Generalized tri-circular projections on some vector-valued spaces of analytic functions

In this paper, we characterize generalized tri-circular projections on the following three vector-valued spaces of analytic functions on the open unit disk $\mathbb{D}$: the Hardy space $H^p(\mathcal{K})$ for a complex separable Hilbert space $\mathcal{K}$ and $1\le p<\mathrm{\infty },p\mathrm{\ne }2$; the space $H^{\mathrm{\infty }}(E)$ of $E$-valued bounded analytic functions for a Banach space $E$ with trivial multiplier algebra; and the little Bloch spaces ${\mathcal{B}}_0(\mathbb{D},E)$ and ${\mathcal{B}}_{\ast }(\mathbb{D},E)$, for $E$ smooth, strictly convex, and reflexive.