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Banach space of analytic functions with infinite-dimensional domains

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The primary interest here is in the interplay between function theory on infinite-dimensional domains, geometric properties of Banach spaces, and Banach and Fréchet algebras. Throughout, will denote a complex Banach space with open unit ball .

Definition and basic properties.

Let denote the space of complex-valued -homogeneous polynomials , i.e. functions to which is associated a continuous -linear function such that for all . Each such polynomial is associated with a unique symmetric -linear form via the polarization formula. For an open subset , one says that is holomorphic, or analytic, if has a complex Fréchet derivative at each point of (cf. also Algebra of functions). Equivalently, is holomorphic if at each point there is a sequence of -homogeneous polynomials such that for all in a neighbourhood of . If , then the algebra of holomorphic functions from to always contains as a proper subset the subalgebra of holomorphic functions which are bounded on bounded subsets such that . The latter space is a Fréchet algebra with metric determined by countably many such subsets, whereas there are a number of natural topologies on .

The natural analogues of the classical Banach algebras of analytic functions are the following:

;

All are Banach algebras with identity when endowed with the supremum norm (cf. also Banach algebra).

Results and problems.

For any of the above algebras of analytic functions, let denote the set of homomorphisms . Since the Michael problem has an affirmative solution [a5], every homomorphism is automatically continuous. For each such , define (noting that, always, ). Basic topics of interest here are the relation between the "fibres" , , and the relation between the geometry of and of .

The spectrum displays very different behaviour in the infinite-dimensional setting, in comparison with the finite-dimensional situation. As an illustration, every element corresponds to a homomorphism on . Indeed, for each there is a linear extension mapping from . Applying this mapping to the Taylor series of a holomorphic function yields a multiplicative linear extension operator, mapping to ; similar results hold for and . For example, each yields an element of via . A complete description of is unknown (1998) for general , although it is not difficult to see that . The question of whether the fourth dual of also provides points of the spectrum is connected with Arens regularity of [a7]. In any case, can be made into a semi-group with identity ; the commutativity of this semi-group is related, once again, to Arens regularity of [a6].

It is natural to look for analytic structure in the spectrum . In fact, every fibre over contains a copy of . In many situations, e.g. when is super-reflexive (cf., also Reflexive space), there is an analytic embedding of the unit ball of a non-separable Hilbert space into . Further information has been obtained by J. Farmer [a8], who has studied analytic structure in fibres in -spaces. However, note that there is a peak set (cf. also Algebra of functions) for which is contained in .

There has also been recent (1998) interest in the following areas:

reflexivity of ;

algebras of weakly continuous holomorphic functions; and

Banach-algebra-valued holomorphic mappings.

Basic references on holomorphic functions in infinite dimensions are [a1], [a2], [a3]; a recent (1998) very helpful source, with an extensive bibliography, is [a4].

References

[a1] S. Dineen, "Complex analysis in localy convex spaces" , North-Holland (1981)
[a2] S. Dineen, "Complex analysis on infinite dimensional spaces" , Springer (1999)
[a3] J. Mujica, "Complex analysis in Banach spaces" , North-Holland (1986)
[a4] T. Gamelin, "Analytic functions on Banach spaces" , Complex Potential Theory (Montreal 1993) , NATO Adv. Sci. Inst. Ser. C Math. Phys. Sci. , 439 , Kluwer Acad. Publ. (1994) pp. 187–233
[a5] B. Stensones, "A proof of the Michael conjecture" preprint (1999)
[a6] R. Aron, B. Cole, T. Gamelin, "Spectra of algebras of analytic functions on a Banach space" J. Reine Angew. Math. , 415 (1991) pp. 51–93
[a7] R. Aron, P. Galindo, D. Garcia, M. Maestre, "Regularity and algebras of analytic functions in infinite dimensions" Trans. Amer. Math. Soc. , 384 : 2 (1996) pp. 543–559
[a8] J. Farmer, "Fibers over the sphere of a uniformly convex Banach space" Michigan Math. J. , 45 : 2 (1998) pp. 211–226
How to Cite This Entry:
Banach space of analytic functions with infinite-dimensional domains. Richard M. Aron (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Banach_space_of_analytic_functions_with_infinite-dimensional_domains&oldid=18803
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098