A Riemann-Hilbert problem with a vanishing coefficient and applications to Toeplitz operators

Abstract We study the homogeneous Riemann-Hilbert problem with a vanishing scalar-valued continuous coefficient. We characterize non-existence of nontrivial solutions in the case where the coefficient has its values along several rays starting from the origin. As a consequence, some results on injectivity and existence of eigenvalues of Toeplitz operators in Hardy spaces are obtained.


Introduction
Coburn's theorem states that a nonzero Toeplitz operator has a trivial kernel or a dense range. It follows that for a continuous symbol : T → C, a point λ in σ (T ) \ σ ess (T ) is an eigenvalue of T if and only if the winding number of λ − , wind(λ − ), is negative. On the other hand, the question of whether λ in σ ess (T ) (= (T)) is an eigenvalue is quite delicate and only very few results are known, most of which require strong restrictions on the behavior of the symbol in the neighborhood of its zeros; see [14] and the references therein. Let us denote the set of all Hölder continuous functions on T by C µ . The following result easily follows from [13,Lemma 4.11] and Proposition 6.

Theorem 1.
Let 1 < < ∞, µ ∈ [0 1], and let : T → R be a continuous function such that while are not eigenvalues of T whenever

Remark 2.
The difference between (1.2) and (1.3) is explained by the simple observation that in the first case − λ gets both positive and negative values; while in the latter case the argument of − λ remains constant.
Recall that σ (T : provided that is continuous real-valued function (see [1,Sec. 2.36]). Therefore, Theorem 1 implies that T has no eigenvalues if is Lipschitz continuous. If ∈ C µ with µ < 1, then we can only say that there are no eigenvalues in the interior of σ (T ); in fact, one can construct a real-valued Hölder continuous function so that the endpoints of σ (T : H 2 → H 2 ) are eigenvalues of T ; see [12]. The next result, which follows from the main result of [15], generalizes Theorem 1 to the case when the values of are located on two rays where, in general, δ 1 ≤ δ 2 < δ 3 < · · · < δ < 2π. We also set E = { ∈ T : ( ) ∈ S }. Theorem 3.
If, in addition, the symbol traverses through S to S at most finitely many times, then 0 in σ ess (T ) is not an eigenvalue Note that the setting of Theorem 1 is invariant under translation by λ ∈ (T), whereas the setting of Theorem 3 is not. The present approach addresses the question of whether 0 is not an eigenvalue of T ; that is, whether T is injective.
However, this bound is not optimal as we see in the following theorem. In what follows, we identify functions defined on T with 2π-periodic functions defined on R. The function is assumed to have values on rays S 1 S 2 S starting 29 Brought to you by | University of Reading Authenticated | 134.225.68.6 Download Date | 10/29/13 4:42 PM from the origin. Let θ( ) = arg ( − 0), which is clearly a is piecewise constant function. We assume that the argument has only finitely many jumps. A point at which goes to zero along a ray S and then returns to the same ray is not considered a jump. Note that since θ( ) = arg ( − 0), the function θ remains constant when traversing over such zeros. Therefore, we allow some cases where there are infinitely many zeros, which is a natural setting in some applications. However, the Riemann-Hilbert problem cannot have a nontrivial solution if has zeros on a set of positive measure. The change in the argument of at 0 is denoted by δ 0 ; that is, and denote the largest contribution from both positive and negative jumps by δ; that is, Note that, by a simple rotation argument, we may assume that we always have δ ≤ π.

Theorem 4.
satisfy (1.1), and suppose δ ± ≤ π. Then the Toeplitz operator T : provided that the symbol traverses through any S to another S at most finitely many times.
Obviously if traverses only along neighboring rays, then (1.8) is no different from (1.5). Also, if there are only two rays S 1 and S 2 , then δ = δ 2 − δ 1 , and the condition δ ≤ π is superfluous. The adjoint of T : H → H is the operator T¯ : H → H (1/ + 1/ = 1). Since the setting of our theorem is invariant under complex conjugation, it is not difficult to construct operators T such that both T and T * are injective. However, since vanishes, such operators cannot be Fredholm. This reflects the inconvenient fact that T with vanishing symbol is often not normally solvable. The proof of Theorem 4 is given in the following section. Our approach is based on that of [15]. The reason we have a strict inequality in (1.8) is related to the properties of some norm inequalities for harmonic conjugation of characteristic functions; see [2, Chap. III, Sec. 2]. We conjecture that the strict inequality in (1.8) may only be needed if the argument of the symbol has infinitely many jumps. However, our aim here is to show how some of the main results of the two-ray case (see [15]) are altered when additional rays are inserted between the two, depending on the order in which the symbol traverses through the rays. This provides more insight into how the geometry of the sets E and S affects the nonexistence of eigenvalues in the essential spectra of Toeplitz operators. It is also of interest to know what happens when the symbol is matrix-valued.

The Riemann-Hilbert problem
Let ∈ L ∞ and 1 < < ∞. The Riemann-Hilbert problem (RHP) in Hardy spaces is the problem of finding ψ ∈ H (D) for which where * denotes the nontangential boundary values of (see [2]). The following well-known result essentially shows that the study of Toeplitz operators is closely related to the RHP in Hardy spaces. We give the proof for completeness because it is not readily available in the literature. Let us first recall a couple of useful results. For ∈ L 1 , define Note that such F is analytic on C \ T and F (∞) = 0. For ∈ T, we denote by F + ( ) the boundary values of F as → nontangentially in D and by F − ( ) as → outside of T. According to the Plemelj formulas, if ∈ L 1 , then F is analytic in C \ T and where Q = I − P is the complementary projection (see [5, Chapter 2, Section 4]).

Proposition 5.
If The one-to-one correspondence of the two problems follows from the fact that F 1 = F 2 if and only if 1 = 2 almost everywhere.

Remark 7.
The assertion of Proposition 6 remains true also if = 1 provided that T is bounded on H 1 . The proof of the case = 1 is analogous to the general case.
where θ( ) = arg ( ). Note that where is the Hilbert transform, defined by

Remark 9.
Theorem 4 shows that T is injective if > 2 We can show that the condition is sharp; that is, if then we can construct symbols ∈ C µ such that the kernel of T is nontrivial. Indeed, let be in C µ such that ( ) ∈ S 1 ∪ S ∪ {0} and | | −1 ∈ L for < µ −1 . Recall the outer functions X and H defined in (2.5) and (2.7). Since |H * | = | | 1/2 and H is outer, we have H ∈ H ∞ and H −1 ∈ H 2 for < µ −1 . Let Then * = ψ * (see (2.8)), and so 0 ∈ (T) = σ ess (T ) is an eigenvalue of T . Using Lemma 8 and Hölder's inequality, we see that ∈ H and ψ ∈ H provided that 1 ≤ < 2/(µ + δ + /π). It is obvious that there are symbols such as those above with an additional property that δ + ≤ δ − . Thus, if (2.10) holds, then the kernel of T may be nontrivial. According to Coburn's lemma, a nonzero bounded Toeplitz operator T on H has a trivial kernel or a dense range. When = 2, this result was proved for Hölder continuous symbols in [7], for continuous symbols in [4,10], and for bounded symbols in [3]. The case 1 < < ∞ for bounded symbols is in [11]. Because of the duality argument used in the proof of Coburn's lemma in the most general case, there seems to be no obvious way to extend the result to the case = 1. However, there is an alternative approach due to Vukotić [17], which we recall next. We write where is the set of all analytic polynomials.

Theorem 10.
Let 1 ≤ < ∞ and suppose that T is a nontrivial bounded Toeplitz operator on H . If T is not one-to-one, then Proof. The proof given in [17] also works when = 2. We only comment on the case = 1. Put χ ( ) = . The main idea is still the observation that the rank of the commutator is at most one; that is, it can be showed (see [17]) that is the analytic extension of T . All the algebraic properties used in [17] remain true in the case 1 ≤ < ∞.
Coburn's lemma for Toeplitz operators on H 1 now follows directly from the preceding theorem.

Corollary 11.
If T is bounded on H 1 , then either its kernel is trivial or its range is dense.
It is well known that continuity of is not sufficient for T to be bounded on H 1 . The most natural substitute for the class of continuous functions is the algebra where V MO log is the space of functions of logarithmic vanishing mean oscillation; see [9,16] for the definition. Observe that  Proof. Apply (3.2), (3.3), and Corollary 11.
As in the case 1 < < ∞, the situation regarding the (non)existence of eigenvalues embedded in the essential spectra of Toeplitz operators is a much more difficult question. One reason that the approach used in the previous section cannot be applied here is related to the role that conjugate exponents play in the proof of Theorem 4. All known results are restricted to real-valued symbols that satisfy Hölder or a slightly weaker condition. We give one condition, which is based on the following result (see [13]). Let : R → R be a 2π-periodic function which satisfies the condition in (1.1) with µ = 1. If changes sign, the Riemann-Hilbert problem (2.1) has no solutions in H 1 .
Let us see what this means in terms of eigenvalues. Using a similar argument as in the proof of Proposition 6, it is easy to see that no point in the interior of the essential spectrum of T can be an eigenvalue of T . In particular, if is Lipschitz continuous and if λ is in the essential spectrum of T , then λ may be an eigenvalue only if it is one of the endpoints of (T).