Suppose A=[aij]∈Mn(C) is a complex n×n matrix and B∈B(H) is a bounded linear operator on a complex Hilbert space H. We show that w(A⊗B)≤w(C), where w(⋅) denotes the numerical radius and C=[cij] with cij=w([0aijaji0]⊗B). This refines Holbrook’s classical bound w(A⊗B)≤w(A)‖B‖ (1969) [31], when all entries of A are non-negative. If moreover aii≠0 ∀i, we prove that w(A⊗B)=w(A)‖B‖ if and only if w(B)=‖B‖. We then extend these and other results to the more general setting of semi-Hilbertian spaces induced by a positive operator. In the reverse direction, we also specialize these results to Kronecker products and hence to Schur/entrywise products, of matrices: (1)(a) We first provide an alternate proof (using w(A)) of a result of Goldberg and Zwas (1974) [24] that if the spectral norm of A equals its spectral radius, then each Jordan block for each maximum-modulus eigenvalue must be 1×1 (“partial diagonalizability”). (b) Using our approach, we further show given m≥1 that w(A∘m)≤wm(A) – we also characterize when equality holds here. (2) We provide upper and lower bounds for the ℓp operator norm and the numerical radius of A⊗B for all A∈Mn(C), which become equal when restricted to doubly stochastic matrices A. Finally, using these bounds we obtain an improved estimation for the roots of an arbitrary complex polynomial.