SOME RESULTS ON VECTOR-VALUED FOCK SPACES

Abstract Let X be a complex Banach space and denote by $$F^p_\alpha (X)$$ F α p ( X ) and $$F^{w,p}_\alpha (X)$$ F α w , p ( X ) the X -valued Fock spaces of entire functions f such that $$\Vert f(z)\Vert e^{-\frac{\alpha }{2}|z|^2}\in L^p(dA)$$ ‖ f ( z ) ‖ e - α 2 | z | 2 ∈ L p ( d A ) and $$x^*(f(z))e^{-\frac{\alpha }{2}|z|^2}\in L^p(dA)$$ x ∗ ( f ( z ) ) e - α 2 | z | 2 ∈ L p ( d A ) for any $$x^*\in X^*$$ x ∗ ∈ X ∗ , respectively. For Hilbert-valued functions, it is shown that $$F^2_\alpha (H)=F^{w,2}_\alpha (H)$$ F α 2 ( H ) = F α w , 2 ( H ) if and only if H is finite dimensional and also that