diff --git a/src/kronecker.jl b/src/kronecker.jl
index f6beefe5..57e1079b 100644
--- a/src/kronecker.jl
+++ b/src/kronecker.jl
@@ -52,6 +52,9 @@ Base.kron(A::KroneckerMap, B::LinearMap) =
     KroneckerMap{promote_type(eltype(A), eltype(B))}(tuple(A.maps..., B))
 Base.kron(A::KroneckerMap, B::KroneckerMap) =
     KroneckerMap{promote_type(eltype(A), eltype(B))}(tuple(A.maps..., B.maps...))
+Base.kron(A::ScaledMap, B::LinearMap) = A.λ * kron(A.lmap, B)
+Base.kron(A::LinearMap{<:RealOrComplex}, B::ScaledMap) = B.λ * kron(A, B.lmap)
+Base.kron(A::ScaledMap, B::ScaledMap) = (A.λ * B.λ) * kron(A.lmap, B.lmap)
 Base.kron(A::LinearMap, B::LinearMap, C::LinearMap, Ds::LinearMap...) =
     kron(kron(A, B), C, Ds...)
 Base.kron(A::AbstractMatrix, B::LinearMap) = kron(LinearMap(A), B)
@@ -104,9 +107,10 @@ Base.:(==)(A::KroneckerMap, B::KroneckerMap) = (eltype(A) == eltype(B) && A.maps
 # multiplication helper functions
 #################
 
-@inline function _kronmul!(y, B, X, At, T)
+@inline function _kronmul!(y, B, x, At, T)
     na, ma = size(At)
     mb, nb = size(B)
+    X = reshape(x, (nb, na))
     v = zeros(T, ma)
     temp1 = similar(y, na)
     temp2 = similar(y, nb)
@@ -119,14 +123,23 @@ Base.:(==)(A::KroneckerMap, B::KroneckerMap) = (eltype(A) == eltype(B) && A.maps
     end
     return y
 end
-@inline function _kronmul!(y, B, X, At::Union{MatrixMap, UniformScalingMap}, T)
+@inline function _kronmul!(y, B, x, At::UniformScalingMap, _)
+    na, ma = size(At)
+    mb, nb = size(B)
+    X = reshape(x, (nb, na))
+    Y = reshape(y, (mb, ma))
+    _unsafe_mul!(Y, B, X, At.λ, false)
+    return y
+end
+@inline function _kronmul!(y, B, x, At::MatrixMap, _)
     na, ma = size(At)
     mb, nb = size(B)
+    X = reshape(x, (nb, na))
     Y = reshape(y, (mb, ma))
     if nb*ma < mb*na
-        _unsafe_mul!(Y, B, Matrix(X*At))
+        _unsafe_mul!(Y, B, X * At.lmap)
     else
-        _unsafe_mul!(Y, Matrix(B*X), _parent(At))
+        _unsafe_mul!(Y, Matrix(B*X), At.lmap)
     end
     return y
 end
@@ -140,18 +153,14 @@ const KroneckerMap2{T} = KroneckerMap{T, <:Tuple{LinearMap, LinearMap}}
 function _unsafe_mul!(y::AbstractVecOrMat, L::KroneckerMap2, x::AbstractVector)
     require_one_based_indexing(y)
     A, B = L.maps
-    X = LinearMap(reshape(x, (size(B, 2), size(A, 2)));
-                    issymmetric = false, ishermitian = false, isposdef = false)
-    _kronmul!(y, B, X, transpose(A), eltype(L))
+    _kronmul!(y, B, x, transpose(A), eltype(L))
     return y
 end
 function _unsafe_mul!(y::AbstractVecOrMat, L::KroneckerMap, x::AbstractVector)
     require_one_based_indexing(y)
     A = first(L.maps)
     B = kron(Base.tail(L.maps)...)
-    X = LinearMap(reshape(x, (size(B, 2), size(A, 2)));
-                    issymmetric = false, ishermitian = false, isposdef = false)
-    _kronmul!(y, B, X, transpose(A), eltype(L))
+    _kronmul!(y, B, x, transpose(A), eltype(L))
     return y
 end
 # mixed-product rule, prefer the right if possible
diff --git a/src/uniformscalingmap.jl b/src/uniformscalingmap.jl
index 350812bf..175c1be5 100644
--- a/src/uniformscalingmap.jl
+++ b/src/uniformscalingmap.jl
@@ -17,7 +17,6 @@ MulStyle(::UniformScalingMap) = FiveArg()
 
 # properties
 Base.size(A::UniformScalingMap) = (A.M, A.M)
-_parent(A::UniformScalingMap) = A.λ
 Base.isreal(A::UniformScalingMap) = isreal(A.λ)
 LinearAlgebra.issymmetric(::UniformScalingMap) = true
 LinearAlgebra.ishermitian(A::UniformScalingMap) = isreal(A)
diff --git a/src/wrappedmap.jl b/src/wrappedmap.jl
index 4631c246..48528c13 100644
--- a/src/wrappedmap.jl
+++ b/src/wrappedmap.jl
@@ -38,7 +38,6 @@ Base.:(==)(A::MatrixMap, B::MatrixMap) =
 
 # properties
 Base.size(A::WrappedMap) = size(A.lmap)
-_parent(A::WrappedMap) = A.lmap
 LinearAlgebra.issymmetric(A::WrappedMap) = A._issymmetric
 LinearAlgebra.ishermitian(A::WrappedMap) = A._ishermitian
 LinearAlgebra.isposdef(A::WrappedMap) = A._isposdef
diff --git a/test/kronecker.jl b/test/kronecker.jl
index 7acd6e78..1c511a6f 100644
--- a/test/kronecker.jl
+++ b/test/kronecker.jl
@@ -8,6 +8,10 @@ using Test, LinearMaps, LinearAlgebra, SparseArrays
         LA = LinearMap(A)
         LB = LinearMap(B)
         LK = @inferred kron(LA, LB)
+        @test kron(LA, 2LB) isa LinearMaps.ScaledMap
+        @test kron(3LA, LB) isa LinearMaps.ScaledMap
+        @test kron(3LA, 2LB) isa LinearMaps.ScaledMap
+        @test kron(3LA, 2LB).λ == 6
         @test_throws ErrorException LinearMaps.KroneckerMap{Float64}((LA, LB))
         @test occursin("6×6 LinearMaps.KroneckerMap{$(eltype(LK))}", sprint((t, s) -> show(t, "text/plain", s), LK))
         @test @inferred size(LK) == size(K)
diff --git a/test/uniformscalingmap.jl b/test/uniformscalingmap.jl
index 8b158701..0a958cdb 100644
--- a/test/uniformscalingmap.jl
+++ b/test/uniformscalingmap.jl
@@ -11,7 +11,6 @@ using Test, LinearMaps, LinearAlgebra, BenchmarkTools
     w = similar(v)
     Id = @inferred LinearMap(I, 10)
     @test occursin("10×10 LinearMaps.UniformScalingMap{Bool}", sprint((t, s) -> show(t, "text/plain", s), Id))
-    @test LinearMaps._parent(Id) == true
     @test_throws ErrorException LinearMaps.UniformScalingMap(1, 10, 20)
     @test_throws ErrorException LinearMaps.UniformScalingMap(1, (10, 20))
     @test size(Id) == (10, 10)
diff --git a/test/wrappedmap.jl b/test/wrappedmap.jl
index 485ea99b..db36fb1b 100644
--- a/test/wrappedmap.jl
+++ b/test/wrappedmap.jl
@@ -7,7 +7,6 @@ using Test, LinearMaps, LinearAlgebra
     SB = B'B + I
     L = @inferred LinearMap{Float64}(A)
     @test occursin("10×20 LinearMaps.WrappedMap{Float64}", sprint((t, s) -> show(t, "text/plain", s), L))
-    @test LinearMaps._parent(L) === A
     MA = @inferred LinearMap(SA)
     MB = @inferred LinearMap(SB)
     @test eltype(Matrix{Complex{Float32}}(LinearMap(A))) <: Complex