Some spaces of entire and nuclearly entire functions on a Banach space. II.

In Part I of "Some Spaces of Entire and Nuclearly Entire Functions on a Banach Space" the topological properties of weighted spaces of nuclearly entire and entire functions on a Banach space were investigated. Part II is principally concerned with the study of convolution operators on these spaces. The notation of Part I is continued in Part II. Part I is published in vol. 270 of this Journal. Convolution operators on $l)Nb(E, @) are investigated in Section III and it is shown that when @ is an increasing -sequence space, the algebra of convolution operators on $&Nb(E, @) can be identified in a natural way with $Q*Nb(E, @). Section IV is devoted to the study of spaces $)Nb(E, @) for which Malgrange-Gupta type Approximation Theorems hold. It is shown that for a wide class of these spaces, the kernel of any convolution operator is the closure of the subspace generated by the exponential polynomial functions in the kernel. Those spaces belongingto this class which are in addition metrizable satisfy the property that any non-zero convolution operator on the space is onto. The section concludes with a partial table of examples, which in particular shows the duality relationship between §Nb(E, @) and $Qb(E', @) when @ is perfect. The final section deals with the growth of entire functions on a Banach space with values in a complex normed linear space relative to growth of composition with linear forms on the ränge space and linear transformations int o the d omain space, äs well äs relative to the growth of one dimensional complex line restrictions. In particular it is shown that an entire function on a Banach space with values in a complex normed linear space is of exponential type **> each of its restrictions to one dimensional subspaces is an entire function of exponential type. Examples are given to emphasize the importance of the condition that the domain of definition be a Banach space äs opposed to an arbitrary complex normed linear space.