PUBLICATIONS DE L’INSTITUT MATHÉMATIQUE Nouvelle série, tome 96 (110) (2014), 85–102 DOI: 10.2298/PIM1410085D

THE COBURN–SIMONENKO THEOREM FOR SOME CLASSES OF WIENER–HOPF PLUS HANKEL OPERATORS

Victor D. Didenko and Bernd Silbermann

Abstract. Wiener–Hopf plus Hankel operators W (a) + H(b) : L^{p}(ℝ^{+} ) → L^{p}(ℝ^{+} ) with generating functions a and b from a subalgebra of L^{∞} (ℝ) containing almost periodic functions and Fourier images of L^{1}(ℝ)-functions are studied. For a and b satisfying the so-called matching condition

a(t)a(−t) = b(t)b(−t), t ∈ ℝ, we single out some classes of operators W (a) + H(b) which are subject to the Coburn–Simonenko theorem.

1. Introduction

The classical Coburn–Simonenko Theorem states that for a Toeplitz or Wiener– Hopf operator A with a scalar nonzero generating function, at least one of the numbers dimkerA or dimcokerA is equal to zero. Thus if it is known that the corresponding operator is Fredholm with index zero, the Coburn–Simonenko Theorem implies that this operator is invertible. Note that Fredholmness of such operators with generating functions from various classes is well understood. On the other hand, for Toeplitz plus Hankel operators T (a) + H(b) with piecewise continuous generating functions a and b their Fredholm properties can be derived by a direct application of results [4, Sections 4.95–4.102], [12, Sections 4.5 and 5.7], [13]. The case of quasi piecewise continuous generating functions has been studied in [15], whereas formulas for the index of the operators T (a) + H(b) considered on various Banach and Hilbert spaces and with various assumptions about the generating functions a and b have been established in [8,14]. It is also worth mentioning that lately a lot of effort has been spent to obtain information concerning the kernel and cokernel dimensions of Toeplitz plus Hankel or Wiener–Hopf plus Hankel operators. Here we are not going to discuss the history of these investigations in much detail, but just mention a few important developments. For example, in the works of Ehrhardt [9, 10] and Ehrhardt and Basor [1–3], Toeplitz plus Hankel operators have been studied in H^{p}-spaces on the unit circle 𝕋 mainly under the assumption that the generating functions of these operators are piecewise continuous, satisfy an algebraic relation, and that the operators are Fredholm. Wiener–Hopf plus Hankel operators have received less attention in the literature and results are scarce (see, for example, [5] and references there). In addition, in most cases the conditions imposed on the generating functions are very restrictive and ensure that the problem can be handled in a more or less straightforward way.

2010 Mathematics Subject Classification: Primary 47B35, 47B38; Secondary 47B33, 45E10.

Key words and phrases: Wiener–Hopf plus Hankel operator, Coburn–Simonenko theorem, invertibility.

Dedicated to Giuseppe Mastroianni, in occasion of his retirement.

Partially supported by the Universiti Brunei Darussalam under Grant UBD/GSR/S&T/19.

Let us now describe the problem studied in the present paper. Consider the set G of all functions of the form

(1.1)  a(t) = ∑_{j=−∞}^{∞} a_{j} e^{iδ_{j} t} + ∫_{−∞}^{∞} k(s)e^{its} ds, −∞ < t < ∞,

where δ_{j} ∈ ℝ are pairwise distinct and

∑_{j=−∞}^{∞} |a_{j} | < ∞, ∫_{−∞}^{∞} |k(s)| ds < ∞.

The set G actually forms a commutative unital Banach algebra under pointwise operations and the norm

∥a∥ := ∑_{j=−∞}^{∞} |a_{j} | + ∫_{−∞}^{∞} |k(s)| ds.

This algebra G contains both the algebra AP_{w} of all almost periodic functions with absolutely convergent Fourier series and the algebra L_{0} of all Fourier transforms of functions from L^{1}(ℝ). Moreover, the algebra G is the direct sum of AP_{w} and L_{0}, and L_{0} is an ideal in G. A function a ∈ G is invertible in G if and only if it satisfies the condition inf_{t∈ℝ} |a(t)| > 0. Moreover, if b ∈ AP_{w}, k ∈ L_{0}, and b + k is invertible in G, then b is also invertible in AP_{w} (see [11, Chapter VII]). Further, let us introduce the subalgebra G^{+} (G^{−} ) of the algebra G, which consists of all functions (1.1) such that all numbers δ_{j} are nonnegative (nonpositive) and function k vanishes on the negative (positive) semi-axis. It is clear that the functions from G^{+} and G^{−} admit holomorphic extensions to the upper and to the lower half-plane, respectively, and the intersection of the sets G^{+} and G^{−} contains constant functions only.

If b ∈ AP_{w}, k ∈ L_{0}, and the element a = b + k is invertible in G, then the numbers

ν(a) := lim_{l→∞} \frac{1}{2l}[argb(t)]_{−l}^{l} , and n(a) := \frac{1}{2π} [arg(1 + b^{−1} (t)k(t)]_{t=−∞}^{∞} ,

are well defined. In particular, the first limit exists because b is an almost periodic function.

COBURN-SIMONENKO THEOREM FOR WIENER–HOPF PLUS HANKEL OPERATORS 87

Let ℝ^{+} := (0, ∞) and let P be the projection operator from L^{p}(ℝ), 1 ⩽ p ⩽ ∞ onto L^{p}(ℝ^{+} ), that is P : f ↦ f|_{ℝ^{+}} . Analogously, Q is the projection operator from L^{p}(ℝ) onto L^{p}(ℝ^{−} ), ℝ^{−} := (−∞, 0). In what follows we will identify the space L^{p}(ℝ^{+} ) (L^{p}(ℝ^{−} )) with the subspace of L^{p}(ℝ) consisting of all functions vanishing on ℝ^{−} (ℝ^{+} ). Note that P^{2} = P and Q^{2} = Q.

Each function a ∈ G,

a(t) = ∑_{j=−∞}^{∞} a_{j} e^{iδ_{j} t} + ∫_{−∞}^{∞} k(s)e^{its} ds,

generates two operators W^{0}(a) : L^{p}(ℝ) → L^{p}(ℝ) and W (a) : L^{p}(ℝ^{+} ) → L^{p}(ℝ^{+} ) defined by

(1.2)  (W^{0}(a)f)(t) := ∑_{j=−∞}^{∞} a_{j} f(t − δ_{j}) + ∫_{−∞}^{∞} k(t − s)f(s) ds,
W (a)f := P W^{0}(a)f.

These operators belong to the spaces L(L^{p}(ℝ)) and L(L^{p}(ℝ^{+} )), respectively, i.e., they are linear bounded operators. Moreover, the mappings G → L(L^{p}(ℝ)) and G → L(L^{p}(ℝ^{+} )) defined, respectively, by

a ↦ W^{0}(a) and a ↦ W (a) ,

are injective linear bounded mappings. The function a is referred to as the generating function, or the symbol, for both operators W^{0}(a) and W (a). The Fredholm theory for the operators W^{0}(a), a ∈ G is relatively simple. An operator W^{0}(a) is semi-Fredholm if and only if a is invertible in G. A proof of this result is implicitly contained in the proof of Theorem 2.4, §2, Chapter VII in [11].

Note that the convolution operator (1.2) is shift invariant that is W^{0}(a)τ_{v} = τ_{v}W^{0}(a) for any v ∈ ℝ, where τ_{v} is the operator defined by (τ_{v}f)(t) := f(t − v). The operator W (a) is called integro-difference operator [11, Chapter VII]. It is shown in [4, Sections 9.4 and 9.21] that integro-difference operators are indeed Wiener–Hopf integral operators. If a does not vanish identically, then W (a) has a trivial kernel or a dense range in L^{p}(ℝ^{+} ) at least for 1 < p < ∞ and this is the Coburn–Simonenko Theorem for such class of operators (see [4, Section 9.5 (d)]).

Now we can formulate the following result.

Theorem 1.1 (Gohberg/Feldman [11]). If a ∈ G, then the operator W (a) is one-sided invertible in L^{p}(ℝ^{+} ) for 1 ⩽ p ⩽ ∞ if and only if a is invertible in G. Further, if a ∈ G is invertible in G, then the following assertions are true.

(i) If ν(a) > 0, then the operator W (a) is invertible from the left and dimcokerW (a) = ∞.

(ii) If ν(a) < 0, then the operator W (a) is invertible from the right and

dimkerW (a) = ∞. (iii) If ν(a) = 0, then the operator W (a) is invertible from the left (right) if

n(a) ⩾ 0 (n(a) ⩽ 0) and

dimcokerW (a) = n(a) (dimkerW (a) = −n(a)).

(iv) If a ∈ G is not invertible in G, then W (a) is not a semi-Fredholm operator.

Remark 1.1. Using the Coburn–Simonenko Theorem, one can show that if W (a) is normally solvable and a ≠ 0, then a is invertible in G, at least in the case where the operator W (a) acts on the space L^{p}(ℝ^{+} ), p ∈ (1, ∞).

Let us introduce Hankel operators. For, consider the operator J : L^{p}(ℝ) → L^{p}(ℝ) defined by Jφ := φ̃, where φ̃(t) := φ(−t). If a ∈ G and 1 ⩽ p ⩽ ∞, then on the space L^{p}(ℝ^{+} ) the Hankel operators H(a) and H(ã) are defined as follows

H(a) : φ ↦ P W^{0}(a)QJφ,
H(ã) : φ ↦ JQW^{0}(a)P φ.

Note that JQW^{0}(a)P = P W^{0}(ã)QJ, and the last identity is the consequence of the following relations

(1.3)  J^{2} = I, JQ = P J, JP = QJ, JW^{0}(a)J = W^{0}(ã).

On the space L^{p}(ℝ), 1 ⩽ p ⩽ ∞ we also consider the operators U and U^{−1} defined by

(Uφ)(t) := φ(t) − 2 ∫_{−∞}^{t} e^{s−t} φ(s) ds, −∞ < t < ∞,
(U^{−1} φ)(t) := φ(t) − 2 ∫_{t}^{∞} e^{t−s} φ(s) ds, −∞ < t < ∞.

It is well known [11] that

U = W^{0}(χ), U^{−1} = W^{0}(χ^{−1} ),

where χ(t) := (t − i)/(t + i), χ^{−1} (t) := (t + i)/(t − i), t ∈ ℝ. Moreover, since W^{0}(χ)W^{0}(χ^{−1} ) = W^{0}(χχ^{−1} ), we get UU^{−1} = U^{−1} U = I.

One of the aims of this work is to establish a Coburn–Simonenko Theorem for the operators W (a)+H(aχ) and W (a)−H(aχ^{−1} ), where a ∈ G is invertible. Recall that the semi-Fredholmness of the operators W (b) + H(c) : L^{p}(ℝ^{+} ) → L^{p}(ℝ^{+} ), b, c ∈ G implies that the element b is invertible in G at least in the case where 1 < p < ∞. Indeed, the proof of Theorem 2.30 in [4] with the shifts U^{±n} and the Toeplitz operators T (a) replaced, respectively, by the translations τ_{±ν} , ν ∈ ℝ^{+} and the operators W (b) + H(c) implies that ∥W^{0}(b)f∥ ⩾ c∥f∥ for all f ∈ L^{p}(ℝ). But then W^{0}(b) is semi-Fredholm and, therefore, b is invertible in G. For p = 1 this proof does not work. Nevertheless, we conjecture that for p = 1 the result is also true. Therefore, the above requirement of the invertibility of the element a is not too restrictive.

Finally, let us also mention that if a, b ∈ G, then W^{0}(ab) = W^{0}(a)W^{0}(b), and if a ∈ G^{−} , c ∈ G^{+} , and b ∈ G, then W (abc) = W (a)W (b)W (c). Moreover, in the following we will make use of the identities

(1.4)  \begin{aligned}W (ab) & = W (a)W (b) + H(a)H(b̃), \\ H(ab) & = W (a)H(b) + H(a)W (b̃). \end{aligned}

2. Kernels of Wiener–Hopf plus Hankel operators. General properties

In this section we establish certain relations between the kernels of Wiener– Hopf plus Hankel operators and matrix Wiener-Hopf operators in the case where the generating functions a, b ∈ G. The corresponding results for Toeplitz plus Hankel operators T (a) + H(b), a, b ∈ L^{∞} have been obtained recently [7]. Taking into account Theorem 1.1 we can always assume that a is invertible in G. Along with the operator W (a) + H(b) let us also consider the Wiener–Hopf minus Hankel operator W (a) − H(b) and the Wiener–Hopf operator W (V (a, b))) defined by the matrix

V (a, b) := ( \begin{matrix}a − bb̃a^{−1} & d \\ −c & a^{−1} \end{matrix} ) ,

where c := b̃a^{−1} , d := ba^{−1} .

The following lemma describes connections between the solutions of homogeneous equations with Wiener–Hopf plus/minus Hankel operators and the solutions of the associated homogeneous equation with the matrix Wiener–Hopf operator W (V (a, b)).

Lemma 2.1. Assume that a, b ∈ G, a is invertible in G, and the operators W (a) ± H(b) are considered on the space L^{p}(ℝ^{+} ), 1 ⩽ p ⩽ ∞.

• If (φ, ψ)^{T} ∈ kerW (V (a, b)), then

(2.1) (Φ, Ψ)^{T} = \frac{1}{2}(φ − JQW^{0}(c)φ + JQW^{0}(a^{−1} )ψ, φ + JQW^{0}(c)φ − JQW^{0}(a^{−1} )ψ)^{T}

∈ kerdiag(W (a) + H(b), W (a) − H(b))

• If (Φ, Ψ)^{T} ∈ kerdiag(W (a) + H(b), W (a) − H(b)), then

(2.2)  (Φ + Ψ, P (W^{0}(b̃)(Φ + Ψ) + W^{0}(ã)JP (Φ − Ψ)))^{T} ∈ kerW (V (a, b)).

Moreover, the operators

E_{1} : kerW (V (a, b)) → kerdiag(W (a) + H(b), W (a) − H(b)),
E_{2} : kerdiag(W (a) + H(b), W (a) − H(b)) → kerW (V (a, b)),

defined, respectively, by relations (2.1) and (2.2) are mutually inverse.

Proof. Consider the operators

(2.3) A := ( \begin{matrix}I & 0 \\ W^{0}(b̃) & W^{0}(ã) \end{matrix}) ( \begin{matrix}I & I \\ J & −J \end{matrix} ) ,  B_{1} := 2 ( \begin{matrix}I & J \\ I & −J \end{matrix} ) ,
B_{2} := diag(I, I) − diag(P, Q) ( \begin{matrix}W^{0}(a) & W^{0}(b) \\ W^{0}(b̃) & W^{0}(ã) \end{matrix}) diag(Q, P),
B_{3} := diag(I, I) + diag(P, P) ( \begin{matrix}W^{0}(a − bb̃a^{−1} ) & W^{0}(d) \\ −W^{0}(c) & W^{0}(a^{−1} ) \end{matrix}) diag(Q, Q).

Elementary but tedious computations show that the the operator

diag(W (a) + H(b) + Q, W (a) − H(b) + Q)

can be represented as the product of three matrix operators, viz.

(2.4) ( \begin{matrix}W (a) + H(b) + Q & 0 \\ 0 & W (a) − H(b) + Q \end{matrix} ) = B(W (V (a, b))) + diag(Q, Q)A ,

where B := B_{1}B_{2}B_{3}. The operator A : L^{p}(ℝ) × L^{p}(ℝ) → L^{p}(ℝ) × L^{p}(ℝ) is invertible because a is invertible in G, and it is well known that all the operators B_{1}, B_{2}, B_{3} are invertible as well. Therefore, relations (2.3)–(2.4) imply that for any (φ, ψ)^{T} ∈ kerW (V (a, b))), the element A^{−1} ((φ, ψ)^{T}) belongs to the set

kerdiag(W (a)+H(b)+Q, W (a)−H(b)+Q) = kerdiag(W (a)+H(b), W (a)−H(b)).

Hence

diag(P, P)A^{−1} ((φ, ψ)^{T}) = A^{−1} ((φ, ψ)^{T}).

Computing the left-hand side of the last equation, one obtains relation (2.1). Analogously, if (Φ, Ψ)^{T} ∈ kerdiag(W (a) + H(b), W (a) − H(b)), then A((Φ, Ψ)^{T}) ∈ kerW (V (a, b)) and diag(P, P) A((Φ, Ψ)^{T}) = A((Φ, Ψ)^{T}), so representation (2.2) follows.

Now let (φ, ψ) and (Φ, Ψ) be as above. Then

diag(P, P)A diag(P, P)A^{−1} ((φ, ψ)^{T}) = AA^{−1} ((φ, ψ)^{T}),
diag(P, P)A^{−1} diag(P, P)A((Φ, Ψ)^{T}) = A^{−1} A((Φ, Ψ)^{T}),

which completes the proof. □

From now on we will always assume that the generating functions a and b satisfy the condition

(2.5)  aã = bb̃.

Analogously to [6], relation (2.5) is called matching condition, and if a and b satisfy (2.5), then the duo (a, b) is called a matching pair. For each matching pair (a, b) one can assign another matching pair (c, d) with c := b̃a^{−1} and d := ba^{−1} . Such a pair (c, d) is called the subordinated pair for (a, b), and it is easily seen that the functions which constitutes a subordinated pair have a specific property, namely cc̃ = 1 = dd̃. Throughout this paper any function g ∈ G satisfying the condition

gg̃ = 1,

is called matching function. In passing note that the matching functions c and d can also be expressed in the form

c = ab^{−1} , d = b̃^{−1} a.

Besides, if (c, d) is the subordinated pair for a matching pair (a, b), then (\overline{d}, \overline{c}) is the subordinated pair for the matching pair (\overline{a},\overline{b̃}) which defines the adjoint operator

(2.6)  (W (a) + H(b))^{∗} = W (\overline{a}) + H(\overline{b̃})

for the operator W (a) + H(b). Further, a matching pair (a, b) is called Fredholm, if the Wiener–Hopf operators W (c) and W (d) are Fredholm.

If (a, b) is a matching pair, then the corresponding matrix–function V (a, b) takes the form

V (a, b) = ( \begin{matrix}0 & d \\ −c & a^{−1} \end{matrix} ) ,

where (c, d) is the subordinated pair for the pair (a, b). Moreover, similarly to the corresponding representation of the matrix Toeplitz operator T (V (a, b)) from [6], the operator W (V (a, b))) can be represented as the product of three matrix Wiener– Hopf operators

W (V (a, b)))= ( \begin{matrix}0 & W (d) \\ −W (c) & W (a^{−1} ) \end{matrix}) (2.7)

= ( \begin{matrix}−W (d) & 0 \\ 0 & I \end{matrix}) ( \begin{matrix}0 & −I \\ I & W (a^{−1} ) \end{matrix}) ( \begin{matrix}−W (c) & 0 \\ 0 & I \end{matrix}) ,

where the operator

D := ( \begin{matrix}0 & −I \\ I & W (a^{−1} ) \end{matrix})

in the right-hand side of (2.7) is invertible and

D^{−1} = ( \begin{matrix}W (a^{−1} ) & I \\ −I & 0 \end{matrix}) .

Note that a useful representation for the kernel of the block Toeplitz operator T (V (a, b))) defined by a matching pair (a, b), has been derived recently. Following [7, Proposition 3.3], one can also obtain a similar result for the block Wiener-Hopf operator W (V (a, b)).

Proposition 2.1. Let (a, b) ∈ G×G be a matching pair such that the operator W (c), c = ab^{−1} , is invertible from the right. Then

kerW (V (a, b))) = Ω(c) ∔ Ω̂(d)

where

Ω(c) := {(φ, 0)^{T} : φ ∈ kerW (c)},
Ω̂(d) := {(W_{r}^{−1} (c)W (a^{−1} )s, s)^{T} : s ∈ kerW (d)},

and W_{r}^{−1} (c) is one of the right inverses for the operator W (c).

Proof. It is clear that Ω(c) and Ω̂(d) are closed subspaces of kerW (V (a, b)) and Ω(c) ∩ Ω̂(d) = {0}.

If (y_{1}, y_{2})^{T} ∈ kerW (V (a, b)), then W (d)y_{2} = 0, and W (c)y_{1} = W (a^{−1} )y_{2}. Since W_{r}^{−1} (c) is left-invertible, the space L^{p}(ℝ^{+} ) is the direct sum of the closed subspaces kerW (c) and imW_{r}^{−1} (c), i.e., L^{p}(ℝ^{+} ) = kerW (c) ∔ imW_{r}^{−1} (c). Consequently, the element y_{1} can be represented in the form y_{1} = y_{10} + y_{11}, where y_{10} ∈ kerW (c) and y_{11} ∈ imW_{r}^{−1} (c). Moreover, there is a unique vector y_{3} ∈ L^{p}(ℝ^{+} ) such that y_{11} = W_{r}^{−1} (c)y_{3}, so we get

W (c)y_{1} = W (c)(W_{r}^{−1} (c)y_{3} + y_{10}) = y_{3} = W (a^{−1} )y_{2}.

It implies that y_{1} = W_{r}^{−1} (c)W (a^{−1} )y_{2} + y_{10}, what leads to the representation

(y_{1}, y_{2})^{T} = (W_{r}^{−1} (c)W (a^{−1} )y_{2}, y_{2})^{T} + (y_{10}, 0)^{T},

with (W_{r}^{−1} (c)W (a^{−1} )y_{2}, y_{2})^{T} ∈ Ω̂(d) and (y_{10}, 0)^{T} ∈ Ω(c). □

Thus φ ∈ kerW (c) implies that (φ, 0)^{T} ∈ kerW (V (a, b))) and by Lemma 2.1

(2.8)  \begin{matrix}φ − JQW^{0}(c)P φ ∈ ker(W (a) + H(b)), \\ φ + JQW^{0}(c)P φ ∈ ker(W (a) − H(b)). \end{matrix}

It is even more remarkable that the functions φ−JQW^{0}(c)P φ and φ+JQW^{0}(c)P φ belong to the kernel of the operator W (c) as well.

Proposition 2.2. Let g ∈ G be a matching function, i.e., gg̃ = 1. If f ∈ kerW (g), then JQW^{0}(g)P f ∈ kerW (g) and (JQW^{0}(g)P)^{2}f = f.

Proof. If gg̃ = 1 and f ∈ kerW (g), then

W (g)(JQW^{0}(g)P f) = P W^{0}(g)P JQW^{0}(g)P f = JQW^{0}(g̃)QW^{0}(g)P f
= JQW^{0}(g̃)W^{0}(g)P f − JQW^{0}(g̃)P W^{0}(g)P f = 0,

and assertion (i) follows. On the other hand, for any f ∈ kerW (g) one has

(JQW^{0}(g)P)^{2}f = JQW^{0}(g)P JQW^{0}(g)P f = P W^{0}(g̃)QW^{0}(g)P f
= P W^{0}(g̃)W^{0}(g)P f − P W^{0}(g̃)P W^{0}(g)P f = f,

which completes the proof. □

Consider now the operator P(g) := JQW^{0}(g)P |_{ker W (g)} . Proposition 2.2 implies that P(g) : kerW (g) → kerW (g) and P^{2}(g) = I. Thus on the space kerW (g) the operators P^{−} (g) := (1/2)(I − P(g)) and P^{+} (g) := (1/2)(I + P(g)) are complementary projections generating a decomposition of kerW (g). Moreover, relations (2.8) lead to the following result.

Corollary 2.1. Let (c, d) be the subordinated pair for a matching pair (a, b) ∈ G × G. Then kerW (c) = imP^{−} (c) ∔ imP^{+} (c), and the following relations hold

(2.9)  imP^{−} (c) ⊂ ker(W (a) + H(b)), imP^{+} (c) ⊂ ker(W (a) − H(b)).

Relations (2.9) show the influence of the operator W (c) on the kernels of the operators W (a) + H(b) and W (a) − H(b). Let us now clarify the role of another operator–viz. the operator W (d), in the structure of the kernels of the operators W (a) + H(b) and W (a) − H(b). Assume additionally that the operator W (c) is invertible from the right. If s ∈ kerW (d), then the element (W_{r}^{−1} (c)W (a^{−1} )s, s)^{T} ∈ kerW (V (a, b))). By Lemma 2.1, the element

2φ^{±} (s) := W_{r}^{−1} (c)W (a^{−1} )s ∓ JQW^{0}(c)P W_{r}^{−1} (c)W (a^{−1} )s ± JQW^{0}(a^{−1} )s

belongs to the null space ker(W (a) ± H(b)) of the corresponding operator W (a) ± H(b).

Lemma 2.2. Let (c, d) be the subordinated pair for a matching pair (a, b) ∈ G × G. If the operator W (c) is right-invertible, then for every s ∈ kerW (d) the following relations

(W (b̃) + H(ã))φ_{+} (s) = P^{+} (d)s, (W (b̃) − H(ã))φ_{−} (s) = P^{−} (d)s,

hold. Thus the corresponding mappings φ_{+} : imP^{+} (d) → imP^{+} (d) and φ_{−} : imP^{−} (d) → imP^{−} (d), are injective operators.

Proof. Assuming that s ∈ kerW (d), one can show that the operator W (b̃) + H(ã) sends φ^{+} (s) into P^{+} (d)s and the operator W (b̃) − H(ã) sends φ^{−} (s) into P^{−} (d)s. The proof of these facts is based on relations (1.3) and runs similarly to the proof of [7, Lemma 3.6]. □

Proposition 2.3. Let (c, d) be the subordinated pair for a matching pair (a, b) ∈ G × G. If the operator W (c) is right-invertible, then

ker(W (a) + H(b)) = φ^{+} (imP^{+} (d)) ∔ imP^{−} (c),
ker(W (a) − H(b)) = φ^{−} (imP^{−} (d)) ∔ imP^{+} (c).

Proof. Using the invertibility of the operator E_{1} and Proposition 2.1, one obtains

kerdiag(W (a) + H(b), W (a) − H(b)) = E_{1}(Ω̂(d)) ∔ E_{1}(Ω(c)).

Apparently, Ω̂(d) = Ω̂_{+} (d) ∔ Ω̂_{−} (d), Ω(c) = Ω_{+} (c) ∔ Ω_{−} (c), where

Ω̂_{+} (d) = {(W_{r}^{−1} (c)W (a^{−1} )s, s)^{T} : s ∈ imP^{+} (d)},
Ω̂_{−} (d) = {(W_{r}^{−1} (c)W (a^{−1} )s, s)^{T} : s ∈ imP^{−} (d)},
Ω_{+} (c) = {(s, 0)^{T} : s ∈ imP^{+} (c)},
Ω_{−} (c) = {(s, 0)^{T} : s ∈ imP^{−} (c)}.

Hence

kerdiag(W (a) + H(b), W (a) − H(b))

= E_{1}(Ω̂_{+} (d)) ∔ E_{1}(Ω̂_{−} (d)) ∔ E_{1}(Ω_{+} (c)) ∔ E_{1}(Ω_{−} (c)) = E_{1}(kerW (V (a, b)).

It is clear that if ϕ ∈ ker(W (a)+H(b)), then (ϕ, 0)^{T} ∈ kerdiag(W (a)+H(b), W (a)− H(b)). Now we want to find that uniquely defined element (α, β)^{T} from the kernel of the operator W (V (a, b)), which is sent into the element (ϕ, 0) by the operator E_{1}. It can be uniquely represented in the form

(α, β)^{T}= (W_{r}^{−1} (c)W (a^{−1} )s_{+} , s_{+} )^{T}+ (W_{r}^{−1} (c)W (a^{−1} )s_{−} , s_{−} )^{T}+ (v_{+} , 0)^{T}+ (v_{−} , 0)^{T},

where s_{±} ∈ imP^{±} (d), v_{±} ∈ imP^{±} (c). Then

(ϕ, 0)^{T} = E_{1}((α, β)^{T})
= E_{1}((W_{r}^{−1} (c)W (a^{−1} )s_{+} , s_{+} )^{T}) + E_{1}((W_{r}^{−1} (c)W (a^{−1} )s_{−} , s_{−} )^{T})
+ E_{1}((v_{+} , 0)^{T}) + E_{1}((v_{−} , 0)^{T})
= (φ_{+} (s_{+} ), φ_{−} (s_{+} ))^{T} + (φ_{+} (s_{−} ), φ_{−} (s_{−} ))^{T} + (0, v_{+} )^{T} + (v_{−} , 0)^{T}.

Thus

ϕ = φ_{+} (s_{+} ) + φ_{+} (s_{−} ) + v_{−} , 0 = φ_{−} (s_{+} ) + φ_{−} (s_{−} ) + v_{+} .

However, since φ_{+} (s_{−} ) ∈ ker(W (a) + H(b)) and E_{2}((φ_{+} (s_{−} ), 0)^{T}) = (φ_{+} (s_{−} ), 0)^{T} according to Lemma 2.2, we get φ_{+} (s_{−} ) ∈ imP^{−} (c). Analogously, one can show that φ_{−} (s_{+} ) ∈ P^{+} (c). It implies that φ_{−} (s_{−} ) = −(φ_{−} (s_{+} ) + v_{+} ) ∈ imP^{+} (c) and E_{2}((0, φ_{−} (s_{−} ))^{T}) = (φ_{−} (s_{−} ), 0)^{T} because φ_{−} (s_{−} ) ∈ imP^{+} (c). On the other hand, due to Lemma 2.2, one has E_{2}((0, φ_{−} (s_{−} ))^{T}) = (φ_{−} (s_{−} ), s_{−} )^{T}. The comparison of the two expressions for the element E_{2}((0, φ_{−} (s_{−} ))^{T}) gives s_{−} = 0, and therefore φ_{−} (s_{−} ) = 0. This implies φ_{−} (s_{+} ) = −v_{+} . Consequently,

(α, β)^{T} = (W_{r}^{−1} (c)W (a^{−1} )s_{+} , s_{+} )^{T} − (φ_{−} (s_{+} ), 0)^{T} + (v_{−} , 0)^{T},

which leads to the relation

E_{1}((α, β)^{T}) = (φ_{+} (s_{+} ), φ_{−} (s_{+} ))^{T} − (0, φ_{−} (s_{+} ))^{T} + (v_{−} , 0)^{T} = (φ_{+} (s_{+} ) + v_{−} , 0)^{T}.

Thus φ_{+} (s_{+} ) + v_{−} ∈ ker(W (a) + H(b)). This result shows that ker(W (a) + H(b)) is the sum of its subspaces φ_{+} (imP^{+} (d)) and imP^{−} (c). Recalling that (W_{r}^{−1} (c)W (a^{−1} )s_{+} , s_{+} )^{T} − (φ_{−} (s_{+} ), 0)^{T} ∈ Ω̂_{+} (d) ∔ Ω_{+} (c) and (v_{−} , 0)^{T} ∈ Ω_{−} (c), one finally obtains

ker(W (a) + H(b)) = φ_{+} (imP^{+} (d)) ∔ imP^{−} (c).

The relation

ker(W (a) − H(b)) = φ_{−} (imP^{−} (d)) ∔ imP^{+} (c)

can be verified analogously. □

Corollary 2.2. Let (c, d) be the subordinated pair for a matching pair (a, b) ∈ G × G satisfying the conditions of Proposition 3.1. Then

dimker(W (a) + H(b)) = dimimP^{+} (d) + dimimP^{−} (c),
dimker(W (a) − H(b)) = dimimP^{−} (d) + dimimP^{+} (c).

Remark 2.1. If (a, b) ∈ G×G is a Fredholm matching pair, i.e., if W (c), W (d) are Fredholm operators, then W (a) ± H(b) are Fredholm operators and

(2.10)  ind(W (a) + H(b)) + ind(W (a) − H(b)) = indW (c) + indW (d).

We conjecture that if one of the operators W (a)+H(b) or W (a)−H(b) is Fredholm, then so is the other and relation (2.10) holds.

3. Kernels of Wiener–Hopf plus Hankel operators. Specification

In this section we study the kernels of Wiener–Hopf plus Hankel operators W (a) + H(b) in the case where the generating functions a, b ∈ G satisfy matching condition (2.5) and W (c), W (d) are mainly Fredholm operators such that

0 ⩽ | indW (c)|, | indW (d)| ⩽ 1.

Recall that a is supposed to be invertible in G. In view of Theorem 1.1, on has ν(c) = ν(d) = 0 and 0 ⩽ |n(c)|, |n(d)| ⩽ 1.

In order to formulate our first result we need the following lemma.

Lemma 3.1. Let χ(t) := (t − i)/(t + i), t ∈ ℝ.

(i) If the function ψ is defined by

ψ(t) := \{ \begin{matrix}e^{−t} & if t > 0, \\ 0 & if t < 0, \end{matrix}

then W^{0}(χ^{−1} )ψ = −ψ̃. (ii) On each space L^{p}(ℝ^{+} ), 1 ⩽ p ⩽ ∞ the operator W (χ^{−1} ) has a onedimensional kernel generated by the function ψ_{0}(t) = e^{−t} , t > 0.

Proof. Assertion (i) can be obtained by using the relation

(W^{0}(χ^{−1} )g)(t) = g(t) − 2 ∫_{t}^{∞} e^{t−s} g(s) ds, −∞ < t < ∞,

which is valid for all g ∈ L^{p}(ℝ), 1 ⩽ p ⩽ ∞, [11].

Assertion (ii) is well known. It can be proved by the differentiation of the identity

φ(t) = 2 ∫_{t}^{∞} e^{t−s} φ(s) ds.

Moreover, one has

(W (χ^{−1} )g)(t) = g(t) − 2 ∫_{t}^{∞} e^{t−s} g(s) ds, 0 < t < ∞,

(see [11]). Note that assertion (ii) also follows from assertion (i). □

Now we can derive the following version of the Coburn–Simonenko Theorem.

Theorem 3.1. Let a ∈ G be invertible and let A denote any of the four operators W (a) − H(aχ), W (a) + H(aχ^{−1} ), W (a) ± H(a). Then at least one of the spaces kerA or cokerA is trivial.

Proof. Part 1: Let us start with the operator W (a) + H(aχ). The function χ satisfies the relation χ̃ = χ^{−1} , so the duo (a, aχ) is a matching pair with the subordinated pair (c, d) with c = χ^{−1} and d = aa^{−1} χ. Moreover, the operator W (χ^{−1} ) is invertible from the right and one of its right inverses is the operator W (χ). Thus the theory of Section 2 applies. As it was pointed out earlier, the kernel of this operator is kerW (χ^{−1} ) = {cψ_{0} : c ∈ ℂ}, where ψ_{0}(t) = e^{−t} , t > 0. In order to apply Proposition 2.3 we have to identify, in particular, the projections P^{±} (χ^{−1} ) acting on the space kerW (χ^{−1} ). But P^{+} (χ^{−1} ) and P^{−} (χ^{−1} ) are complimentary projections on the one-dimensional space kerW (χ^{−1} ). Therefore, one of these projections is just the identity operator whereas the other one is the zero operator. Consider next the expression JQW^{0}(χ^{−1} )P ψ_{0}. By Lemma 3.1(i) one has

JQW^{0}(χ^{−1} )P ψ_{0} = JQW^{0}(χ^{−1} )ψ = −JQψ̃,

so that JQW^{0}(χ^{−1} )P ψ_{0} = −ψ_{0} and P^{−} (χ^{−1} ) = I on kerW (χ^{−1} ).

According to Proposition 2.3, the kernels of the operators W (a) + H(aχ) and W (a) − H(aχ) can be represented in the form

(3.1)

ker(W (a) − H(aχ)) = φ^{−} (imP^{−} (d)),
ker(W (a) + H(aχ)) = φ^{+} (imP^{+} (d)) ∔ {cψ_{0} : c ∈ ℂ}.

If dimkerW (d) > 0, then coker(W (a) ± H(aχ)) = {0}. Indeed, relation (2.7) and the familiar Coburn–Simonenko Theorem for the operator W (d) show that cokerW (V (a, aχ)) = {0}. Taking into account representation (2.4), one obtains that the cokernel of each of the operators W (a)+H(aχ) and W (a)−H(aχ) contains the zero element only.

Let us now assume that kerW (d) = {0}. Then the first relation (3.1) implies that ker(W (a) − H(aχ)) = 0. Hence, the operator W (a) − H(aχ) is subject to Coburn–Simonenko Theorem.

Part 2: Consider the operator W (a) + H(aχ^{−1} ) and note that W (c) = W (χ) is not right-invertible, so that Proposition 2.3 cannot be directly used in this situation. Nevertheless, the case at hand can be reduced to the operators studied. Thus the operators W (a) ± H(aχ^{−1} ) can be represented in the form

(3.2)  W (a) ± H(aχ^{−1} ) = (W (aχ^{−1} ) ± H(aχ^{−1} χ))W (χ).

The proof of (3.2) follows from (1.4) and relation H(χ)W (χ) = 0. Setting α := aχ^{−1} , we get

(3.3)  W (a) ± H(aχ^{−1} ) = (W (α) ± H(αχ))W (χ).

The operators of the form W (α) ± H(αχ) in the right-hand side of (3.2) have been just studied, and we already know that the function ψ_{0} belongs to the kernels of both operators W (α)+H(αχ) and W (χ^{−1} ). Since W (χ^{−1} )W (χ) = I it follows that ψ_{0} ∉ imW (χ). Consider now the projection Q_{0} := W (χ)W (χ^{−1} ) which projects the space L^{p}(ℝ^{+} ), 1 ⩽ p ⩽ ∞ onto imW (χ) parallel to kerW (χ^{−1} ).

Assume first that kerW (d) = {0} and note that for the matching pairs (a, aχ^{−1} ) and (α, αχ), the corresponding subordinated pairs (c, d) have the same element d, namely, d = aa^{−1} χ. Then (3.3) shows that ker(W (a) + H(aχ^{−1} )) = {0}. Further, if dimkerW (d) > 0, then the space ker(W (α) + H(αχ)) decomposes as follows

ker(W (α) + H(αχ)) = kerW (χ^{−1} ) ⊕ Q_{0}(ker(W (α) + H(αχ))).

However, as was already shown, the operator W (α) − H(αχ) is right-invertible and

kerW (χ^{−1} ) ⊂ kerW (α) + H(αχ).

Therefore, relation (3.2) implies that the operator W (a) + H(aχ^{−1} ) maps L^{p}(ℝ^{+} ) onto L^{p}(ℝ^{+} ), so it is subject to the Coburn–Simonenko Theorem.

Part 3: It remains to consider the operators W (a) ± H(a). For these operators the element c in the corresponding subordinated pair is either 1 or −1, and our claim follows immediately from the Coburn–Simonenko Theorem for scalar Wiener–Hopf operators and from relations (2.7) and (2.4). □

Remark 3.1. The proof of Theorem 3.1 shows that this theorem remains true for more general generating functions, for instance, in the case where a and b belong to the algebras G_{p}, 1 ⩽ p ⩽ ∞ studied in [11, Chapter VII].

The reader can also observe that, in fact, we have proved a bit more than Theorem 3.1 states. A more detailed result can be formulated as follows.

Corollary 3.1. Let a ∈ G be invertible. Then (i) If dimkerW (d) = 0, then

ker(W (a) − H(aχ)) = {0}, ker(W (a) + H(aχ)) = {cψ_{0} : c ∈ ℂ},

and if dimkerW (d) > 0, then coker(W (a) ± H(aχ)) = {0}. (ii) If dimkerW (d) = 0, then ker(W (a) ± H(aχ^{−1} )) = {0},

and if dimkerW (d) > 0, then coker(W (a) + H(aχ^{−1} )) = {0}.

Let us emphasize that the description of the projections P^{±} (χ^{−1} ) did play an important role in our considerations. In the general case one has to study the projections P^{±} (g) for the functions g satisfying the relation gg̃ = 1. Because of the space restriction, we are not going to pursue this matter here. Nevertheless, let us consider the case where ν(g) = 0 and n(g) = −1, which is one of the simplest generalization of the situation g = χ^{−1} . In order to handle this case we need a result from [11, Chapter VII].

Proposition 3.1. Each invertible function g ∈ G admits the factorization of the form

(3.4)  g(t) = g_{−} (t)e^{iνt} ( \frac{t − i}{t + i} )^{n}g_{+} (t), −∞ < t < ∞,

where g_{+}^{±1} ∈ G^{+} , g_{−}^{±1} ∈ G^{−} , ν = ν(g) and n = n(g). Moreover, under the agreement g_{−} (0) = 1, the factorization factors g_{+} and g_{−} are uniquely defined.

Note that the proof of Theorem 1.1 is based on Proposition 3.1.

Definition 3.1. Suppose that g ∈ G satisfies the condition gg̃ = 1 and set

ξ(g) = (−1)^{n}g(0), n = n(g).

Theorem 3.2. If g ∈ G and gg̃ = 1, then ξ(g) = ±1 and the factorization (3.4) takes the form

(3.5)  g(t) = (ξ(g)g_{+}^{−1} (t))e^{iνt} ( \frac{t − i}{t + i} )^{n}g_{+} (t)

with g_{+}^{±1} (t) ∈ G^{−} and g_{−} (t) = ξ(g)g_{+}^{−1} (t).

Proof. Using the condition g^{−1} = g̃, we get from (3.4) that

g_{+}^{−1} (t)e^{−iνt} ( \frac{t − i}{t + i} )^{−n} g_{−}^{−1} (t) = g̃_{−} (t)e^{−iνt} ( \frac{t − i}{t + i} )^{−n} g̃_{+} (t),

where ν = ν(g), n = n(g).

Note that g_{−}^{±1} ∈ G^{+} , g_{+}^{±1} ∈ G^{−} , as easy computations show. Therefore,

g_{+}^{−1} g_{−}^{−1} = g_{−} g̃_{+} ,

and g_{+}^{−1} g_{−}^{−1} ∈ G^{+} , g_{−} g̃_{+} ∈ G^{−} . It follows that there is a constant ξ ∈ ℂ such that g_{+}^{−1} g_{−}^{−1} = ξ = g_{−} g̃_{+} , and g_{−} = ξ g_{+}^{−1} . For the function g_{0} = g_{+} g_{−} we have g_{0} g̃_{0} = 1. Therefore,

1 = g_{0} g̃_{0} = (ξg_{+} g_{+}^{−1} )(ξg̃_{+} g_{+}^{−1} ) = ξ^{2}.

For t = 0, which is one of the fixed points of the operator J, the equation g_{0} = ξg_{+} g_{+}^{−1} implies g_{0}(0) = ξ, and g_{0}(0) = g(0)(−1)^{n} (see (3.5)). Thus we obtain that ξ = g(0)(−1)^{n} which completes the proof. □

Now we again use the notation

χ^{±1} (t) = ( \frac{t − i}{t + i} )^{±1} , t ∈ ℝ.

Theorem 3.3. Let g ∈ G, gg̃ = 1, ν(g) = 0 and n(g) = −1. Then

imP^{±} (g) = \{ c( \frac{1 ∓ ξ(g)}{2} ) W (g_{+}^{−1} )ψ_{0} : c ∈ ℂ\} .

Proof. It is easily seen that kerW (g) = {cW (g_{+}^{−1} )ψ_{0} : c ∈ ℂ} , According to the definition of projections P^{±} (g) we have to compute the expression

JQW^{0}(g)P W (g_{+}^{−1} )ψ_{0}.

We have

JQW^{0}(g)P W (g_{+}^{−1} )= JQW^{0}(g_{−} )W^{0}(χ^{−1} )W^{0}(g_{+} )W^{0}(g_{+}^{−1} )P
= JQW^{0}(g_{−} )W^{0}(χ^{−1} )P.

Recall that by Lemma 3.1, W^{0}(χ^{−1} )P ψ_{0} = W^{0}(χ^{−1} )ψ = −ψ̃, and using Theorem 3.2 we get

JQW^{0}(g)P W (g_{+}^{−1} )ψ_{0} = −JQW^{0}(g_{−} )ψ̃ = −W^{0}(g̃_{−} )ψ
= −P ξ(g)W^{0}(g_{+}^{−1} )P ψ_{0} = −ξ(g)W^{0}(g_{+}^{−1} )ψ_{0},

and we are done. □

The next result is a generalization of Theorem 3.1.

Theorem 3.4. Let a, b ∈ G constitute a matching pair, a be invertible in G and let (c, d) be the subordinated pair for (a, b). If A denotes one of the following operators

(i) W (a) ± H(b) with ν(c) = 0, n(c) = 1 and ξ(c) = ±1;

(ii) W (a) ∓ H(b) with ν(c) = 0, n(c) = −1 and ξ(c) = ±1;
(iii) W (a) ± H(b) with ν(c) = 0 and n(c) = 0

considered on the space L^{p}(ℝ^{+} ), then at least one of the spaces kerA or cokerA is trivial.

Proof. The proof mimics that of Theorem 3.1 with minor modifications. First, we note that the case ξ(c) = −1 can be reduced to the case ξ(c) = 1 via rearrangements W (a) + H(b) = W (a) − H(−b) and W (a) − H(b) = W (a) + H(−b). Therefore, we only consider the situation ξ(c) = 1 in the cases (i) and (ii). Further, one has to use Theorem 3.3 instead of the description of the projections P^{±} (χ^{±1} ). Consider the operator W (a) + H(b) in the case where ν(c) = 0 and n(c) = 1. Representing the operator W (a) ± H(b) in the form

W (a) ± H(b) = (W (aχ^{−1} ) ± H(bχ))W (χ),

we observe that (aχ^{−1} , bχ) is a matching pair with the subordinated pair (cχ^{−2} , d) and indW (cχ^{−2} ) = −1, imP^{+} (cχ^{−2} ) = kerW (cχ^{−2} ) = {cW (c_{+}^{−1} )ψ_{0} : c ∈ ℂ}. Let us also note that kerW (cχ^{−2} ) = kerW (c_{+} χ^{−1} ) and W (c_{+} χ^{−1} ) W (c_{+}^{−1} χ) = I. Hence, kerW (cχ^{−2} )∩imW (c_{+}^{−1} χ) = {0}. Since obviously imW (c_{+}^{−1} χ) = imW (χ), we obtain kerW (cχ^{−1} ) ∩ imW (χ) = {0}.

Now one can proceed similarly to Part 2 in the proof of Theorem 3.1. □

Corollary 3.2. Assume that a, b ∈ G constitute a matching pair with the subordinated pair (c, d) such that ξ(c) = 1. Then

(i) If dimkerW (d) = 0, and indW (c) = 1, then

ker(W (a) − H(b)) = {0}, ker(W (a) + H(b)) = {cW (c_{+}^{−1} )ψ_{0} : c ∈ ℂ},

and if dimkerW (d) > 0, then coker(W (a) ± H(b)) = {0}. (ii) If dimkerW (d) = 0, and indW (c) = −1, then ker(W (a) ± H(b)) = {0},

and if dimkerW (d) > 0, then coker(W (a) + H(b)) = {0}.

An interesting and important subclass of the operators considered in this paper comprises the identity plus Hankel operators. Let us specify the above results in this situation

Corollary 3.3. If b ∈ G is a matching function, then (1, b) is a matching pair with the subordinated pair (b̃, b), and if A denotes any of the operators

(i) I − H(b) with ν(b̃) = 0, n(b̃) = −1 and ξ(b̃) = 1;

(ii) I + H(b) with ν(b̃) = 0, n(b̃) = 1 and ξ(b̃) = 1;
(iii) I ± H(b) with ν(b̃) = 0 and n(b̃) = 0,

considered on the space L^{p}(ℝ^{+} ), then kerA or cokerA is trivial.

Now we revisit Theorem 3.1 and consider the operators W (a) ± H(aχ) and W (a) ± H(aχ^{−1} ) under additional assumptions.

1^{0} . Suppose that ν(a) = n(a) = 0 and b = aχ. The subordinated pair (c, d) is given by the elements c = χ^{−1} and d = aa^{−1} χ. Thus

indW (c) = 1, indW (d) = −1, ξ(c) = ξ(d) = 1.

According to (2.10) we have

(3.6)  ind(W (a) + H(aχ)) + ind(W (a) − H(aχ)) = 0.

Further, by Corollary 3.1(i) we also have

ker(W (a) − H(aχ)) = 0, ker(W (a) + H(aχ)) = {cψ_{0} : c ∈ ℂ}.

In order to describe the cokernels of the above operators we make use of the adjoint operators. If p ∈ [1, ∞), then according to (2.6) the adjoint operators have the form W (\overline{a}) ± H(\overline{ã}χ), and the duo (\overline{a}, \overline{ã}χ) is a matching pair with the subordinated pair (\overline{d}, \overline{c}), so that indW (\overline{d}) = 1, indW (\overline{c}) = −1 and ξ(\overline{d}) = 1. By Corollary 3.2(ii), ker(W (\overline{a}) − H(\overline{ã}χ)) = \{0\}, which finally proves that the operator W (a) − H(aχ) is invertible. Note that this result is also true for the space L^{∞} (ℝ^{+} ). Indeed, the operator W (\overline{a}) − H(\overline{ã}χ) acts on the space L^{1}(ℝ^{+} ) and the above considerations show that dimker(W (\overline{a}) − H(\overline{ã}χ)) = 0. The adjoint of this operator acts on the space L^{∞} (ℝ^{+} ) and is equal to the operator W (a) − H(aχ), the kernel of which is trivial. Therefore, the operator W (\overline{a}) − H(\overline{ã}χ) is invertible on the space L^{1}(ℝ^{+} ). Consequently, its adjoint W (a) − H(aχ) is invertible on L^{∞} (ℝ^{+} ). Then relation (3.6) immediately implies that ind(W (a) + H(aχ)) = 0. Note that the operator W (a) + H(aχ) provides an example of operators where both spaces ker(W (a) + H(aχ)) and coker(W (a) + H(aχ)) are nontrivial.

2^{0} . Suppose that ν(a) = 0, n(a) = −1 and b = aχ. For the subordinated pair (c, d) we have c = χ^{−1} and d = aa^{−1} χ so that indW (c) = 1, indW (d) = 1, ξ(d) = 1. Since indW (d) = 1, Corollary 3.1(i) indicates that coker(W (a) ± H(aχ)) = {0}. besides, dimker(W (a) ± H(aχ)) = 1 by Proposition 2.3.

3^{0} . Suppose that ν(a) = n(a) = 0 and b = aχ^{−1} . Since c = χ, the operator W (c) is not invertible from the right. Write

(3.7)  W (a) ± H(aχ^{−1} ) = (W (aχ^{−1} ) ± H(aχ^{−1} χ))W (χ),

and set α := aχ^{−1} . The operators W (α) ± H(αχ) are considered in 2^{0} , so we have

dimker(W (α) ± H(αχ)) = 1
dimcoker(W (α) ± H(αχ)) = 0.

According to the Part 2 in the proof of Theorem 3.1, one has ker(W (a)+H(aχ^{−1} )) = {0}. This and the relation dimcoker(W (α) + H(αχ)) = 0 show the invertibility of the operator W (a) + H(aχ^{−1} ). Due to Proposition 2.3 (see also (3.1)) we know that the kernel of the operator W (α) − H(αχ) is spanned on the element

(3.8) κ = W (χ)W (α^{−1} )W (d_{+}^{−1} )ψ_{0} + JQW^{0}(χ^{−1} )P W^{0}(χ)P W (α^{−1} )W (d_{+}^{−1} )ψ_{0}

− JQW^{0}(α^{−1} )P W (d_{+}^{−1} )ψ_{0},

where we used the fact that W (χ) is a right inverse for the operator W (χ^{−1} ) and where d_{+}^{−1} arises from the factorization (3.5) of the function d = aa^{−1} χ^{−1} . Note that the first term in (3.8) belongs to the set imW (χ), whereas the second one is equal to zero. Thus the operator W (a) − H(aχ^{−1} ) is invertible if and only if H(α^{−1} )W (d_{+}^{−1} )ψ_{0} ∉ imW (χ). On the other hand, if this condition is not satisfied, the operator W (a) − H(aχ^{−1} ) presents an example of a Wiener–Hopf plus Hankel operator with one-dimensional kernel and cokernel.

4^{0} . Suppose that ν(a) = 0, n(a) = 1 and b = aχ^{−1} . Let us use representation (3.7) and set α = aχ^{−1} . It follows from Part 1^{0} that W (α) − H(αχ) is invertible whereas the operator W (α)+H(αχ) has one-dimensional kernel and cokernel. Since

ker(W (α) + H(αχ)) = {cψ_{0} : c ∈ ℂ} ∩ imW (χ) = {0},

we conclude that the operator W (a) + H(aχ^{−1} ) has trivial kernel and a cokernel of dimension 1. Of course, the same conclusion is valid for the operator W (a)−H(aχ).

It is worth noting that a similar consideration with natural amendments can be used in the contest of Theorem 3.4. Let us restrict ourselves to the operators I + H(b) with the generating function b satisfying the condition bb̃= 1. Then (1, b) is a matching pair with the subordinated pair (b̃, b).

5^{0} . Suppose that ν(b) = n(b) = 0. Then the operators W (b) and W (b̃) are invertible and relations (2.4), (2.7) already show that I + H(b) and I − H(b) are invertible operators.

Assume next that ν(b) = 0 but n(b) = 1 and ξ(b̃) = 1. Then indW (b̃) = 1 and indW (b) = −1. By Corollary 3.2(i), one has

ker(I − H(b)) = {0}, ker(I + H(b)) = {cW (b_{+} )ψ_{0} : c ∈ ℂ}.

Similarly to Part 1^{0} one shows that the operator I − H(b) is invertible and ind(I + H(b)) = 0.

Finally, let us assume that ν(b) = 0, n(b) = −1 and ξ(b̃) = 1. Since indW (b̃) = −1, we will use the relation I ± H(b) = (W (χ^{−1} ) ± H(bχ))W (χ). It is clear that (χ^{−1} , bχ) is a matching pair with the subordinated pair (b̃χ^{−2} , b) and indW (b̃χ^{−2} )= indW (b) = 1. Analogously to Part 2^{0} we obtain that

coker(W (χ^{−1} ) ± H(bχ)) = {0}.

Moreover, by Proposition 2.3, dimker(W (χ^{−1} ) ± H(bχ)) = 1 and since

ker(W (χ^{−1} ) + H(bχ)) = {cW (b_{+} )ψ_{0} : c ∈ ℂ} ∩ imW (χ) = {0},

the operator I + H(b) is invertible. If kerW (χ^{−1} ) − H(bχ) ∩ imW (χ) = {0}, then I − H(b) is invertible. Otherwise, ind(I − H(b)) = 0, but this operator is not invertible.

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