We develop several ℓ p -operator norm inequalities for k × k block matrices defined on the ℓ p -sum of Banach spaces. Using these inequalities, we obtain p-numerical radius and spectralradius bounds for k × k block matrices. We deduce a p-numerical radius bound for the Kroneckerproduct A ⊗ B, where A ∈ Mk (C) is a k × k complex matrix and B ∈ L (H) is a bounded linearoperator on a complex Hilbert space H. This improves and extends the Holbrook’s bound w ( A ⊗B) ≤ w ( A) ∥ B∥. If ∥ A∥ℓ p and wp ( A) denote the ℓ p -operator norm and p-numerical radius ofA ∈ Mk (C), respectively, then it is shown that