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This work focuses on the properties of vector-valued Fock spaces associated with complex Banach spaces and entire functions.

March 14, 2026Open Access

Some Results on Vector-Valued Fock Spaces

Key Points

This work focuses on the properties of vector-valued Fock spaces associated with complex Banach spaces and entire functions.
Define X-valued Fock spaces with specific growth conditions on entire functions.
Establish criteria for equivalence of Fock spaces in Hilbert settings.
Analyze the behavior of functionals in relation to Fock spaces.
Demonstrated that certain growth conditions on entire functions lead to inclusion in L^p spaces.
Showed equivalence between Fock spaces for Hilbert-valued functions under finite dimensional conditions.
Provided insights into the relationship between functionals and Fock space properties.

Abstract

Abstract Let X be a complex Banach space and denote by Fᵖ_ (X) F α p (X) and F^w, p_ (X) F α w, p (X) the X -valued Fock spaces of entire functions f such that f (z) e^- {2|z|²} Lᵖ (dA) ‖ f (z) ‖ e - α 2 | z | 2 ∈ L p (d A) and x^* (f (z) ) e^- {2|z|²} Lᵖ (dA) x ∗ (f (z) ) e - α 2 | z | 2 ∈ L p (d A) for any x^* X^* x ∗ ∈ X ∗, respectively. For Hilbert-valued functions, it is shown that F²_ (H) =F^w, 2_ (H) F α 2 (H) = F α w, 2 (H) if and only if H is finite dimensional and also that <jats: tex-math

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Blasco et al. (Thu,) studied this question.

synapsesocial.com/papers/69b4fc1fb39f7826a300ccdd https://doi.org/https://doi.org/10.1007/s10958-026-08265-x

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