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On types of polynomials and holomorphic functions on Banach spaces. (English) Zbl 0773.46020

In 1966 L. Nachbin introduced the notion of a holomorphy type to consider certain types of polynomials (f.i. compact, nuclear, absolutely summing) in a uniform way. Holomorphy types with special properties were studied by S. Dineen in 1971. Using the well developed theory of linear operator ideals various methods of the construction of holomorphy types were presented in the dissertation of the first named author.
These methods work also in the case of \(p\)-normed quasinormed ideals. After introducing the basic notions the factorization method will be studied here in more details.
Of special interest are multilinear operators of type \({\mathcal L}({\mathcal I}_ p)\) where \({\mathcal I}_ p\) denotes the usual Schatten class of linear operators in Hilbert spaces. These multilinear operators can be characterized by the summability of their eigenvalues or some other sort of associated sequences of reals. The results will be applied to multilinear operators defined by kernels or convolutions and to holomorphic functions of ideal type.

MSC:

46G20 Infinite-dimensional holomorphy
47L20 Operator ideals
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