diff --git a/src/blockmap.jl b/src/blockmap.jl
index 11430249..1c53db57 100644
--- a/src/blockmap.jl
+++ b/src/blockmap.jl
@@ -81,7 +81,7 @@ julia> L * ones(Int, 6)
  6
 ```
 """
-function Base.hcat(As::Union{LinearMap, UniformScaling, AbstractVecOrMatOrQ}...)
+function Base.hcat(As::Union{LinearMap, UniformScaling, AbstractArray, AbstractQ}...)
     T = promote_type(map(eltype, As)...)
     nbc = length(As)
 
@@ -119,7 +119,7 @@ julia> L * ones(Int, 3)
  3
 ```
 """
-function Base.vcat(As::Union{LinearMap,UniformScaling,AbstractVecOrMatOrQ}...)
+function Base.vcat(As::Union{LinearMap, UniformScaling, AbstractArray, AbstractQ}...)
     T = promote_type(map(eltype, As)...)
     nbr = length(As)
 
@@ -162,10 +162,7 @@ julia> L * ones(Int, 6)
  6
 ```
 """
-Base.hvcat
-
-function Base.hvcat(rows::Tuple{Vararg{Int}},
-                    As::Union{LinearMap, UniformScaling, AbstractVecOrMatOrQ}...)
+function Base.hvcat(rows::Tuple{Vararg{Int}}, As::Union{LinearMap, UniformScaling, AbstractArray, AbstractQ}...)
     nr = length(rows)
     T = promote_type(map(eltype, As)...)
     sum(rows) == length(As) ||
diff --git a/src/fillmap.jl b/src/fillmap.jl
index a527850b..2903cc6f 100644
--- a/src/fillmap.jl
+++ b/src/fillmap.jl
@@ -18,7 +18,7 @@ FillMap(λ, m::Int, n::Int) = FillMap(λ, (m, n))
 
 # properties
 Base.size(A::FillMap) = A.size
-MulStyle(A::FillMap) = FiveArg()
+MulStyle(::FillMap) = FiveArg()
 LinearAlgebra.issymmetric(A::FillMap) = A.size[1] == A.size[2]
 LinearAlgebra.ishermitian(A::FillMap) = isreal(A.λ) && A.size[1] == A.size[2]
 LinearAlgebra.isposdef(A::FillMap) = (size(A, 1) == size(A, 2) == 1 && isposdef(A.λ))
diff --git a/src/getindex.jl b/src/getindex.jl
index bffacc45..f88535a2 100644
--- a/src/getindex.jl
+++ b/src/getindex.jl
@@ -47,7 +47,7 @@ _getindex(A::LinearMap, I::Indexer, J::Indexer) =
     error("partial two-dimensional slicing of LinearMaps is not supported, consider using A[:,J][I] or A[I,:][J] instead")
 
 _getindex(A::LinearMap, ::Base.Slice, j::Integer) = A*unitvec(A, 2, j)
-function _getindex(A::LinearMap, i::Integer, J::Base.Slice)
+function _getindex(A::LinearMap, i::Integer, ::Base.Slice)
     try
         # requires adjoint action to be defined
         return vec(unitvec(A, 1, i)'A)
diff --git a/src/left.jl b/src/left.jl
index 90159efa..f2bf2ea0 100644
--- a/src/left.jl
+++ b/src/left.jl
@@ -92,7 +92,7 @@ julia> mul!(C, A, B, 1, 1)
  5.0  7.0
 ```
 """
-function mul!(X::AbstractMatrix, Y::AbstractMatrix, A::LinearMap, α, β)
+function mul!(X::AbstractMatrix, Y::AbstractMatrix, A::LinearMap, α::Number, β::Number)
     check_dim_mul(X, Y, A)
     _unsafe_mul!(X, Y, A, α, β)
 end
diff --git a/src/uniformscalingmap.jl b/src/uniformscalingmap.jl
index 86d6ba2a..d162c840 100644
--- a/src/uniformscalingmap.jl
+++ b/src/uniformscalingmap.jl
@@ -42,9 +42,9 @@ Base.:(*)(J::UniformScalingMap, α::RealOrComplex) = UniformScalingMap(J.λ * α
 Base.:(*)(J::UniformScalingMap, x::AbstractVector) =
     length(x) == J.M ? J.λ * x : throw(DimensionMismatch("*"))
 # multiplication with matrix
-Base.:(*)(J::UniformScalingMap, B::AbstractMatrix) =
+Base.:(*)(J::UniformScalingMap, B::Union{AbstractMatrix,AbstractQ}) =
     size(B, 1) == J.M ? J.λ * LinearMap(B) : throw(DimensionMismatch("*"))
-Base.:(*)(A::AbstractMatrix, J::UniformScalingMap) =
+Base.:(*)(A::Union{AbstractMatrix,AbstractQ}, J::UniformScalingMap) =
     size(A, 2) == J.M ? LinearMap(A) * J.λ : throw(DimensionMismatch("*"))
 # disambiguation
 Base.:(*)(xc::LinearAlgebra.AdjointAbsVec, J::UniformScalingMap) = xc * J.λ
diff --git a/src/wrappedmap.jl b/src/wrappedmap.jl
index abe9bafa..e5f2d220 100644
--- a/src/wrappedmap.jl
+++ b/src/wrappedmap.jl
@@ -108,13 +108,13 @@ _unsafe_mul!(Y, A::VecOrMatMap, s::Number) = _unsafe_mul!(Y, A.lmap, s)
 _unsafe_mul!(Y, A::VecOrMatMap, s::Number, α, β) = _unsafe_mul!(Y, A.lmap, s, α, β)
 
 # combine LinearMap and Matrix objects: linear combinations and map composition
-Base.:(+)(A₁::LinearMap, A₂::AbstractMatrix) = +(A₁, WrappedMap(A₂))
-Base.:(+)(A₁::AbstractMatrix, A₂::LinearMap) = +(WrappedMap(A₁), A₂)
-Base.:(-)(A₁::LinearMap, A₂::AbstractMatrix) = -(A₁, WrappedMap(A₂))
-Base.:(-)(A₁::AbstractMatrix, A₂::LinearMap) = -(WrappedMap(A₁), A₂)
+Base.:(+)(A₁::LinearMap, A₂::Union{AbstractMatrix,AbstractQ}) = +(A₁, WrappedMap(A₂))
+Base.:(+)(A₁::Union{AbstractMatrix,AbstractQ}, A₂::LinearMap) = +(WrappedMap(A₁), A₂)
+Base.:(-)(A₁::LinearMap, A₂::Union{AbstractMatrix,AbstractQ}) = -(A₁, WrappedMap(A₂))
+Base.:(-)(A₁::Union{AbstractMatrix,AbstractQ}, A₂::LinearMap) = -(WrappedMap(A₁), A₂)
 
 """
-    *(A::LinearMap, X::AbstractMatrix)::CompositeMap
+    *(A::LinearMap, X::Union{AbstractMatrix,AbstractQ})::CompositeMap
 
 Return the `CompositeMap` `A*LinearMap(X)`, interpreting the matrix `X` as a linear
 operator, rather than a collection of column vectors. To compute the action of `A` on each
@@ -129,10 +129,10 @@ julia> A*X isa LinearMaps.CompositeMap
 true
 ```
 """
-Base.:(*)(A₁::LinearMap, A₂::AbstractMatrix) = *(A₁, WrappedMap(A₂))
+Base.:(*)(A₁::LinearMap, A₂::Union{AbstractMatrix,AbstractQ}) = *(A₁, WrappedMap(A₂))
 
 """
-    *(X::AbstractMatrix, A::LinearMap)::CompositeMap
+    *(X::Union{AbstractMatrix,AbstractQ}, A::LinearMap)::CompositeMap
 
 Return the `CompositeMap` `LinearMap(X)*A`, interpreting the matrix `X` as a linear
 operator. To compute the right-action of `A` on each row of `X`, call `Matrix(X*A)`
@@ -146,4 +146,4 @@ julia> X*A isa LinearMaps.CompositeMap
 true
 ```
 """
-Base.:(*)(A₁::AbstractMatrix, A₂::LinearMap) = *(WrappedMap(A₁), A₂)
+Base.:(*)(A₁::Union{AbstractMatrix,AbstractQ}, A₂::LinearMap) = *(WrappedMap(A₁), A₂)