Regular Holomorphic Functions On Complex Banach Lattices
Regular Holomorphic Functions on ComplexBanach LatticesRay RyanNational University of Ireland GalwayWorkshop on Infinite Dimensional AnalysisBuenos Aires 20141 / 21
1. Holomorphy — some of the historyIHilbert (1909): Holomorphic functions on CN defined locallyby monomial expansions:Xf(z) cα (z a)αα2 / 21
1. Holomorphy — some of the historyIHilbert (1909): Holomorphic functions on CN defined locallyby monomial expansions:Xf(z) cα (z a)ααIFréchet, Gâteaux, Michael, Taylor,. . . : Power series ofhomogeneous polynomials:Xf(z) Pn (z a)nin a neighbourhood of a.Pn : (bounded) n-homogeneous polynomials.2 / 21
1. Holomorphy — some of the historyIHilbert (1909): Holomorphic functions on CN defined locallyby monomial expansions:Xf(z) cα (z a)ααIFréchet, Gâteaux, Michael, Taylor,. . . : Power series ofhomogeneous polynomials:Xf(z) Pn (z a)nIin a neighbourhood of a.Pn : (bounded) n-homogeneous polynomials.Grothendieck, Nachbin, Gupta (1950’s and 60’s): Duality interms of nuclear functions/tensor products: 0P(n E 0 ) PN (n E)(subject to AP)2 / 21
IBoland, Dineen (1970’s): Holomorphic functions on nuclearlocally convex spaces. For suitable nuclear spaces withbasis, the monomials are a basis for the space of holomorphicfunctions.3 / 21
IBoland, Dineen (1970’s): Holomorphic functions on nuclearlocally convex spaces. For suitable nuclear spaces withbasis, the monomials are a basis for the space of holomorphicfunctions.IMatos, Nachbin (1980’s and 90’s): Monomial expansions forholomorphic functions on a Banach space with unconditionalbasis.Hν (E): the space of holomorphic functions representablelocally by unconditionally convergent monomial expansions.3 / 21
IBoland, Dineen (1970’s): Holomorphic functions on nuclearlocally convex spaces. For suitable nuclear spaces withbasis, the monomials are a basis for the space of holomorphicfunctions.IMatos, Nachbin (1980’s and 90’s): Monomial expansions forholomorphic functions on a Banach space with unconditionalbasis.Hν (E): the space of holomorphic functions representablelocally by unconditionally convergent monomial expansions.IDefant, Dı́az, Garcı́a, Kalton, Maestre (2001 and 2005)proved Dineen’s conjecture: if E is a Banach space and n 2,then P(n E) has an unconditional basis if and only if E is finitedimensional.3 / 21
2. The Matos-Nachbin Holomorphy TypeE: a Banach space with unconditional Schauder basis (ej ).Every P P(n E) has a monomial expansion:XP(z) A(z, . . . , z), z zj ejjP(z) Xcα z α α nBut this expansion is only conditionally convergent in general.4 / 21
2. The Matos-Nachbin Holomorphy TypeE: a Banach space with unconditional Schauder basis (ej ).Every P P(n E) has a monomial expansion:XP(z) A(z, . . . , z), z zj ejjP(z) Xcα z α α nBut this expansion is only conditionally convergent in general.Pν (n E): the subspace of polynomials for which the monomialexpansion is unconditionally convergent at every point.4 / 21
If P Pν (n E), thenP̃(z) : X cα zα α nalso belongs to Pν (n E). A norm is defined on Pν (n E) byν(P) : kP̃k supX cα zα : kzk 6 1Pν (n E) is a Banach space with this norm.5 / 21
If P Pν (n E), thenXP̃(z) : cα zα α nalso belongs to Pν (n E). A norm is defined on Pν (n E) byν(P) : kP̃k supX cα zα : kzk 6 1Pν (n E) is a Banach space with this norm.PN (n E) Pν (n E) P(n E)Extreme cases:1. E c0 : Pν (n E) PN (n E)2. E 1 : Pν (n E) P(n E)(with equivalent norms in each case.)5 / 21
Holomorphic functions.In the complex case, we have a holomorphy type:for P Pν (n E) and z E, 1 νd̂k P(z) 6 4n kzkn k ν(P)k!6 / 21
Holomorphic functions.In the complex case, we have a holomorphy type:for P Pν (n E) and z E, 1 νd̂k P(z) 6 4n kzkn k ν(P)k!Theorem (Matos-Nachbin): A holomorphic function f on adomain U E belongs to Hν (U) if and only if f is representablelocally in U by unconditionally pointwise convergent monomialexpansions.6 / 21
Matos-Nachbin (1992):E a complex Banach space with an unconditional basis. Let U be aReinhardt domain in E containing 0.the following are equivalent:1. U is the domain of convergence of a multiple power seriesaround 0.2. U is modularly decreasing and logarithmically convex.3. U is the domain of existence of some f Hν (()U).4. U is a domain of ν-holomorphy.5. U is a domain of holomorphy.6. U is pseudo-convex.7 / 21
3. RegularityRiesz spaces (vector lattices):A real vector space E with a compatible lattice structure:x, y E x y,x yFor every x E,x x x wherex x 0,x ( x) 0Absolute values: x : x ( x) x x Normed Lattice: Riesz space with a norm satisfyingk x k kxkDedekind complete: every order bounded set has a supremum.8 / 21
Regular operatorsAn operator T : E F between Riesz spaces is regular if it can bewritten as the difference of two positive operators.9 / 21
Regular operatorsAn operator T : E F between Riesz spaces is regular if it can bewritten as the difference of two positive operators.If F is Dedekind complete, then regularity of an operator T isequivalent to order boundedness and in this case, the spaceLr (E; F) of regular operators is a Dedekind complete Riesz space. T (x) sup{ T (y) : y 6 x} for x E T (x) 6 T ( x ) xOrder dual:E Lr (E; R)9 / 21
Banach Lattices10 / 21
Banach Lattices1. Regular operators are bounded (Automatic continuity ofpositive operators.)10 / 21
Banach Lattices1. Regular operators are bounded (Automatic continuity ofpositive operators.)2. The space of regular operators is a Banach lattice with theregular norm:kT kr : k T k10 / 21
Banach Lattices1. Regular operators are bounded (Automatic continuity ofpositive operators.)2. The space of regular operators is a Banach lattice with theregular norm:kT kr : k T k3. E E 0 , the Banach dual, with equality of norms.10 / 21
Banach Lattices1. Regular operators are bounded (Automatic continuity ofpositive operators.)2. The space of regular operators is a Banach lattice with theregular norm:kT kr : k T k3. E E 0 , the Banach dual, with equality of norms.4. Principal ideals: for u 0, the principal idealEu : {x E : x 6 ku for some k N}with norm given by the Minkowski functional of the orderinterval [ u, u] is an AM-space with unit. It is latticeisometric to a C(K) space.Eu C(K)10 / 21
Regular polynomials on Banach latticesMultilinear forms:A L(n E1 , E2 , . . . , En ) is positive ifA(x1 , . . . , xn ) 0for all x1 , . . . , xn 0and A is regular if it is the difference of two positive forms.11 / 21
Regular polynomials on Banach latticesMultilinear forms:A L(n E1 , E2 , . . . , En ) is positive ifA(x1 , . . . , xn ) 0for all x1 , . . . , xn 0and A is regular if it is the difference of two positive forms.Lr (n E1 , . . . , En ): the Banach lattice of regular n-linear forms withthe regular norm. Lr (E1 , Lr (n 1 E2 , . . . , En ))Lr (n E1 , . . . , En ) 11 / 21
Regular polynomials on Banach latticesMultilinear forms:A L(n E1 , E2 , . . . , En ) is positive ifA(x1 , . . . , xn ) 0for all x1 , . . . , xn 0and A is regular if it is the difference of two positive forms.Lr (n E1 , . . . , En ): the Banach lattice of regular n-linear forms withthe regular norm. Lr (E1 , Lr (n 1 E2 , . . . , En ))Lr (n E1 , . . . , En ) Homogeneous polynomials:The n-homogeneous polynomial P Â is positive if A is positive.Pr (n E): the Banach lattice of regular n-homogeneous polynomialswith the regular normkPkr : k P k11 / 21
If P P(n E) is positive, then1. P(x) 0 for every x 0.2. P is monotone on the positive cone:if 0 6 x 6 ythenP(x) 6 P(y)12 / 21
If P P(n E) is positive, then1. P(x) 0 for every x 0.2. P is monotone on the positive cone:if 0 6 x 6 ythenP(x) 6 P(y)But for n 3, these conditions are not sufficient to ensure P 0.12 / 21
If P P(n E) is positive, then1. P(x) 0 for every x 0.2. P is monotone on the positive cone:if 0 6 x 6 ythenP(x) 6 P(y)But for n 3, these conditions are not sufficient to ensure P 0.Absolute value: P(x) 6 P ( x ) x E P is the smallest positive n-homogeneous polynomial satisfyingthis.12 / 21
Every Banach space with a 1-unconditional basis is a Banachlattice,P where the lattice operations are defined coordinatewise: ifx xj ej , thenX x xj ejj13 / 21
Every Banach space with a 1-unconditional basis is a Banachlattice,P where the lattice operations are defined coordinatewise: ifx xj ej , thenX x xj ejjGrecu–Ryan (2004): If E is a Banach space with a1-unconditional basis, thenPν (n E) Pr (n E)with equality of norms.13 / 21
4. Tensor Products14 / 21
4. Tensor ProductsThe Fremlin Tensor Product (1972)For (archimedean) Riesz spaces E and F, the Fremlin tensorproduct E F linearizes regular bilinear forms on E F.14 / 21
4. Tensor ProductsThe Fremlin Tensor Product (1972)For (archimedean) Riesz spaces E and F, the Fremlin tensorproduct E F linearizes regular bilinear forms on E F.The Fremlin-Wittstock Banach Lattice Tensor Products π F and E ε F: Banach lattice versions of the projective andE injective Banach space tensor products.14 / 21
4. Tensor ProductsThe Fremlin Tensor Product (1972)For (archimedean) Riesz spaces E and F, the Fremlin tensorproduct E F linearizes regular bilinear forms on E F.The Fremlin-Wittstock Banach Lattice Tensor Products π F and E ε F: Banach lattice versions of the projective andE injective Banach space tensor products.Perez-Villanueva (2005):None of Grothendieck’s 14 natural tensor norms are Banach latticenorms.14 / 21
4. Tensor ProductsThe Fremlin Tensor Product (1972)For (archimedean) Riesz spaces E and F, the Fremlin tensorproduct E F linearizes regular bilinear forms on E F.The Fremlin-Wittstock Banach Lattice Tensor Products π F and E ε F: Banach lattice versions of the projective andE injective Banach space tensor products.Perez-Villanueva (2005):None of Grothendieck’s 14 natural tensor norms are Banach latticenorms.Labuschagne (2004): If E and F are Banach lattices and α is areasonable crossnorm on E F, then there is a reasonable α F is a Banach lattice withcrossnorm α on E F such that E respect to the ordering induced by the α -closure of the Fremlincone of E F.14 / 21
Regular multilinear forms on C(K) spaces15 / 21
Regular multilinear forms on C(K) spaces π C(L) C(K) ε C(L)C(K) with equality of norms.15 / 21
Regular multilinear forms on C(K) spaces π C(L) C(K) ε C(L)C(K) with equality of norms.Theorem (Fremlin): Every regular multilinear form on a productof C(K) spaces is integral.15 / 21
Regular multilinear forms on C(K) spaces π C(L) C(K) ε C(L)C(K) with equality of norms.Theorem (Fremlin): Every regular multilinear form on a productof C(K) spaces is integral.Linearizing regular polynomialsLoane (2007) : Construction of a symmetric n-fold Fremlintensor product satisfying O 0E Pr (n E)n, π ,sfor Banach lattices E.15 / 21
5. Complex Banach LatticesMittelmeyer-Wolff (1974) Axiomatization:E a complex vector space, with a function m : E R , satisfying1. m(λx) λ m(x);2. m(m(m(x) m(y)) m(x y)) m(x) m(y) m(x y);3. m(m(y) km(x)) m(y) km(x) k 0 implies x 0;4. E is the R-linear span of m(E).This structure is called a Complex Riesz Space. E is thealgebraic complexification of the real vector space ER m(E) andthis space has a vector lattice structure with m as absolute value.16 / 21
The Krivine Functional Calculus for Banach LatticesFix x1 , . . . , xn E.Let Cn be the vector lattice of all continuous, positivelyhomogeneous real functions on Rn , withkfk sup{ f(t1 , . . . , tn ) : t1 · · · tn 1}17 / 21
The Krivine Functional Calculus for Banach LatticesFix x1 , . . . , xn E.Let Cn be the vector lattice of all continuous, positivelyhomogeneous real functions on Rn , withkfk sup{ f(t1 , . . . , tn ) : t1 · · · tn 1}There exists a unique mapτ : Cn EsatisfyingIτ(tj ) xj for each j 1, . . . , n.Iτ is linear and preserves the lattice operations.Ikτ(f)k 6 kfk x1 · · · xn .17 / 21
Complex Banach Lattices:For z x iy EC , the modulus is defined byq z x 2 y 2 sup x cos θ y sin θ06θ62πEC is a Banach space with the normkzk k z k18 / 21
6. Regular HolomorphyLet E be a complex Banach lattice. The spaces of regularhomogeneous polynomials form a holomorphy type, with acorresponding space Hr (U) of regular holomorphic functions foreach domain U in E.19 / 21
6. Regular HolomorphyLet E be a complex Banach lattice. The spaces of regularhomogeneous polynomials form a holomorphy type, with acorresponding space Hr (U) of regular holomorphic functions foreach domain U in E.PDefinition: A power series n Pn of n-homogeneouspolynomials on E is regularly convergent at z E if each Pn isregular andX Pn ( z ) n19 / 21
6. Regular HolomorphyLet E be a complex Banach lattice. The spaces of regularhomogeneous polynomials form a holomorphy type, with acorresponding space Hr (U) of regular holomorphic functions foreach domain U in E.PDefinition: A power series n Pn of n-homogeneouspolynomials on E is regularly convergent at z E if each Pn isregular andX Pn ( z ) nTheorem: f H(U) is a regular holomorphic function if andonly if f is representable locally in U by regularly pointwiseconvergent power series.19 / 21
Properties of the domain of regular convergencePLet n Pn be a power series whose terms are all regular. Let Dbe the set of points at which the series is regularly convergent.20 / 21
Properties of the domain of regular convergencePLet n Pn be a power series whose terms are all regular. Let Dbe the set of points at which the series is regularly convergent.D is a solid set: if z D and w 6 z , then w D.20 / 21
Properties of the domain of regular convergencePLet n Pn be a power series whose terms are all regular. Let Dbe the set of points at which the series is regularly convergent.D is a solid set: if z D and w 6 z , then w D.The finite dimensional case:When E Ck , the domain of convergence is logarithmicallyconvex:if z, w D, then z θ w 1 θ D for every θ (0, 1)20 / 21
A Hölder Inequality for homogeneous polynomials:Let P be a regular homogeneous polynomial on the (real orcomplex) Banach lattice E and let a, b be positive elements of E.Then θ 1 θP(aθ b1 θ ) 6 P (a) P (b)for every θ (0, 1).Proof:P restricts to a regular polynomial on the principal ideal generatedby u a b.By Fremlin’s theorem, P is an integral polynomial on En C(K).Apply the Hölder inequality.21 / 21
A Hölder Inequality for homogeneous polynomials:Let P be a regular homogeneous polynomial on the (real orcomplex) Banach lattice E and let a, b be positive elements of E.Then θ 1 θP(aθ b1 θ ) 6 P (a) P (b)for every θ (0, 1).Proof:P restricts to a regular polynomial on the principal ideal generatedby u a b.By Fremlin’s theorem, P is an integral polynomial on En C(K).Apply the Hölder inequality.Corollary: The domain of regular convergence of a power seriesof regular homogeneous polynomials is logarithmically convex.21 / 21
terms of nuclear functions/tensor products: P(nE0) P N(nE) 0 (subject to AP) 2/21. I Boland, Dineen (1970’s): Holomorphic functions on nuclear locally convex spaces. For suitable nuclear spaces with basis, the mono
Chapter 4 Holomorphic and real analytic differential geometry In this chapter we develop the basic theory of holomorphic and real analytic man-ifolds. We will be assuming that the reader has a solid background in basic smooth . v 2V: We make the following definition. 4.1.4Definition (Holomorphic and antiholomorphic subspace) Let J be a .
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underpinnings of complex analysis andtoaccentuate thegeometricviewpoint whenever possible. This is evident from the following sample of special topics . holomorphic removability, conformal metrics and Ahlfors’s generalization of the Schwarz lemma, holomorphic (branched) covering maps, and the uniformization theorem for spherical domains. To .
complex analysis notes 3 Laplace’s Equation in 2D - Complex Methods Harmonic functions are solutions of Laplace’s equation. We have seen that the real and imaginary parts of a holomorphic function are harmonic. So, there must be a connection between com-plex functions and solutions of the two-dimensional Laplace equa-tion.
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The Lagrangian Grassmannian 20 The Maslov Index 21 Lecture 4. The Nonsqueezing Theorem and Capacities 23 Preliminaries on J-Holomorphic Curves 26 Lecture 5. Sketch Proof of the Nonsqueezing Theorem 29 Fredholm Theory 30 Compactness 31 Bibliography 33 Helmut Hofer, Holomorphic Curves and Dynamics in Dimension Three 35 Lecture 1.
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