diff --git a/src/Spaces/PolynomialSpace.jl b/src/Spaces/PolynomialSpace.jl
index e632fbc..d2d8f9b 100644
--- a/src/Spaces/PolynomialSpace.jl
+++ b/src/Spaces/PolynomialSpace.jl
@@ -151,20 +151,33 @@ See https://github.com/JuliaLinearAlgebra/BandedMatrices.jl/blob/master/LICENSE
 end
 
 _view(::Any, A, b) = view(A, b)
-_view(::Val{true}, A::BandedMatrix, b) = dataview(view(A, b))
+function _view(::Val{true}, A::BandedMatrix, b::Band)
+    l, u = bandwidths(A)
+    -l <= b.i <= u || throw(ArgumentError("invalid band $b for bandwidths $((-l,u))"))
+    dataview(view(A, b))
+end
 
-function _get_bands(B, C, bmk, f, ValBC)
+function _get_bands(B, C, bmk, f, valB)
     Cbmk = _view(Val(true), C, band(bmk*f))
     Bm = _view(Val(true), B, band(flipsign(bmk-1, f)))
     B0 = _view(Val(true), B, band(flipsign(bmk, f)))
-    Bp = _view(ValBC, B, band(flipsign(bmk+1, f)))
+    Bp = _view(valB, B, band(flipsign(bmk+1, f)))
     Cbmk, Bm, B0, Bp
 end
 
-function _jac_gbmm!(α, J, B, β, C, b, (Cn, Cm), n, ValJ, ValBC)
-    Jp = _view(ValJ, J, band(1))
-    J0 = _view(ValJ, J, band(0))
-    Jm = _view(ValJ, J, band(-1))
+# Fast implementation of C[:,:] = α*J*B+β*C where the bandediwth of B is
+# specified by b, not by the parameters in B
+function jac_gbmm!(α, J, B, β, C, b, valB)
+    if β ≠ 1
+        lmul!(β,C)
+    end
+
+    n = size(J,1)
+    Cn, Cm = size(C)
+
+    Jp = _view(Val(true), J, band(1))
+    J0 = _view(Val(true), J, band(0))
+    Jm = _view(Val(true), J, band(-1))
 
     kr = intersect(-1:b-1, b-Cm+1:b-1+Cn)
 
@@ -172,7 +185,7 @@ function _jac_gbmm!(α, J, B, β, C, b, (Cn, Cm), n, ValJ, ValBC)
     # this might also help with cache localization
     k = -1
     if k in kr
-        Cbmk, Bm, B0, Bp = _get_bands(B, C, b-k, 1, ValBC)
+        Cbmk, Bm, B0, Bp = _get_bands(B, C, b-k, 1, valB)
         for i in 1:n-b+k
             Cbmk[i] += α * Bm[i+1] * Jp[i]
         end
@@ -180,14 +193,14 @@ function _jac_gbmm!(α, J, B, β, C, b, (Cn, Cm), n, ValJ, ValBC)
 
     k = 0
     if k in kr
-        Cbmk, Bm, B0, Bp = _get_bands(B, C, b-k, 1, Val(true))
+        Cbmk, Bm, B0, Bp = _get_bands(B, C, b-k, 1, valB)
         for i in 1:n-b+k
             Cbmk[i] += α * (Bm[i+1] * Jp[i] + B0[i] * J0[i])
         end
     end
 
     for k in max(1, first(kr)):last(kr)
-        Cbmk, Bm, B0, Bp = _get_bands(B, C, b-k, 1, Val(true))
+        Cbmk, Bm, B0, Bp = _get_bands(B, C, b-k, 1, valB)
         Cbmk[1] += α * (Bm[2] * Jp[1] + B0[1] * J0[1])
         for i in 2:n-b+k
             Cbmk[i] += α * (Bm[i+1] * Jp[i] + B0[i] * J0[i] + Bp[i-1] * Jm[i-1])
@@ -198,7 +211,7 @@ function _jac_gbmm!(α, J, B, β, C, b, (Cn, Cm), n, ValJ, ValBC)
 
     k = -1
     if k in kr
-        Ckmb, Bp, B0, Bm = _get_bands(B, C, b-k, -1, ValBC)
+        Ckmb, Bp, B0, Bm = _get_bands(B, C, b-k, -1, valB)
         for (i, Ji) in enumerate(b-k:n-1)
             Ckmb[i] += α * Bp[i] * Jm[Ji]
         end
@@ -206,7 +219,7 @@ function _jac_gbmm!(α, J, B, β, C, b, (Cn, Cm), n, ValJ, ValBC)
 
     k = 0
     if k in kr
-        Ckmb, Bp, B0, Bm = _get_bands(B, C, b-k, -1, Val(true))
+        Ckmb, Bp, B0, Bm = _get_bands(B, C, b-k, -1, valB)
         Ckmb[1] += α * Bp[1] * Jm[b-k]
         for (i, Ji) in enumerate(b-k+1:n-1)
             Ckmb[i] += α * B0[i] * J0[Ji]
@@ -238,21 +251,6 @@ function _jac_gbmm!(α, J, B, β, C, b, (Cn, Cm), n, ValJ, ValBC)
     return C
 end
 
-# Fast implementation of C[:,:] = α*J*B+β*C where the bandediwth of B is
-# specified by b, not by the parameters in B
-function jac_gbmm!(α, J, B, β, C, b, valJ, valBC)
-    if β ≠ 1
-        lmul!(β,C)
-    end
-
-    n = size(J,1)
-    Cn, Cm = size(C)
-
-    _jac_gbmm!(α, J, B, β, C, b, (Cn, Cm), n, valJ, valBC)
-
-    C
-end
-
 function BandedMatrix(S::SubOperator{T,ConcreteMultiplication{C,PS,T},
                                      NTuple{2,UnitRange{Int}}}) where {PS<:PolynomialSpace,T,C<:PolynomialSpace}
     M=parent(S)
@@ -285,7 +283,6 @@ function BandedMatrix(S::SubOperator{T,ConcreteMultiplication{C,PS,T},
 
     #Multiplication is transpose
     J=Operator{T}(Recurrence(M.space))[jkr,jkr]
-    valJ = all(>=(1), bandwidths(J)) ? Val(true) : Val(false)
 
     B=n-1  # final bandwidth
 
@@ -293,15 +290,16 @@ function BandedMatrix(S::SubOperator{T,ConcreteMultiplication{C,PS,T},
     Bk2 = BandedMatrix(Zeros{T}(size(J)), (B,B))
     dataview(view(Bk2, band(0))) .= a[n]/recβ(T,sp,n-1)
     α,β = recα(T,sp,n-1),recβ(T,sp,n-2)
-    Bk1 = (-α/β)*Bk2
+    Bk1 = lmul!(-α/β, copy(Bk2))
     dataview(view(Bk1, band(0))) .+= a[n-1]/β
-    jac_gbmm!(one(T)/β,J,Bk2,one(T),Bk1,0,valJ, Val(true))
+    jac_gbmm!(one(T)/β,J,Bk2,one(T),Bk1,0, Val(true))
     b=1  # we keep track of bandwidths manually to reuse memory
     for k=n-2:-1:2
+        # b goes from 1:
         α,β,γ=recα(T,sp,k),recβ(T,sp,k-1),recγ(T,sp,k+1)
         lmul!(-γ/β,Bk2)
         dataview(view(Bk2, band(0))) .+= a[k]/β
-        jac_gbmm!(1/β,J,Bk1,one(T),Bk2,b,valJ,Val(true))
+        jac_gbmm!(1/β,J,Bk1,one(T),Bk2,b,Val(true))
         LinearAlgebra.axpy!(-α/β,Bk1,Bk2)
         Bk2,Bk1=Bk1,Bk2
         b+=1
@@ -309,7 +307,7 @@ function BandedMatrix(S::SubOperator{T,ConcreteMultiplication{C,PS,T},
     α,γ=recα(T,sp,1),recγ(T,sp,2)
     lmul!(-γ,Bk2)
     dataview(view(Bk2, band(0))) .+= a[1]
-    jac_gbmm!(one(T),J,Bk1,one(T),Bk2,b,valJ,Val(false))
+    jac_gbmm!(one(T),J,Bk1,one(T),Bk2,b,Val(false))
     LinearAlgebra.axpy!(-α,Bk1,Bk2)
 
     # relationship between jkr and kr, jr