diff --git a/ext/KrylovKitExt.jl b/ext/KrylovKitExt.jl
index b18565d23..025175524 100644
--- a/ext/KrylovKitExt.jl
+++ b/ext/KrylovKitExt.jl
@@ -6,12 +6,13 @@ using CommonSolve: CommonSolve
 using Setfield: Setfield, @set
 using NamedTupleTools: NamedTupleTools, delete
 
-using Rimu: Rimu, AbstractDVec, AbstractHamiltonian, IsDeterministic, PDVec, DVec,
-    PDWorkingMemory, scale!!, working_memory, zerovector, dimension, replace_keys
+using Rimu: Rimu, AbstractDVec, AbstractHamiltonian, AbstractOperator, IsDeterministic,
+    starting_address, PDVec, DVec, PDWorkingMemory,
+    scale!!, working_memory, zerovector, dimension, replace_keys
 
 using Rimu.ExactDiagonalization: MatrixEDSolver, KrylovKitSolver,
     KrylovKitDirectEDSolver,
-    LazyDVecs, EDResult, LazyCoefficientVectorsDVecs
+    LazyDVecs, EDResult, LazyCoefficientVectorsDVecs, Multiplier
 
 const U = Union{Symbol,EigSorter}
 
@@ -55,6 +56,45 @@ function KrylovKit.eigsolve(
     )
 end
 
+function _prepare_multiplier(
+    ham, vec; basis=nothing, starting_address=starting_address(ham), full_basis=true
+)
+    if issymmetric(ham) && (isnothing(vec) || isreal(vec))
+        eltype = Float64
+    else
+        eltype = ComplexF64
+    end
+    if isnothing(basis)
+        prop = Multiplier(ham, starting_address; full_basis, eltype)
+    else
+        prop = Multiplier(ham, basis; eltype)
+    end
+end
+
+function KrylovKit.eigsolve(
+    ham::AbstractOperator, vec::Vector, howmany::Int=1, which::U=:LR;
+    basis=nothing, starting_address=starting_address(ham), full_basis=true, kwargs...
+)
+    # Change the type of `vec` to float, if needed.
+    v = scale!!(vec, 1.0)
+    prop = _prepare_multiplier(ham, v; basis, starting_address, full_basis)
+    return eigsolve(
+        prop, v, howmany, which;
+        ishermitian=ishermitian(ham), issymmetric=issymmetric(ham), kwargs...
+    )
+end
+function KrylovKit.eigsolve(
+    ham::AbstractOperator, howmany::Int=1, which::U=:LR;
+    basis=nothing, starting_address=starting_address(ham), full_basis=true, kwargs...
+    )
+    prop = _prepare_multiplier(ham, nothing; basis, starting_address, full_basis)
+    v = rand(eltype(prop), size(prop, 1))
+    return eigsolve(
+        prop, v, howmany, which;
+        ishermitian=ishermitian(ham), issymmetric=issymmetric(ham), kwargs...
+    )
+end
+
 # solve for KrylovKit solvers: prepare arguments for `KrylovKit.eigsolve`
 function CommonSolve.solve(s::S; kwargs...
 ) where {S<:Union{MatrixEDSolver{<:KrylovKitSolver},KrylovKitDirectEDSolver}}
@@ -92,10 +132,6 @@ function _kk_eigsolve(s::MatrixEDSolver{<:KrylovKitSolver}, howmany, which, kw_n
     # solve the problem
     vals, vecs, info = eigsolve(s.basissetrep.sparse_matrix, x0, howmany, which; kw_nt...)
     success = info.converged ≥ howmany
-    if !success
-        @warn "KrylovKit.eigsolve did not converge for all requested eigenvalues:" *
-              " $(info.converged) converged out of $howmany requested value(s)."
-    end
 
     return EDResult(
         s.algorithm,
@@ -113,13 +149,20 @@ end
 
 # solve with KrylovKit direct
 function _kk_eigsolve(s::KrylovKitDirectEDSolver, howmany, which, kw_nt)
-
-    vals, vecs, info = eigsolve(s.problem.hamiltonian, s.v0, howmany, which; kw_nt...)
-    success = info.converged ≥ howmany
-    if !success
-        @warn "KrylovKit.eigsolve did not converge for all requested eigenvalues:" *
-              " $(info.converged) converged out of $howmany requested value(s)."
+    prop = _prepare_multiplier(s.problem.hamiltonian, s.v0#=TODO: new args go here=#)
+    if isnothing(s.v0)
+        x0 = rand(size(prop, 1))
+    else
+        x0 = zeros(eltype(prop), size(prop, 1))
+        for (k, v) in pairs(s.v0)
+            x0[prop.mapping[k]] = v
+        end
     end
+    vals, vecs, info = eigsolve(
+        prop, x0, howmany, which;
+        issymmetric=issymmetric(prop), ishermitian=ishermitian(prop), kw_nt...
+    )
+    success = info.converged ≥ howmany
 
     basis = keys(vecs[1])
 
@@ -128,7 +171,7 @@ function _kk_eigsolve(s::KrylovKitDirectEDSolver, howmany, which, kw_nt)
         s.problem,
         vals,
         vecs,
-        LazyCoefficientVectorsDVecs(vecs, basis),
+        LazyDVecs(vecs, basis),
         basis,
         info,
         howmany,
diff --git a/src/ExactDiagonalization/ExactDiagonalization.jl b/src/ExactDiagonalization/ExactDiagonalization.jl
index 805b4121b..f0ddf1a9f 100644
--- a/src/ExactDiagonalization/ExactDiagonalization.jl
+++ b/src/ExactDiagonalization/ExactDiagonalization.jl
@@ -19,7 +19,7 @@ provided by external packages.
 """
 module ExactDiagonalization
 
-using LinearAlgebra: LinearAlgebra, eigen!, ishermitian, Matrix
+using LinearAlgebra: LinearAlgebra, eigen!, issymmetric, ishermitian, Matrix, dot
 using SparseArrays: SparseArrays, nnz, nzrange, sparse
 using CommonSolve: CommonSolve, solve, init
 using VectorInterface: VectorInterface, add
@@ -29,13 +29,12 @@ using StaticArrays: setindex
 
 using Rimu: Rimu, DictVectors, Hamiltonians, Interfaces, BitStringAddresses, replace_keys,
     clean_and_warn_if_others_present
-using ..Interfaces: AbstractDVec, AbstractHamiltonian, AdjointUnknown,
+using ..Interfaces: AbstractDVec, AbstractHamiltonian, AbstractOperator, AdjointUnknown,
     diagonal_element, offdiagonals, starting_address, LOStructure, IsHermitian
 using ..BitStringAddresses: AbstractFockAddress, BoseFS, FermiFS, CompositeFS, near_uniform
 using ..DictVectors: FrozenDVec, PDVec, DVec
 using ..Hamiltonians: allows_address_type, check_address_type, dimension,
-    ParitySymmetry, TimeReversalSymmetry
-
+    ParitySymmetry, TimeReversalSymmetry, AbstractOperator
 
 export ExactDiagonalizationProblem, KrylovKitSolver, LinearAlgebraSolver
 export ArpackSolver, LOBPCGSolver
@@ -47,6 +46,7 @@ export sparse # from SparseArrays
 include("basis_breadth_first_search.jl")
 include("basis_fock.jl")
 include("basis_set_representation.jl")
+include("multiplier.jl")
 include("algorithms.jl")
 include("exact_diagonalization_problem.jl")
 include("init_and_solvers.jl")
diff --git a/src/ExactDiagonalization/multiplier.jl b/src/ExactDiagonalization/multiplier.jl
new file mode 100644
index 000000000..8a642f004
--- /dev/null
+++ b/src/ExactDiagonalization/multiplier.jl
@@ -0,0 +1,162 @@
+"""
+    Multiplier(::AbstractOperator{T}, basis; eltype=T)
+    Multiplier(::AbstractHamiltonian{T}, [address]; full_basis=true, eltype=T)
+
+Wrapper for an [`AbstractOperator`](@ref) and a basis that allows multiplying regular Julia
+vectors with the operator.
+
+The `eltype` argument can be used to change the eltype of the internal buffers, e.g. for
+multiplying complex vectors with real operators.
+
+If an [`AbstractHamiltonian`](@ref) with no `basis` is passed, the basis is constructed
+automatically. In that case, when `full_basis=true` the entire basis is constructed from an
+address as [`build_basis`](@ref)`(address)`, otherwise it is constructed as
+[`build_basis`](@ref)`(hamiltonian, address)`. You may want to set `full_basis=false` when
+dealing with Hamiltonians that block, such as [`HubbardMom1D`](@ref).
+
+Supports calling, `Base.:*`, `mul!` and the three-argument `dot`.
+
+## Example
+
+```julia
+julia> H = HubbardReal1D(BoseFS(1, 1, 1, 1));
+
+julia> bsr = BasisSetRepresentation(H);
+
+julia> v = ones(length(bsr.basis));
+
+julia> w1 = bsr.sparse_matrix * v;
+
+julia> mul = ExactDiagonalization.Multiplier(H, bsr.basis);
+
+julia> w2 = mul * v;
+
+julia> w1 ≈ w2
+true
+
+julia> dot(w1, bsr.sparse_matrix, v) ≈ dot(w1, mul, v)
+true
+```
+"""
+struct Multiplier{T,H<:AbstractOperator,A,I}
+    hamiltonian::H
+    basis::Vector{A}
+    mapping::Dict{A,I}
+    size::Tuple{Int,Int}
+    buffer::Matrix{T}
+    indices::Vector{UnitRange{Int}}
+end
+function Multiplier(
+    hamiltonian::H, basis::Vector{A}; eltype=eltype(H)
+) where {A,H<:AbstractOperator}
+    I = length(basis) > typemax(Int32) ? Int64 : Int32
+    T = eltype
+    mapping = Dict(b => I(i) for (b, i) in zip(basis, eachindex(basis)))
+    threads = Threads.nthreads()
+    buffer = zeros(T, (length(basis), threads))
+
+    chunk_size = length(basis) ÷ threads
+    prev = 0
+    indices = UnitRange{Int}[]
+    for t in 1:threads - 1
+        push!(indices, prev+1:prev+chunk_size)
+        prev += chunk_size
+    end
+    push!(indices, prev+1:length(basis))
+
+    return Multiplier{T,H,A,I}(
+        hamiltonian, basis, mapping, (length(basis), length(basis)), buffer, indices
+    )
+end
+function Multiplier(
+    hamiltonian::AbstractHamiltonian,
+    address::AbstractFockAddress=starting_address(hamiltonian);
+    full_basis=true, eltype=eltype(hamiltonian),
+)
+    if full_basis
+        basis = build_basis(address)
+    else
+        basis = build_basis(hamiltonian, address)
+    end
+    return Multiplier(hamiltonian, basis)
+end
+function Base.show(io::IO, mul::Multiplier{T}) where {T}
+    print(io, "Multiplier{$T}($(mul.hamiltonian))")
+end
+
+Base.size(mul::Multiplier) = mul.size
+Base.size(mul::Multiplier, i) = mul.size[i]
+Base.eltype(::Type{Multiplier{T}}) where {T} = T
+Base.eltype(::Multiplier{T}) where {T} = T
+LinearAlgebra.issymmetric(mul::Multiplier) = issymmetric(mul.hamiltonian)
+LinearAlgebra.ishermitian(mul::Multiplier) = ishermitian(mul.hamiltonian)
+
+function Base.adjoint(mul::Multiplier{T,<:Any,A,I}) where {T,A,I}
+    hamiltonian = mul.hamiltonian'
+    return Multiplier{T,typeof(hamiltonian),A,I}(
+        hamiltonian, mul.basis, mul.mapping, mul.size,
+    )
+end
+
+function LinearAlgebra.mul!(dst, mul::Multiplier{T}, src) where {T}
+    @boundscheck begin
+        length(src) == size(mul, 2) || throw(DimensionMismatch("operator has size $(size(mul)), vector has length $(length(src))"))
+        length(dst) == size(mul, 1) || throw(DimensionMismatch("operator has size $(size(mul)), output vector has length $(length(dst))"))
+        @assert size(mul.buffer, 1) == length(src)
+    end
+    H = mul.hamiltonian
+    basis = mul.basis
+    mapping = mul.mapping
+    buffer = mul.buffer
+    indices = mul.indices
+
+    @inbounds Threads.@threads for t in 1:size(mul.buffer, 2)
+        buffer[:, t] .= zero(T)
+        for i in indices[t]
+            addr1 = mul.basis[i]
+            val1 = src[i]
+            buffer[i, t] += diagonal_element(H, addr1) * val1
+            for (addr2, elem) in offdiagonals(H, addr1)
+                j = get(mapping, addr2, 0)
+                !iszero(j) && (buffer[j, t] += elem * val1)
+            end
+        end
+    end
+    return sum!(dst, buffer)
+end
+
+function (mul::Multiplier)(src)
+    dst = zeros(length(src))
+    return LinearAlgebra.mul!(dst, mul, src)
+end
+
+Base.:*(mul, src) = mul(src)
+
+function LinearAlgebra.dot(dst, mul::Multiplier, src)
+    @boundscheck begin
+        length(src) == size(mul, 2) || throw(DimensionMismatch("operator has size $(size(mul)), vector has length $(length(src))"))
+        length(dst) == size(mul, 1) || throw(DimensionMismatch("operator has size $(size(mul)), output vector has length $(length(dst))"))
+        @assert size(mul.buffer, 1) == length(src)
+    end
+
+    H = mul.hamiltonian
+    basis = mul.basis
+    mapping = mul.mapping
+    buffer = mul.buffer
+    indices = mul.indices
+
+    @inbounds Threads.@threads for t in 1:size(mul.buffer, 2)
+        buffer[1, t] = result = zero(eltype(buffer))
+        for i in indices[t]
+            addr1 = mul.basis[i]
+            val1 = src[i]
+            result += conj(dst[i]) * diagonal_element(H, addr1) * val1
+            for (addr2, elem) in offdiagonals(H, addr1)
+                j = get(mapping, addr2, 0)
+                result += conj(get(dst, j, 0.0)) * elem * val1
+            end
+        end
+        buffer[1, t] = result
+    end
+    return sum(buffer[1, t] for t in 1:size(mul.buffer, 2))
+end