In this paper, we characterize the \(d\times d\) matrix-valued weights W on the complex plane \({\mathbb {C}}\) such that the Fock projection \(P_{\alpha }\) is bounded on the vector-valued spaces \(L^2_{\alpha ,W}({\mathbb {C}}^d)\) induced by W. It is proved that \(P_{\alpha }\) is bounded on \(L^2_{\alpha ,W}({\mathbb {C}}^d)\) if and only if W satisfies a restricted \({\textbf{A}}_2\) -condition. Then we establish some function-theoretic and operator-theoretic properties for the Fock spaces \(F^2_{\alpha ,W}({\mathbb {C}}^d)\) induced by the \(d\times d\) matrix-valued weights W satisfying the restricted \({\textbf{A}}_2\) -condition: we show that the \({\mathbb {C}}^d\) -valued polynomials are dense in \(F^2_{\alpha ,W}({\mathbb {C}}^d)\) ; a Littlewood–Paley formula for \(F^2_{\alpha ,W}({\mathbb {C}}^d)\) is established; the bounded differentiation and integration operators \(D^{(n)}:F^2_{\alpha ,W}({\mathbb {C}}^d)\rightarrow L^2(\Phi dA;{\mathbb {C}}^d)\) are characterized, where \(\Phi \) is a nonnegative matrix-valued function; we also investigate the boundedness of the Volterra type integration operator \(T_G\) acting on \(F^2_{\alpha ,W}({\mathbb {C}}^d)\) , where G is a matrix-valued entire function. In particular, it is shown that for \(d\ge 2\) , \(T_{G}\) may be unbounded on \(F^2_{\alpha ,W}({\mathbb {C}}^d)\) when G is a linear polynomial, and \(T_{G}\) may be compact on \(F^2_{\alpha ,W}({\mathbb {C}}^d)\) when G is a polynomial of degree greater than 3. These phenomena are in sharp contrast with the case \(d=1\) , where \(T_{G}\) is bounded (resp. compact) if and only if G is a polynomial of degree not more than 2 (resp. a linear polynomial).
