diff --git a/Project.toml b/Project.toml
index c790519..6c99f47 100644
--- a/Project.toml
+++ b/Project.toml
@@ -1,7 +1,7 @@
 name = "KroneckerArrays"
 uuid = "05d0b138-81bc-4ff7-84be-08becefb1ccc"
 authors = ["ITensor developers <support@itensor.org> and contributors"]
-version = "0.1.14"
+version = "0.1.15"
 
 [deps]
 Adapt = "79e6a3ab-5dfb-504d-930d-738a2a938a0e"
@@ -10,6 +10,7 @@ DiagonalArrays = "74fd4be6-21e2-4f6f-823a-4360d37c7a77"
 FillArrays = "1a297f60-69ca-5386-bcde-b61e274b549b"
 GPUArraysCore = "46192b85-c4d5-4398-a991-12ede77f4527"
 LinearAlgebra = "37e2e46d-f89d-539d-b4ee-838fcccc9c8e"
+MapBroadcast = "ebd9b9da-f48d-417c-9660-449667d60261"
 MatrixAlgebraKit = "6c742aac-3347-4629-af66-fc926824e5e4"
 
 [weakdeps]
@@ -28,5 +29,6 @@ DiagonalArrays = "0.3.5"
 FillArrays = "1.13.0"
 GPUArraysCore = "0.2.0"
 LinearAlgebra = "1.10"
+MapBroadcast = "0.1.9"
 MatrixAlgebraKit = "0.2.0"
 julia = "1.10"
diff --git a/src/cartesianproduct.jl b/src/cartesianproduct.jl
index c734253..c77a2f5 100644
--- a/src/cartesianproduct.jl
+++ b/src/cartesianproduct.jl
@@ -75,3 +75,12 @@ for f in (:+, :-)
     end
   end
 end
+
+using Base.Broadcast: axistype
+function Base.Broadcast.axistype(
+  r1::CartesianProductUnitRange, r2::CartesianProductUnitRange
+)
+  prod = axistype(arg1(r1), arg1(r2)) × axistype(arg2(r1), arg2(r2))
+  range = axistype(unproduct(r1), unproduct(r2))
+  return cartesianrange(prod, range)
+end
diff --git a/src/fillarrays/kroneckerarray.jl b/src/fillarrays/kroneckerarray.jl
index 689f1e4..0a39106 100644
--- a/src/fillarrays/kroneckerarray.jl
+++ b/src/fillarrays/kroneckerarray.jl
@@ -1,3 +1,13 @@
+using FillArrays: FillArrays, Zeros
+function FillArrays.fillsimilar(
+  a::Zeros{T},
+  ax::Tuple{
+    CartesianProductUnitRange{<:Integer},Vararg{CartesianProductUnitRange{<:Integer}}
+  },
+) where {T}
+  return Zeros{T}(arg1.(ax)) ⊗ Zeros{T}(arg2.(ax))
+end
+
 using FillArrays: RectDiagonal, OnesVector
 const RectEye{T,V<:OnesVector{T},Axes} = RectDiagonal{T,V,Axes}
 
@@ -208,3 +218,17 @@ end
 function Base.map!(f::Base.Fix2{typeof(*),<:Number}, dest::EyeEye, a::EyeEye)
   return error("Can't write in-place.")
 end
+
+using Base.Broadcast:
+  AbstractArrayStyle, AbstractArrayStyle, BroadcastStyle, Broadcasted, broadcasted
+
+struct EyeStyle <: AbstractArrayStyle{2} end
+EyeStyle(::Val{2}) = EyeStyle()
+function _BroadcastStyle(::Type{<:Eye})
+  return EyeStyle()
+end
+Base.BroadcastStyle(style1::EyeStyle, style2::EyeStyle) = EyeStyle()
+
+function Base.similar(bc::Broadcasted{EyeStyle}, elt::Type)
+  return Eye{elt}(axes(bc))
+end
diff --git a/src/kroneckerarray.jl b/src/kroneckerarray.jl
index 5bbae8c..05f9b26 100644
--- a/src/kroneckerarray.jl
+++ b/src/kroneckerarray.jl
@@ -226,6 +226,8 @@ end
 for op in (:+, :-)
   @eval begin
     function Base.$op(a::KroneckerArray, b::KroneckerArray)
+      iszero(a) && return $op(b)
+      iszero(b) && return a
       if a.b == b.b
         return $op(a.a, b.a) ⊗ a.b
       elseif a.a == b.a
@@ -241,8 +243,15 @@ for op in (:+, :-)
   end
 end
 
-using Base.Broadcast: AbstractArrayStyle, BroadcastStyle, Broadcasted
+# Allows for customizations for FillArrays.
+_BroadcastStyle(x) = BroadcastStyle(x)
+
+using Base.Broadcast: Broadcast, AbstractArrayStyle, BroadcastStyle, Broadcasted
 struct KroneckerStyle{N,A,B} <: AbstractArrayStyle{N} end
+arg1(::Type{<:KroneckerStyle{<:Any,A}}) where {A} = A
+arg1(style::KroneckerStyle) = arg1(typeof(style))
+arg2(::Type{<:KroneckerStyle{<:Any,B}}) where {B} = B
+arg2(style::KroneckerStyle) = arg2(typeof(style))
 function KroneckerStyle{N}(a::BroadcastStyle, b::BroadcastStyle) where {N}
   return KroneckerStyle{N,a,b}()
 end
@@ -253,30 +262,69 @@ function KroneckerStyle{N,A,B}(v::Val{M}) where {N,A,B,M}
   return KroneckerStyle{M,typeof(A)(v),typeof(B)(v)}()
 end
 function Base.BroadcastStyle(::Type{<:KroneckerArray{<:Any,N,A,B}}) where {N,A,B}
-  return KroneckerStyle{N}(BroadcastStyle(A), BroadcastStyle(B))
+  return KroneckerStyle{N}(_BroadcastStyle(A), _BroadcastStyle(B))
 end
 function Base.BroadcastStyle(style1::KroneckerStyle{N}, style2::KroneckerStyle{N}) where {N}
-  return KroneckerStyle{N}(
-    BroadcastStyle(style1.a, style2.a), BroadcastStyle(style1.b, style2.b)
-  )
+  style_a = BroadcastStyle(arg1(style1), arg1(style2))
+  (style_a isa Broadcast.Unknown) && return Broadcast.Unknown()
+  style_b = BroadcastStyle(arg2(style1), arg2(style2))
+  (style_b isa Broadcast.Unknown) && return Broadcast.Unknown()
+  return KroneckerStyle{N}(style_a, style_b)
 end
 function Base.similar(bc::Broadcasted{<:KroneckerStyle{N,A,B}}, elt::Type) where {N,A,B}
-  ax_a = map(ax -> ax.product.a, axes(bc))
-  ax_b = map(ax -> ax.product.b, axes(bc))
+  ax_a = arg1.(axes(bc))
+  ax_b = arg2.(axes(bc))
   bc_a = Broadcasted(A, nothing, (), ax_a)
   bc_b = Broadcasted(B, nothing, (), ax_b)
   a = similar(bc_a, elt)
   b = similar(bc_b, elt)
   return a ⊗ b
 end
+# Fallback definition of broadcasting falls back to `map` but assumes
+# inputs have been canonicalized to a map-compatible expression already,
+# for example by absorbing scalar arguments into the function.
 function Base.copyto!(dest::AbstractArray, bc::Broadcasted{<:KroneckerStyle})
-  return throw(
-    ArgumentError(
-      "Arbitrary broadcasting is not supported for KroneckerArrays since they might not preserve the Kronecker structure.",
-    ),
-  )
+  allequal(axes, bc.args) || throw(ArgumentError("Broadcasted axes must be equal."))
+  map!(bc.f, dest, bc.args...)
+  return dest
 end
 
+# Broadcast rewrite rules. Canonicalize inputs to absorb scalar inputs into the
+# function.
+function Base.broadcasted(style::KroneckerStyle, ::typeof(*), a::Number, b::KroneckerArray)
+  return broadcasted(style, Base.Fix1(*, a), b)
+end
+function Base.broadcasted(style::KroneckerStyle, ::typeof(*), a::KroneckerArray, b::Number)
+  return broadcasted(style, Base.Fix2(*, b), a)
+end
+function Base.broadcasted(style::KroneckerStyle, ::typeof(/), a::KroneckerArray, b::Number)
+  return broadcasted(style, Base.Fix2(/, b), a)
+end
+using MapBroadcast: MapBroadcast, MapFunction
+function Base.broadcasted(
+  style::KroneckerStyle,
+  f::MapFunction{typeof(*),<:Tuple{<:Number,MapBroadcast.Arg}},
+  a::KroneckerArray,
+)
+  return broadcasted(style, Base.Fix1(*, f.args[1]), a)
+end
+function Base.broadcasted(
+  style::KroneckerStyle,
+  f::MapFunction{typeof(*),<:Tuple{MapBroadcast.Arg,<:Number}},
+  a::KroneckerArray,
+)
+  return broadcasted(style, Base.Fix2(*, f.args[2]), a)
+end
+function Base.broadcasted(
+  style::KroneckerStyle,
+  f::MapFunction{typeof(/),<:Tuple{MapBroadcast.Arg,<:Number}},
+  a::KroneckerArray,
+)
+  return broadcasted(style, Base.Fix2(/, f.args[2]), a)
+end
+
+# TODO: Define by converting to a broadcast expession (with MapBroadcast.jl)
+# and then constructing the output with `similar`.
 function Base.map(f, a1::KroneckerArray, a_rest::KroneckerArray...)
   return throw(
     ArgumentError(
@@ -312,6 +360,8 @@ for f in [:+, :-]
     function Base.map!(
       ::typeof($f), dest::KroneckerArray, a::KroneckerArray, b::KroneckerArray
     )
+      iszero(b) && return map!(identity, dest, a)
+      iszero(a) && return map!($f, dest, b)
       if a.b == b.b
         map!($f, dest.a, a.a, b.a)
         map!(identity, dest.b, a.b)
@@ -350,6 +400,15 @@ for op in [:*, :/]
     end
   end
 end
+for f in [:+, :-]
+  @eval begin
+    function Base.map!(::typeof($f), dest::KroneckerArray, src::KroneckerArray)
+      map!($f, dest.a, src.a)
+      map!(identity, dest.b, src.b)
+      return dest
+    end
+  end
+end
 
 using DiagonalArrays: DiagonalArrays, diagonal
 function DiagonalArrays.diagonal(a::KroneckerArray)
diff --git a/src/linearalgebra.jl b/src/linearalgebra.jl
index 2218363..95bed45 100644
--- a/src/linearalgebra.jl
+++ b/src/linearalgebra.jl
@@ -15,6 +15,15 @@ using LinearAlgebra:
   svdvals,
   tr
 
+using LinearAlgebra: LinearAlgebra
+function KroneckerArray(J::LinearAlgebra.UniformScaling, ax::Tuple)
+  return Eye{eltype(J)}(arg1.(ax)) ⊗ Eye{eltype(J)}(arg2.(ax))
+end
+function Base.copyto!(a::KroneckerArray, J::LinearAlgebra.UniformScaling)
+  copyto!(a, KroneckerArray(J, axes(a)))
+  return a
+end
+
 using LinearAlgebra: LinearAlgebra, pinv
 function LinearAlgebra.pinv(a::KroneckerArray; kwargs...)
   return pinv(a.a; kwargs...) ⊗ pinv(a.b; kwargs...)
diff --git a/test/test_basics.jl b/test/test_basics.jl
index 45f6403..47460d3 100644
--- a/test/test_basics.jl
+++ b/test/test_basics.jl
@@ -87,14 +87,15 @@ elts = (Float32, Float64, ComplexF32, ComplexF64)
   a′ = similar(a)
   @test_throws "not supported" a′ .= sin.(a)
   a′ = similar(a)
-  @test_broken a′ .= 2 .* a
+  a′ .= 2 .* a
+  @test collect(a′) ≈ 2 * collect(a)
   bc = broadcasted(+, a, a)
   @test bc.style === style
   @test similar(bc, elt) isa KroneckerArray{elt,2,typeof(a.a),typeof(a.b)}
-  @test_broken copy(bc)
+  @test collect(copy(bc)) ≈ 2 * collect(a)
   bc = broadcasted(*, 2, a)
   @test bc.style === style
-  @test_broken copy(bc)
+  @test collect(copy(bc)) ≈ 2 * collect(a)
 
   # Mapping
   a = randn(elt, 2, 2) ⊗ randn(elt, 3, 3)
diff --git a/test/test_blocksparsearrays.jl b/test/test_blocksparsearrays.jl
index 0f2ea3a..178e6bd 100644
--- a/test/test_blocksparsearrays.jl
+++ b/test/test_blocksparsearrays.jl
@@ -64,12 +64,11 @@ arrayts = (Array, JLArray)
     @test_broken inv(a)
   end
 
-  if (VERSION ≤ v"1.11-" && arrayt === Array && elt <: Complex) ||
-    (arrayt === Array && elt <: Real)
+  if arrayt === Array
     u, s, v = svd_compact(a)
     @test Array(u * s * v) ≈ Array(a)
   else
-    # Broken on GPU and for complex, investigate.
+    # Broken on GPU.
     @test_broken svd_compact(a)
   end
 
@@ -135,14 +134,17 @@ end
     @test_broken exp(a)
   end
 
-  if VERSION < v"1.11-" && elt <: Complex
-    # Broken because of type stability issue in Julia v1.10.
-    @test_broken svd_compact(a)
-  elseif arrayt === Array
+  ## if VERSION < v"1.11-" && elt <: Complex
+  ##   # Broken because of type stability issue in Julia v1.10.
+  ##   @test_broken svd_compact(a)
+  if arrayt === Array
     u, s, v = svd_compact(a)
     @test u * s * v ≈ a
-    @test blocktype(u) === blocktype(a)
-    @test blocktype(v) === blocktype(a)
+    @test blocktype(u) >: blocktype(u)
+    @test eltype(u) === eltype(a)
+    @test blocktype(v) >: blocktype(a)
+    @test eltype(v) === eltype(a)
+    @test eltype(s) === real(eltype(a))
   else
     @test_broken svd_compact(a)
   end
@@ -150,4 +152,16 @@ end
   # Broken operations
   @test_broken inv(a)
   @test_broken a[Block.(1:2), Block(2)]
+
+  @testset "Block deficient" begin
+    d = Dict(Block(1, 1) => Eye{elt}(2, 2) ⊗ dev(randn(elt, 2, 2)))
+    a = @constinferred dev(blocksparse(d, r, r))
+
+    d = Dict(Block(2, 2) => Eye{elt}(3, 3) ⊗ dev(randn(elt, 3, 3)))
+    b = @constinferred dev(blocksparse(d, r, r))
+
+    @test_broken a + b
+    # @test Array(a + b) ≈ Array(a) + Array(b)
+    # @test Array(2a) ≈ 2Array(a)
+  end
 end