : : : : bor for the compact-open topology tf : D Ñ F : u f P AvpDq; @u P F 1u Subspace of HG;vpDq with compact closed unit ball for the compact-open topology tf : D Ñ F : u f P AG;vpDq; @u P F 1u Closed, balanced, convex hull D Family of convex, balanced, closed, bounded subsets in E Set of continuous semi-norms on F Linear hull of B Dual space of locally convex space E Space E1 equipped with the bornological topology associated with the strong topology Space of all holomorphic F -valued functions de ned on D Space of all holomorphic C-valued functions de ned on D Space of all G^ateaux holomorphic F -valued functions de ned on D HGpD; F q Space of all G^ateaux holomorphic C-valued functions de ned on D HGpDq Space of holomorphic functions from D into C HbpDq which are bounded on every bounded set in D : Space of holomorphic functions which are bounded on rU HubpEq for some neighbourhood U of P E and for all r ¡ HvpD; F q : tf P HpD; F q : pv:fqpDq is bounded on Du : tf P HpDq : pv:fqpDq is bounded on Du HvpDq HG;vpD; F q : tf P HGpD; F q : pv:fqpDq is bounded on Du : tf P HGpDq : pv:fqpDq is bounded on Du HG;vpDq HolpD; Xq : Space of holomorphic mappings from D into F KpEq : Family of compact, absolutely convex subsets of E HpD; F q HpDq : : : : : OX P SHpDq Uk : Sheaf of germs of holomorphic functions on X : Set of all plurisubharmonic functions on D 1u : tx P E : }x}k : tz P C : }z} 1u Contents Introduction Chapter The rapid convergence of Tauberian type for holomophic func- tions 1.1 Some basic concepts and results 7 1.2 Overview of Zorn space 1.3 The Zorn property of the dense subspace 1.4 The rapid convergence of Tauberian type and holomorphic extension 10 Chapter Tauberian Convergences in weighted spaces of holomorphic functions 2.1 Some basic concepts and results 13 13 2.2 Linearization of weighted (G^ateaux) holomorphic functions 14 2.3 Tauberian Convergences in weighted spaces of holomorphic functions 15 2.4 Linearization of weighted (G^ateaux) holomorphic functions Chapter Vitali spaces and weak tautness 17 18 3.1 Some basic concepts 18 3.2 Weak tautness 19 3.3 Vitali Property, weak tautness and tautness 21 3.4 Weak tautness of Hartogs domains and balanced domains 21 Conclusion 22 References 24 Introduction The research problem of \the spread of certain properties" is one of the classical problems in Mathematical Analysis The problem is to nd the largest domain such that a certain analytic property of a mathematical object de ned on a subdomain is still satis ed on this domain For example, given a holomorphic function f de ned on some n domain of C , one investigates its holomorphic extensions to larger domains More general, let E; F be locally convex spaces on C and D be a subdomain of E One looks for additional conditions to make sure that every sequence of holomorphic functions taking values in F , de ning and pointwise converging on a subset of D is also uniformly convergent almost everywhere on D, etc Such results are called \the convergence of Tauberian type" in the literature Among the above-mentioned results, Vitali's Theorem is an interesting convergence theorem This is a kind of the convergences of Tauberian type where the requirement conditions are that the considered subsets admit a limit point in D and the sequence of holomorphic funtions is locally uniformly convergent on this subset A classical theorem of Vitali asserts that if the sequence of holomorphic functions pf mqm¥1 is uniformly n bounded on the compact subsets of a domain D of C and if the sequence is pointwise convergent to a function f on a subset X of D which is not contained in any complex hypersurface then pfmqm¥1 uniformly converges on every compact subset of D: Note that the vector-valued version of Vitali's Theorem plays an important role in Semi-group Theory For the case that E; F are nite-dimensional spaces, it seems that the easiest proof of Vitali's Theorem was derived with the help of Montel's Theorem In contrast to the scalar case, it is very hard to establish similar results for the case of vector-valued holomorphic functions since Montel's Theorem is no longer valid It was not until 1957 that a quite complicated proof of the theorem was provided by Hille and Phillips [40] for the case where range spaces are in nite-dimensional Banach spaces However, in 2000, by employing a very-weak holomorphicity notion and Uniqueness Theorem, Arendt and Nikolski proved Vitali's Theorem for the nets of Banach-valued one-complex variable holomorphic functions in the case that the small subsets in which the net converges contain an accumulation point [2] Thirteen years later, Quang et al proposed convergence theorems of Vitali type for a locally bounded sequence of holomorphic functions which de ne on a subdomain of a Frechet space and take values in a Frechet space as well as for a sequence of holomorphic functions between Frechet-Schwartz spaces which is bounded on bounded subsets [66] Most recently, Dieu et al [14] achieved analogues of Vitali Theorem for the case that the uniform boundedness of the sequences under consideration is omitted A feasible approach is imposing a stronger requirement on the convergences and/or on the size of small subsets In such research approach, several versions of Vitali Theorem have been obtained in their works for bounded holomorphic functions and rational functions that are rapidly pointwise n convergent on a non-pluripolar subset of a domain in C To continue this line of research, the rst problem that we are interested in is as follows Problem Under which conditions on Frechet (or locally convex) spaces E and F , can every F -valued function f that is de ned, continuous, and fast-enough pointwise approximated by a sequence of F -valued polynomials pp mqm¥1 on a non-pluripolar balanced convex compact (or not too small) subset B of E extend to an entire func-tion? Next, we will study the Tauberian convergence in weighted spaces of holomorphic functions Let D be a domain D of E: A function v : D Ñ p0; 8q is called a weight if it is continuous and strictly positive De ne HvpD; F q : tf P HpD; F q : pv:fqpDq is bounded on Du; HG;vpD; F q : tf P HGpD; F q : pv:fqpDq is bounded on Du; where HpD; F q; HGpD; F q stand for the space of all F -valued holomorphic and G^ateaux holomorphic functions de ned on D; respectively In particular of F C, instead of HvpD; Cq and HG;vpD; Cq we shortly write HvpDq and HG;vpDq, respectively Let AvpDq € HvpDq be a subspace with the closed unit ball is compact in the compact-open topology 0: We de ne the space of vector-valued functions in a weak sense AvpD; F q : tf : D Ñ F : u f P AvpDq; @u P F 1u: If E and F are Banach spaces, Jorda [45] proved the following result Theorem 1.1 Let AvpDq be a subspace of H vpDq which has a 0-compact closed unit ball, let D0 be a set of uniqueness for A vpDq: If pfiqiPI is a bounded net in AvpD; F q such that pfipxqqiPI is convergent for each x P D0; then pfiqiPI is convergent to a function f P AvpD; F q uniformly on the compact subsets of D: From the above results, we have the problem as follows Problem Extending Theorem 1.1 to more general cases: (i) E is a metrizable locally convex space and F is a locally complete space; (ii) the sequence pfmqm¥1 belongs to di erent classes of functions Due to the above setting, it is natural that solving this problem will carry out in the cooperation with the study of weakly holomorphic functions In functional analysis, it seems that two essential approaches to study analyticity of vector-valued functions are based on the notions of weakly holomorphic and (strongly) holomorphic functions The rst approach is easier to check in practical examples since one can use the nice tools of scalar-valued holomorphic functions Recall that a function f : D Ñ F is weakly holomorphic if u f is holomorphic for each u P F 1: It is obvious that every holomorphic function is weakly holomorphic Under which conditions, does the inverse hold? The rst answer for this question is derived by Dunford [23] who proved that the inverse holds for the class of Banach-valued weakly holomorphic functions de ned on a domain of C Later, Grothendieck [34] extended the result to the case that the rage spaces are quasi-complete It turns out that the assertion is true in the case that E; F are Hausdor locally convex spaces and E is metrizable (see [59]) Thus, it is reasonable to question (addressed by Grosse-Erdmann [32, 33], and Arendt{Nikolski [2]) that whether or not a (proper/smallest) subset W of the dual of range spaces such that a function f is holomorphic once it is very-weakly holomorphic (named, p; W q-holomorphic), i.e., u f is holomorphic for all u P W: In other words, we are looking for minimal assumptions such that a weakly holomorphic function is (strongly) holomorphic Arendt and Nikolski dealt with this problem for the case that D € C and F is a complex Banach space Let W be a subset of F and pF; W q be the W -topology of F (the weak topology on F induced by W ) A result of [2] says that a pF; W q-holomorphic function f : D Ñ F is holomorphic if and only if W determines boundedness, i.e each pF; W q-bounded set in F is bounded If f : D Ñ F is additionally assumed to be locally bounded then f is holomorphic once W is a separating-points subspace of F 1: A generalization of this result to the case that F is a locally complete and convex space was obtained by Grosse-Erdmann [33] Hai [35] extended the results of Arendt and Nikolski in [2] to the case that f de nes on an open set D either in a Schwartz-Frechet space E P p q with values in a Schwartz-Frechet space F P pLB8q, or in C with values in a Frechet space F P pLB8q In 2013, Quang et al [66] consider the above problem for the case where E P p q and F P pLB8q or F P pDNq; but \the locally boundedness" of f is dropped to a weaker property, i.e, the boundedness on bounded sets in D: The problem of weakly holomorphic functions is closely related to the problem of holomorphic extensions Speci cally, the following problem of extension of weakly holo-morphic functions (EWH) is one of many issues that has been recently studied (EWH) Let E and F be locally convex spaces Let A € D € E; W € F 1; and f : A Ñ F be such that for every ' P W the function ' f : A Ñ C has an extension in HpDq: When does this imply that there is an extension g P HpD; F q of f? One of the results related to Problem (EWH) was given by Bogdanowicz [9] The result says that a function f de ned on a domain D in C taking values in a complete locally convex complex Hausdor space F such that uf can be holomorphically extended to a domain D2 € D1 for each u P F 1; must admit a holomorphic extension to D 2: Recently, Grosse-Erdmann [33], Arendt, Nikolski [2], Bonet, Frerick, Jorda [10], Frerick, Jorda [29], Frerick, Jorda, Wengenroth [30] have given results in this way, but with requiring extensions of u f only for a proper subset W € F and the conditions on D1 are smoother Also, Laitila and Tylli [50] have recently discussed the di erence be-tween strong and weak de nitions for important spaces of vector-valued functions Most recently, Problem (EWH) is also investigated for more general cases, for instance, on Frechet spaces with the linear topological invariants by Quang, Lam, Dai [66], Quang, Dai [62, 63]; and for p; W q-meromorphic functions by Quang, Lam [67, 68] Naturally and generally, solving Problem (EWH) in weighted spaces of holomorphic functions is also set The next problem that we are interested is: Problem Study weighted versions of the problem (EWH), especially, of the main results of [66, 62, 63] concerning the holomorphic extension of p; W qholomorphic functions Beside achievements for the class of vector-valued holomorphic functions (called holomorphic function), problem of Tauberian convergence for the sequences of holomorphic functions taking values in spaces which not have any vector structure (called holomorphic mappingss), such as complex manifolds, complex/Banach analytic spaces, has been received much attentions in recent years Here, complex analytic space (respectively, Banach analytic space) is understood in the sense that a connected topological space in which every point has a neighborhood that is isomophism with a analytic set in n some nite-dimensional space C (respectively, in a Banach space) such that transition mappings are holomorphic between open sets Therefore, analytic space contains two di erent objects: complex space ( nite-dimensional) and Banach analytic space (in nitedimensional) As we have known, hyperbolic and taut spaces play very important roles in nite-dimensional complex analytic geometry Thus, it is natural to consider analogues of these notions in in nite-dimensional setting In 1960s, Wu [80] introduced the taut and tight manifolds In [18], Dineen gave a notion of tautness for a Banach manifold equipped with a Hausdor topology A Banach analytic space X is taut if every sequence p fnqn¥1 € Holp ; Xq; the space of all X-valued holomorphic mappings on the unit disc € C; contains a subsequence pfnk qk¥1 such that one of the following two conditions holds: pfnk qk¥1 is convergent in Holp ; Xq; pfnk qk¥1 is compactly divergent, i.e for every compact subset K € and L € X there exists k0 such that fnk pKq X L ? for k ¡ k0: Thus, the next problem is to study the Tauberian convergence in the space Holp ; Xq: Speci cally, we are interested in the following problem Problem Study Banach analytic space X having the Vitali Property in the sense that every sequence pfnqn¥1 € Holp ; Xq is convergent if the set Zpfnq t has a limit point in P : lim fnp q existsu n : However, there are a few existing results related to these spaces in in nitedimensional complex analysis More than 10 years ago, Thai and Giao [72] proved the holomorphic extension theorem of Kwack [46] for Banach hyperbolic spaces It seems that in in nite-dimensional complex analysis it is di cult to study taut spaces because we not have local compactness Therefore, one of the issues is considered in this thesis is to overcome the above di culties Beside the introduction and the inclusion, the thesis consists of chapters and refer-ences 2.2 Linearization of weighted (G^ateaux) holomorphic functions The main idea of the linearization problem is to permit identi cation classes of F - valued functions de ned on an open set in E with F -valued continuous linear maps from a certain space where E; F are locally convex spaces This is very important because it can help us in simplifying calculations on spaces of vector-valued functions Recent works of Carando and Zalduendo [12], and of Mujica [55] are devoted to get linearization results for (unweighted) spaces of continuous/holomorphic functions between locally convex spaces; and of Beltran [7] for weighted (LB)-spaces of entire functions on Banach spaces Let AvpDq (resp AG;vpDq) be a subspace of HvpDq (resp HG;vpDq) such that the closed unit ball is compact for the compact open topology 0: First, we study linearization theorem in space of F -valued functions in a weak sense AvpD; F q : tf : U Ñ F : u f P AvpDq @u P F 1u: We obtain the following Theorem 2.2.1 Let D be a domain in a metrizable locally convex space E and v be a weight on D: Let AvpDq be a subspace of H vpDq such that the closed unit ball is 0compact Then there exist a Banach space P AvpDq and a mapping D P HpD; PAvpDqq with the following universal property: For each complete locally convex space F; a function f P AvpD; F q if and only if there is a unique mapping T f P LpPAvpDq; F q such that Tf D f: This property characterize PAvpDq uniquely up to an isometric isomorphism Since J is a topological isomorphism the space P AvpDq is called the predual of AvpDq: Now we consider the above result for weighted G^ateaux holomorphic functions Let D be an open subset of metrizable locally convex space E: Denote FpEq the family of all nite dimensional subspaces of E: By Theorem 2.2.1, for each Y P FpEq there exists a unique map p Y P LpPAvpDXY q; PAvpDqq such that the following diagram is commutative id DXY /D DXY P (2.1) D pY /P AvpDXY q AvpDq where id is the identity mapping and PAvpDXY q denotes the predual of AvpD X Y q: If Y; Z P FpEq such that Y € Z; then by Theorem 2.2.1 again, there exists a unique map pZY P LpPAvpDXY q; PAvpDXZqq such that the following diagram is commutative id DXY / DXZ DXY DXZ p P ZY AvpDXY q 14 /P AvpDXZq It follows that pZ pZY pY whenever Y € Z: Denote P Ôp : A vpDq Y Y and equip PA vpDq pP AvpDXY qq PFpEq with the topology induced by PAvpDq: Let AG;vpDq be a subspace of HG;vpDq such that the closed unit ball is compact for the topology 0: For each complete locally convex space F; we put AG;vpD; F q : tf : D Ñ F : u f P AG;vpDq @u P F 1u: We obtain Theorem 2.2.2 Let D be a domain in a metrizable locally convex space E: Then (a) PA (b) D vpDq is a dense subspace of PAvpDq; P HpD; P ; A vpDqq (c) For each complete locally convex space F; the function f P A G;vpD; F q if and only if there exists a unique linear mapping T f : PA vpDq Ñ F such that Tf D f: Moreover, Tf is continuous if and only if f is continuous Lemma 2.2.3 Let D be a domain in a metrizable locally convex space E and F a com-plete locally convex space Then family pf jqjPI € AvpD; F q is bounded if and only if the corresponding family T fj € LpPAvpDq; F q is equicontinuous The following is a consequence of Theorem 2.2.2 and Lemma 2.2.3 Corollary 2.2.4 Let D be a domain in a metrizable locally convex space E and F a complete locally convex space Then family pf jqjPI € AvpD; F q is bounded if and only if the corresponding family pTfj qjPI € LpPA vpDq; F q is equicontinuous 2.3 Tauberian Convergences in weighted spaces of holomorphic functions We will apply the results of the Section 2.2 to investigate the problem on Tauberian convergences for sequences/nets in weighted spaces H vpE; F q and AvpD; F q: A subset M € D is said to be a set of uniqueness for A vpDq if each f P AvpDq such that f M then f 0: A subset M € D is said to be sampling for A vpDq if there exists a constant C ¥ such that for every f P AvpDq we have sup vpxq|fpxq| Ô C sup vpzq|fpzq|: xPD zPM 15 (2.2) Remark 2.3.1 For M € D; denote M : tvpxq x : x P Mu € BP v D p q pD q where BP Av Av denotes the unit ball of PAvpDq: Since the closed unit ball BAvpDq of the space AvpDq is 0-compact, by the HahnBanach theorem, it is easy to check that the following are equivalent: (i) M is separating in AvpDq; v (ii) xMy v : spanM is pPA v v pDq; AvpDqq-dense; (iii) M is a set of uniqueness for AvpDq: For the norm given by }f}M;v : supzPM vpzq|fpzq| on AvpDq; it is obvious that the following are equivalent: (i) M is a sampling for AvpDq; (ii) } }v € } }M;v on A pDq: v Obviously, if M is sampling for AvpDq then M is separating in AvpDq; hence, M v is a set of uniqueness for AvpDq: The rst results of this section are concerned with the weighted Tauberian convergence of sequences of G^ateaux holomorphic functions in a space pE K ; Eq Theorem 2.3.2 Let E; F be Frechet spaces and v be a weight on E: Assume that E is nuclear with the topology E r and E P p q: Then there exists a non-pluripolar set K P KpEq satisfying the following: if pf mqm¥1 is a bounded sequence in H G;vppEK ; Eq; F q such that pfmqm¥1 is convergent at each x P K to a function f which is continuous at some x0 P K then f has an extension r r f P HvpE; F q and pfmqm¥1 is convergent to f uniformly on the compact subsets of pEK ; Eq: From Theorem 1.3.4, as Theorem 2.3.2, we have Theorem 2.3.3 Let E; F be Frechet spaces and v be a weight on E: Assume that E is Schwartz with an absolute Schauder basis and the topology E r and E P p q: Then there exists a non-pluripolar set K P KpEq satisfying the following: if pf mqm¥1 is a bounded sequence in HG;vppEK ; Eq; F q such that pfmqm¥1 is convergent at each x P K to a function f which is continuous at r some x0 P K then f has an extension f P HvpE; F q and pf mqm¥1 is convergent to a function f uniformly on the compact subsets of pEK ; Eq: r Now, the remain of this section deals with the Tauberian convergence of nets in weighted spaces AvpD; F q: Theorem 2.3.4 Let D be a domain in a metrizable locally convex space E: Let A vpDq € HvpDq be a subspace such that the closed unit ball B AvpDq is 0-compact, M € D be a set of uniqueness for AvpDq and F be a complete locally convex space If pf jqjPI is a bounded net in AvpD; F q such that pfjqjPI is convergent for each x P M then pf jqjPI is convergent to a function f P AvpD; F q uniformly on the compact subsets of D: 16 2.4 Application to the problem of weighted holomorphic extensions Using the results in the previous section, we investigate the holomorphic extension of weak-type holomorphic functions in weighted spaces of holomorphic functions Given a Frechet space E; an increasing sequence pBnqn¥1 of bounded subsets of E1 is said to x the topology if the polars pBnqn¥1 taken in E form a fundamental system of 0-neighbourhoods of E: Theorem 2.4.2 Let v be a weight on a domain D in a metrizable locally convex space E and AvpDq a subspace of HvpDq such that the closed unit ball BAvpDq is 0-compact Let M € D be a set of uniqueness for A vpDq: Let F be a complete locally convex space and W € F be a subspace which determines boundedness in F: If f : M Ñ F is a function such that u f has an extension f u P AvpDq for each u P W then f admits a unique extension f P AvpD; F q: r N In the case where D is either a domain in C ; or a Banach space, by Montel's theorem, the closed unit ball BHvpDq of the space HvpDq is 0-compact Therefore, from Theorem 2.2.1, we get the following: N Corollary 2.4.3 Let D be either a domain in C ; or a Banach space E and v be a weight on D and M € D be a set of uniqueness for HvpDq: Let F be a complete locally convex space and W € F be a subspace which determines boundedness in F: If f : M Ñ F is a function such that u f has an extension f u P HvpDq for each u P W then f admits a unique extension f P HvpD; F q: r Theorem 2.4.4 Let v be a weight on a domain D in a metrizable locally convex space E and AvpDq a subspace of HvpDq such that the closed unit ball BAvpDq is 0-compact Let M € D be a set of uniqueness for A vpDq: Let F be a Frechet space and W YnBn € F where tBnun xes the topology of F: If f : M Ñ F is a function such that u f has an extension f u P AvpDq for each u P W and pfuquPBn is bounded in AvpDq for all n then f admits a unique extension f P r AvpD; F q: The following result is a version of Theorem 2.4.4 in case M be a sampling Theorem 2.4.5 Let v be a weight on a domain D in a metrizable locally convex space E and AvpDq a subspace of HvpDq such that the closed unit ball BAvpDq is 0-compact Let M be a sampling set for AvpDq and W be a pF 1; F q-dense subspace of the dual F of a locally complete locally convex space F: If f : M Ñ F is a function such that sup vpxqppfpxqq for all p P cspF q xPM (2.3) r and u f has an extension fu P AvpDq for each u P W then f admits a unique extension f P AvpD; F q: 17 Chapter Vitali spaces and weak tautness The main content of this chapter is to study the problem Tauberian convergence for sequence of holomorphic mappings on open unit disc € C: The new results presented in this chapter are extracted from the article [38] 3.1 Some basic concepts A ringed space pX; OX q is called complex analytic space (shortly, complex space) if X is Hausdor and for each x P X there exists a neighborhood U of x such that pU; OX U q is isomorphism to a local model (as a ringed space) An in nite-dimensional extension version of complex analytic spaces appearing in the context of studying variant forms of analytic structure is the concept of Banach analytic space Here, a local model is a Banach analytic set De nition 3.1.2 Let U be an open set in a Banach space A subset A € U is called Banach analytic set if A is the set of common zero-points of a nite family of holomorphic functions on U with values in some Banach space A ringed space pX; OX q is called Banach analytic space if X is Hausdor and for each x P X there exists a neighborhood U of x such that pU; OX U q is isomorphism to a Banach analytic set De nition 3.1.3 Let D be an open set in C n and X be a complex space (respectively, Banach analytic space) A mapping f : D Ñ X is said to be holomorphic at x P D if there exists a neighborhood V of x and an open set U of X such that fpV q € U and f r r f : V Ñ Cm (respectively, f : V Ñ F ) holomorphic, where is a isomorphic between pU; OX U q and a local model (Banach analytic set in the Banach space F ) Denote by HolpD; Xq the space of all holomorphic mappings from D to X and endow it a topology of uniform convergence on compact subsets of D 18 De nition 3.1.4 Let X be a Banach analytic space and p, q be arbitrary points of X We call a holomorphic chain connecting p to q the set ta1; a2; : : : ; an P ; f1; f2; : : : ; fn P Holp ; Xqu such that ° De ne L ln f1p0q p; fipaiq fi 1p0q; fnpanq q: n i1 |ai| |ai| and X pp; qq : inf L , where the in mum is taken over the set of all holomorphic chains connecting p to q Then, the function called Kobayashi pseudodistance on the Banach analytic space X X : X X Ñ r0; 8q is De nition 3.1.5 Let X be a Banach analytic space We call X is hyperbolic if X is a distance X is completely hyperpolic if pX; 3.2 X q is a complete metric space Weak tautness De nition 3.2.1 A Banach analytic space X is taut if every sequence pf nq € Holp ; Xq contains a subsequence pfnk q such that one of the following two conditions holds: (i) (ii) pfnk qk¥1 is convergent in Holp ; Xq; pfnk qk¥1 is compactly divergent, i.e for every compact subset K € and L € X there exists k0 such that fnk pKq X L ?; @k ¡ k0: However, every open subsets of a in nite dimensional Banach space are not a taut domain in the usual sense as in nite dimensional complex analysis We consider a following example Example 3.2.2 Let E be an in nite dimensional Banach space and Bp0; rq tx P E : } x} ru a ball centered at with radius r ¡ 0: Take a sequence pxnqn¥1 € Bp0; rq for which inf }xn xm} ¡ 0: nm For each n ¥ 1; de ne hn P Holp ; Bp0; rqq by hnp q xn; P : We have hnp0q for n ¥ and }hnp q hmp q} | |}xn xm} Û as n; m Ñ for || r: This implies that no subsequence of phnqn¥1 is either convergent or compactly divergent in Holp ; Bp0; rqq: Hence Bp0; rq is not taut in the usual sense Note, however, that phn zt0uq is compactly divergent 19 This phenomenon happens due to the fact that every in nite-dimensional Banach space is not locally compact This is also the main reason leading to di culties in studying the taut property in complex analysis for in nite-dimensional spaces Therefore, it is necessary to have a extended concept for the taut property of Banach analytic spaces We consider the following extended concept De nition 3.2.3 A Banach analytic space X is weakly taut if every sequence pf nqn¥1 € Holp ; Xq contains a subsequence pfnk qk¥1 such that one of the following two conditions holds: (i) (ii) pfnk qk¥1 is convergent in Holp ; Xq; there exists a discrete subset S of such that pf nk zSqk¥1 is compactly divergent, i e for every compact subset K € zS and L € X there exists k0 such that fnk pKq X L ?; @k ¡ k0: In order to prove the main results of this section, rst we discuss some properties of complete hyperbolic Banach analytic spaces which will be used in the proofs Lemma 3.2.4 Let X denote a complete hyperbolic Banach analytic space and let pf nqn¥1 € Holp ; Xq: Then Z : Zpfnq Zpfnq;U t P U : lim fnp q existsu n is a closed subset of : Lemma 3.2.5 Let X denote a hyperbolic Banach analytic space and let pf nqn¥1 € HolpU; Xq where U is an open subset of C: If pf nqn¥1 is not compactly divergent then there exists pg nqn¥1 € pfnqn¥1 such that Zpgnq ?: Equivalently, if Zpgnq ?: for all pgnqn¥1 € pfnqn¥1 then pfnqn¥1 is compactly divergent on U: Lemma 3.2.6 Let X denote a complete hyperbolic Banach analytic space and let pf nqn¥1 € Holp ; Xq: If Z1 : Zp1fnq denotes the set of limit points of Z : Z pfnq in then Zpfnq is a neighbourhood of Zp1fnq: It is known (see [49]) that every nite dimensional complex space, which is complete hyperbolic, is taut However, for the case of Banach analytic spaces we have Theorem 3.2.7 Let X be a Banach analytic space which is complete hyperbolic Then X is weakly taut Corollary 3.2.8 Let X denote a hyperbolic Banach analytic space If pf nqn¥1 € Holp ; Xq and Z p1gnq ? for every pgnqn¥1 € pfnqn¥1; then there exists a discrete subset S of and a subsequence ph nqn¥1 € pfnqn¥1 such that phnqn¥1 is compactly divergent on zS: Considering property completely hyperbolic of bounded convex domains, we obtain following result 20 Proposition 3.2.9 Every bounded convex domain D in a Banach space E is completely hyperbolic and, hence, is weakly taut Now in the next section we establish the equivalence between weakly taut and taut notions in the nite dimensional case 3.3 Vitali Property, weak tautness and tautness De nition 3.3.1 We say that a Banach analytic space X has the Vitali Property if p fnqn¥1 € Holp ; Xq va Zpfnq has a limit point in then pfnqn¥1 converges The rst result we obtain is the following theorem Theorem 3.3.2 Let X be a complex space Then the following are equivalent: (i) (ii) (iii) X has the Vitali Property; X is weakly taut; X is taut Now we consider relation between weakly taut spaces and Banach analytic spaces having Vitali's Property and we obtain the following Theorem 3.3.3 Let X be a Banach analytic space Then X is weakly taut if and only if X is hyperbolic and has the Vitali Property 3.4 Weak tautness of Hartogs domains and balanced domains Next we establish conditions under which the Hartogs domains 'pXq associated to a Banach analytic space X and balanced domains in a Banach space are weakly taut De nition 3.4.1 Let X be a Banach analytic space X and ' an upper-semicontinuous function on X: Then the Hartogs domain 'pXq is de ned by 'pXq : tpx; q P X C : | | u e 'pxq : Theorem 3.4.2 'pXq is weakly taut if and only if X is weakly taut and ' is continuous plurisubharmonic De nition 3.4.3 Let X be a balanced domain in a Banach space E We say that h be a gauge functional of X given by hpxq inft ¡ : x P Xu: Theorem 3.4.5 Let X be a balanced domain in a Banach space pE; } }q: Then X is weakly taut if and only if X is bounded and the gauge functional h is plurisubharmonic and continuous 21 Conclusion The thesis studies the problems of Tauberian convergences The new contributions of the thesis include: Prove Zorn Property of the domain DK : D X pEK ; Eq where D is a domain in a Frechet space, K is a closed, compact, absolutely convex subsets of E; E K is the linear hull of K; and E is the topology of EK induced by the topology of E: Note that pEK ; Eq is dense in E: Holomorphic extensions from pEK ; Eq to E are also considered simultaneously here (Theorem 1.3.3, Theorem 1.3.4) Assert the existence of non-pluripolar, balanced, convex, compact subsets B of Frechet r space E P p q (nuclear space or Schwartz space with an absolute Schauder basis) such that every Frechet-valued function f which is de ned, continuous, and approximated fast enough by a sequence of Frechet-valued polynomials ppmqm¥1 on a non-pluripolar balanced convex compact subset B of E can be extended to an entire function (Theorem 1.4.7, Theorem 1.4.8) Derive conditions to ensure the existence of non-pluripolar, balanced, convex, compact subsets K of Frechet space E such that every bounded sequence pfmqm¥1 of holomorphic functions in HG;vppEK ; Eq; F q is convergent to a function f P HG;vppEK ; Eq; F q uniformly on the compact subsets of pEK ; Eq whenever pfmqm¥1 converges at every point of K: Moreover, the function f admits a holomorphic exten-sion in HvpE; F q if it is continuous at a simple point in K (Theorem 2.3.2, Theorem 2.3.3) Establish a new version of Jorda's theorem, saying that if E is a metrizable locally convex space and F is a complete locally convex space, every bounded net of func-tions in AvpD; F q will be convergent uniformly on the compact subsets of D to a function in AvpD; F q whenever it converges at every point of a subset of uniqueness for AvpDq (Theorem 2.3.4) Introduce the notion of \weakly taut", which is a generalization of notion of \taut" This helps us to overcome some di cults whenever study holomorphic mappings with values in in nite dimensional complex spaces Establish some relations between hyperbolic Banach analytic spaces, weakly taut spaces and Banach analytic spaces having Vitali's Property (Theorem 3.2.7, Propo-sition 3.2.9, and Theorem 3.3.3); at the same time the equivalence of three classes of Banach analytic spaces in the nite dimensional case is also proved (Theorem 3.3.2) Apply the main results of the thesis to deal with the holomorphic extension problem in the weighted space AvpD; F q € HvpD; F q of F -valued holomorphic functions from 22 a (thin) subset of uniqueness and from a (fat) sampling subset (Theorem 2.4.2, Theorem 2.4.4, Theorem 2.4.5) The above-mentioned results are really new contributions in the research the problem on convergence of Tauberian type They are scienti c signi cance and receive much attention from researchers in the eld of study For future research, we plan to continue with the following open problems: Study to discover more classes of spaces that have the Zorn property and then nd applications of the archievements in the thesis Study problems on the convergence of Tauberian type in the weighted spaces of meromorphic functions and applications In recent paper [65], we established a representation of the space of vector-valued holomorphic functions as the tensor product of the space of scalar-valued holomorphic functions with its range Exactly, we can present the space of Frechet-valued holomorphic functions rHpU; F q; s as rpHpUq; qsb F , where U is an open set in Frechet space and P t 0; !; u: Moreover, as an application, we employ this presentation in dealing with the following problems: (1) Exponential laws for the topologies 0; ! on the space HpU V q where U and V are open sets in locally convex spaces; (2) Coinciding of topologies 0; !; on the space of locally-convex-valued holomorphic functions (germs) HpU; F q (HpK; F q); p (3) Inheritance of analytic properties via we consider the space of holomorphic functions (germs) Employing the tensor presentation idea, we believe that the following problems would be solved in the future: Establishing tensor presentation for weighted spaces of vector-valued holomorphic functions, i.e presenting rAvpU; F q; s in the form of rpAvpUq; qsbp F: If the above presentation holds, we will generalize existing results on the Tauber type convergence for the sequences of scalar-valued functions to the sequences of vector-valued functions in both weighted and non-weighted cases 23 References [2] W Arendt, N Nikolski, Vector-valued holomorphic functions revisited, Math Z.,234 (2000), 777{805 [4] T J Barth, The Kobayashi Distance Induces the Standard Topology, Proc of the Amer Math Soc., 35(2) (1972), 439-441 [7] I A Berezanskii, Inductively re exive, locally convex spaces, Dokl Akad Nauka SSSR 182 (1968), 20{22, English Translation in Soviet Math., (1968), 1080-1082 [9] W M Bogdanowicz, Analytic continuation of holomorphic functions with 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Dineen, Holomorphic functions on strong duals of Frechet-Montel spaces, In nite Dimensional Holomorphy and Applications (Ed.: M C Matos), North-Holland Math Stud., 12 (1977), 147{166 [18] S Dineen, The Schwarz Lemma, The Clarendon Press, Oxford University Press, (1989) 24 [19] S Dineen, M.L Lourenco, Holomorphic functions on strong duals of Frechet-Montel spaces II, Arch Math., 53 (1989), 590{598 [23] N Dunford, Uniformity in linear spaces, Trans Amer Math Soc., 44(2) (1938), 305-356 [29] L Frerick, E Jorda, Extension of vector-valued functions, Bull Belg Math Soc., Simon Stevin, 14(3) (2007), 499-507 [30] L Frerick, E Jorda, J Wengenroth, Extension of bounded vector-valued func-tions, Math Nach., 282(5) (2009), 690-696 [32] K G Grosse-Erdmann, The Borel-Okada Theorem Revisited, Habilitationss-chrift Fernuniversit•at in Hagen, Hagen 1992 [33] K G Grosse-Erdmann, A weak criterion for vector-valued holomorphy, Math Proc Cambridge Philos Soc., 136 (2004), 399-411 [34] A Grothendieck, 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holomorphic functions in weighted spaces and its applications, (submitted) [65] T T Quang, D Q Huy, D T Vy, Tensor representation of spaces of holomorphic functions and applications, Complex Anal Oper Theory, 11(3) (2017) , 611-626 [66] T T Quang, L V Lam, N V Dai, On p; W q-holomorphic functions and theorems of Vitali-type, Complex Anal Oper Theory, 7(1) (2013), 237{259 [67] T T Quang, L V Lam, Levi extension theorems for meromorphic functions of weak type in in nite dimension, Complex Anal Oper Theory, 10 (2016), 1619-1654 [68] T T Quang, L V Lam, Cross the orems for separately p; W qmeromorphic functions, Taiwanese J Math., 20(5) (2016), 1009-1039 [69] T T Quang, D T Vy, L T Hung, P H Bang, The Zorn property for holomorphic functions, Ann Polon Math., 120(2) (2017), 115-133 [72] D D Thai, T N Giao, The convergence-extension theorem of Noguchi in in nite dimension, Proc Amer Math Soc., 130(2) (2002), 477-482 [74] E Vesentini, Invariant distance and invariant di erential metric in 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Sbornik) ... of the dense subspace 1.4 The rapid convergence of Tauberian type and holomorphic extension 10 Chapter Tauberian Convergences in weighted spaces of holomorphic functions 2.1... convergence of Tauberian type" in the literature Among the above-mentioned results, Vitali's Theorem is an interesting convergence theorem This is a kind of the convergences of Tauberian type... results on convergence of Tauberian type for a sequence of Banach-valued polynomials on a locally convex space are introduced We also establish convergent theorems of Tauberian type for a sequence