On the Spectrum of Hilbert Matrix Operator

The Hilbert matrix Hλ=1n+m+λn,m=0∞,λ≠0,-1,-2,…\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {H}}_\lambda =\left( \frac{1}{n+m+\lambda }\right) _{n,m=0}^{\infty }, \quad \lambda \ne 0,-1,-2, \ldots \, \end{aligned}$$\end{document}generates a bounded linear operator in the Hardy spaces Hp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$H^p$$\end{document} and in the lp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$l^p$$\end{document}-spaces. The aim of this paper is to study the spectrum of this operator in the spaces mentioned. In a sense, the presented investigation continues earlier works of various authors. More information concerning the history of the topic can be found in the introduction.


Introduction
The classical Hilbert matrix (1.1) was introduced by Hilbert in 1894 while working on a problem in theory of orthogonal polynomials. In particular, he studied finite sections of the matrix H 1 and observed that they are strongly ill-conditioned. This fact plays an important role in numerical analysis. In this work however, we are merely interested in the matrix H 1 itself and in its generalization , λ = 0, −1, −2, . . . (1.2) thought as linear operators acting on the spaces l p of p-summable sequences and on Hardy spaces H p . In what follows, unless specified otherwise, we assume that 1 < p < ∞. and π/(sin(π/p)) is the best possible constant in the inequality (1.3).
Thus the operator H 1 is bounded on the space l p . On the other hand, a lot of effort has been spent on studying Hilbert matrix operator H λ acting in spaces constituted by subspaces of the space A(D) of all holomorphic functions on D. The idea is to let the matrix H λ act on the Taylor coefficient sequences of functions from the corresponding subspace. The so-defined operator is also denoted by H λ and the reader can consult [23] for more information concerning this matter.
In the period from 1950 to 1960 the spectral properties of the operator H 1 and its generalization H λ have been vigorously studied. In particular, remarkable results are obtained for the spectrum of H 1 in l 2 and its latent roots. Definition 1.2. A latent root for H λ , λ = Z − is a complex number α such that there is a non-zero sequence (x n ) ∞ n=0 , x n ∈ C with the property ∞ m=0 x m n + m + λ = αx n , n = 0, 1, . . . .
We note that it is not supposed the operator H λ acts on a Banach space. Latent roots of H λ also attracted considerable attentions and the most general results are obtained by Rosenblum [18] and Hill [9]. For instance, the following remarkable result is true. Theorem 1.3. (cf. Hill [9]). Let μ be a complex number such that 0 < Re μ ≤ 1/2 and x n = x n (λ, μ) be the sequence defined by x m n + m + λ = π sin(πμ) x n , n = 0, 1, . . . , i.e. π/(sin(πμ)) is a latent root of H λ .
Some other prominent results to be mentioned here are related to the spectrum of Hilbert matrix operators.
On the other hand, taking into account that for real λ the operator H λ : l 2 → l 2 is self-adjoint, Rosenblum explicitly diagonalized H λ and described its spectrum. Theorem 1.5. (cf. Rosenblum [19]). Let λ = 1 − k, where k is a real number such that k / ∈ N. Then the following assertions hold: (i) For any real λ the operator H λ : l 2 → l 2 has a continuous spectrum of multiplicity one located on the interval [0, π]. (ii) If k ≤ 1/2, then the operator H λ has no point spectrum. (iii) If k > 1/2 and u and v are, respectively, the largest non-negative integers such that then π/ sin(πk) and −π/ sin(πk) are the only eigenvalues of the operator H λ and they have multiplicity u + 1 and v + 1, respectively.
Another important work is due to Aleman et al. [1] who, in particular, described the point spectrum of the operator H λ considered on Hardy spaces H p . Besides, Gonsales [6] draw the attention to the spectrum description for the Hilbert matrix operator acting on spaces l p and to the essential spectrum of H λ . The main tool to achieve this aim is the theory of Toeplitz and Hankel operators with piecewise continuous generating functions. Therefore, in Sect. 2, we recall some features of these operators acting on H p or l p spaces. Section 3 contains the description of the essential spectrum of H λ in the spaces H p and l p , 1 < p < ∞. Section 4 deals with the point spectrum of H λ with respect to H p , 1 < p < ∞ and λ ∈ C \ Z. If λ ∈ N, the spectrum of H λ with respect to H p is studied in Sect. 5. The results of Sects. 3-5 are summarized in the Main Theorem formulated in Sect. 6. Section 7 is devoted to the description of the spectrum of H λ acting in l p -spaces. It is shown that this description follows from the Main Theorem formulated in Sect. 6. Section 8 provides an outlook for extension of the present work results.

Spaces and Operators
For every Banach space X, let L(X) stand for the Banach algebra of all bounded linear operators acting on X. We denote by K(X) the closed twosided ideal of all compact operators in L(X). The quotient algebra L(X)/K(X) is called Calkin algebra. For us its importance lies in the fact that the invertibility of the coset A + K(X), A ∈ L(X), in the Calkin algebra is equivalent to the Fredholmness of A, i.e. to the finite-dimensionality of the spaces ker A := {x ∈ X : Ax = 0} and coker A := X/ im A. The number ind A := dim ker A − dim coker A is called the index of the Fredholm operator A. It is also well-known that the condition dim coker A < ∞ is equivalent to the fact that im A is closed and 2.1. The Spaces L p and H p Let T := {z ∈ C : |z| = 1} be the unit circle in the complex plane. We denote by L p , 1 ≤ p < ∞ the Banach space of all classes of complex-valued measurable functions on T summable in the p-th power with respect to the Lebesgue measure on T. Further, let C = C(T) and L ∞ = L ∞ (T) refer to the C * -algebras of all continuous functions and all classes of measurable and bounded functions on T, respectively. The norm in L p ,1 ≤ p ≤ ∞ is denoted by · p . It is clear that C is a C * -subalgebra of L ∞ . If we define functions χ n (t), n ∈ Z by χ n (t) = t n , t ∈ T, then {χ n (t)} n∈Z becomes a Schauder basis for any space L p , 1 < p < ∞.
Besides, we say that f ∈ A(D) belongs to the Hardy space The so-defined spaces H p are Banach spaces. Let us recall some of their properties.
1. If f ∈ H p then the limit q f (e iθ ) := lim exists for almost all θ ∈ [0, 2π] and for p < ∞ the resulting function q f is p-integrable on T. Moreover, projection P , which acts on the set P of all Laurent polynomials by The famous Riesz Theorem states that P extends to a bounded linear operator on the whole space L p , 1 < p < ∞. Besides, it is also clear that P 2 = P and H p is the image of L p for the Riesz projection, i.e. H p = P (L p ).
3. Fejer-Riesz inequality: We recall that Note that η β is the limit on T of that branch of (1−z) β , which is analytic for |z| < 1 and takes the value 1 at z = 0. In particular, for t ∈ T \ {1} we have with a function b invertible in L ∞ . It follows that η β ∈ L p if and only if hence η β ∈ H p if and only if β satisfies the inequality (2.2).

Toeplitz and Hankel Operators on H p .
Let a ∈ L ∞ . Then the Toeplitz operator T (a):  We also need a flip operator J : L p → L p defined by where bar denotes the complex conjugation. This operator J changes the orientation of the unit circle T, which is assumed to be counterclockwise This operator is bounded and a is also called generating function, but unlike Toeplitz operators, the mapping a → H(a) is not one-to-one. It is worth noting here that the matrix representations of the operators T (a) and H(a) in the Schauder basis {t n } n∈Z+ of H p , respectively, are and ( a j+i+1 ) where a k , k ∈ Z are the Fourier coefficients of a. A function a : T → C is called piecewise continuous if it has one-sided limits at any point of T. We will, respectively, write a(t − 0) and a(t + 0) for the left and right limits of f at t ∈ T, recalling that the unit circle T is counter-clockwise oriented. The set of all piecewise continuous functions f on T is denoted by P C. To describe the essential spectrum of H 1 , one can use a highly non-trivial fact that H 1 can be identified with a Hankel operator H(b), which belongs to the smallest closed subalgebra T p (P C) ⊂ L(H p ) containing all Toeplitz operators generated by functions a ∈ P C. More exactly, the matrix Hilbert operator H 1 on H p can be identified with the Hankel operator generated by the function b : exp(ix) → −ix + iπ, x ∈ [0, 2π). Indeed, it is easily seen that the matrix representation of H(b) coincides with H 1 and the action of H(b) can be precisely described by H 1 . For instance, and since the operators H 1 is bounded on H p (original definition) and on im P = H p (equivalent definition). Later on we will sketch the proof of the fact H(b) ∈ T p (P C). This result is important because for the algebra T p (P C) the Fredholm theory is well-developed and the essential spectrum of its elements can be determined (see [7] or [2]). Let us report a few essential features of that theory. Notice that the ideal K(H p ) of all compact operators is contained in T p (P C), i.e.

K(H p ) ⊂ T p (P C).
Therefore, we can consider the quotient algebra T π p (P C) := T p (P C)/K(H p ). Usually, for A ∈ T p (P C) the invertibility of the coset A + K(H p ) in T π p (P C) implies its invertibility in the Calkin algebra L(H p )/K(H p ), and hence the Fredholmness of A. The reverse question, does the invertibility of the coset A + K(H p ) in the Calkin algebra yields its invertibility in T π p (P C) is more interesting. If this is true for every A ∈ T p (P C), we say that T π Let R stand for the two-point compactification of R and let μ p and ν p be the functions defined on R by Notice that when ξ runs through the real line from −∞ to +∞, then μ p (ξ) runs along a circular arc joining the points z 1 = 0 to z 2 = 1 and passes through the point (1 − coth(π/p))/2.
is homeomorphic to the cylinder T × R provided with an exotic non-Euclidian topology. 3. For the algebra T π p (P C), the Gelfand transform G : Let smb :

Corollary 2.3. An operator
It is worth noting that for A ∈ T p (P C) the function smb A is not only responsible for the Fredholmness of A but also allows to determine the index of the the set of all complex numbers λ such that λI − A is not Fredholm, is called the essential spectrum of A and is denoted by ess sp A.

Corollary 2.6. For any operator A ∈ T p (P C), one has
range smb A = ess sp A.

Toeplitz and Hankel operators on l p .
Let l p (Z), 1 ≤ p < ∞ denote the complex Banach space of all sequences (x n ) n∈Z of complex numbers with the norm Replacing Z by the set of all non-negative integers Z + , we get another Banach space, denoted by l p := l p (Z + ). The space l p can be seen as a subspace of l p (Z). Let P be the canonical projection of l p (Z) onto l p and J the bounded linear operator on l p (Z) acting by (Jx) n := x −n−1 . We also consider the operator Q = I − P . For a function a ∈ L ∞ the Laurent operator L(a) associated with a acts on the space l 0 (Z) of all finitely supported sequences on Z by We note that for every k, there are only finitely many non-vanishing summands in this sum. The element a is called a multiplier on l p (Z) if the term It is well-known that M 2 = L ∞ . Moreover, every function a with bounded total variation Var (a) is in M p for every p, and the Stechkin inequality holds with a constant c p independent of a. In particular, every trigonometric polynomial and every piecewise constant function on T are multipliers for every p. Let P = P(T) and P K = P K(T) be, respectively, the algebra of all trigonometric polynomials and the algebra of all piecewise constant functions on T. By C p and P C p we, respectively, denote the closures of the sets P and P K in the algebra M p . Note that C 2 is just the algebra of all continuous functions on T, and P C 2 is the algebra of all piecewise continuous functions on T. Let us finally mention that the Wiener algebra W , which consists of all complex-valued functions defined on T and having absolutely convergent Fourier series, is a subalgebra of M p and For these and other properties of multipliers the reader can consult [2]. It is well-known that For a subalgebra A of M p , let T (A) refer to the smallest closed subalgebra of L(l p ), which contains all operators T (a) with a ∈ A. The Fredholm properties of operators from T (P C p ) are well-understood thanks to the work of R. Duduchava and I. Gohberg/N. Krupnik (see [2] and references therein). Despite Theorems 2.2 and 2.7 are in a sense analogous, their proofs are quite different.
Theorem 2.7. The following assertions are true: where 1/p + 1/q = 1 and μ q is defined by (2.4) with p replaced by q.
We also note that the conclusions mentioned after Theorem 2.2 are also valid for the algebra T (P C p ). In particular, if A ∈ T (P C p ) is Fredholm, then

The Essential Spectrum of H λ in H p and l p
We start this section with the following simple lemma.
we note that the sequence belongs to the Wiener algebra. It is easily seen that the Hankel operator H(v λ ) is compact on both spaces H p and l p . Now we set is bounded not only on H p , but also on l p . As a consequence, we get that H(b λ ) is bounded on both spaces H p and l p , 1 < p < ∞. Clearly, H λ is the matrix representation of H(b λ ) assumed as acting on H p or l p , and the lemma is proved.
In what follows we shall identify H λ with H(b λ ), the latter acting on H p or l p .
The proof of Lemma 3.1 implies the following corollary.
with the function ν p defined by (2.5). (b) For the operator H λ ∈ T (P C p ) the Gelfand transform of the coset H λ + K(l p ) is given by (c) The essential spectrum of H λ acting on H p or l p , 1 < p < ∞ is, respectively, equal to the range of πiν p or πiν q where 1/p + 1/q = 1, and for p = 2 we have ess sp H λ = [0, π].
Sketch of the proof. It is well-known that for bounded operators the Fredholm property and index are invariant under compact perturbations. Since K(H p ) ⊂ T p (P C) and K(l p ) ⊂ T (P C p ), the proof of Theorem 3.3 will follow from Theorems 2.2, 2.7 and Corollary 3.2, but some auxiliary results are needed before that. Therefore, f f is continuous at the point 1 and f f (1) = −1.
Since T (f f ) and T (f )T ( f ) are elements of T p (P C) (and also of T (P C p )), the well-known formula The Gelfand transform of the coset H(f ) 2 + K(H p ) equals It is easy to compute that Hence Notice that these considerations are taken from [22]. What we need to prove now is that the essential spectrum of H(f ) coincides with the range of ν p and H(f ) ∈ T p (P C). The former follows from the results in [8,21,22]. In explicit form they can be found in [20,Section 4.5], which lead to Theorem 4.5.2 in that section and to the description of the essential spectrum of H(f ). In order to show that H(f ) ∈ T p (P C) , one can use, for instance, [20,Theorem 2.3.4] and the fact that T p (P C) can be considered as an subalgebra of B(T, J) defined in [20,Section 4.4].
Thus H(f ) ∈ T p (P C) and Since H(b) = πiH(f ), the assertion (a) of Theorem 3.3 follows.
The assertion (b) of Theorem 3.3 can be proved as follows. Since and H(f ) ∈ T (P C p ), 1/p + 1/q = 1, one can use the results of [17], where the essential spectrum of H(f ) in l p is shown to coincide with the range of the function ν q . An alternative proof of these facts can be also found in [ The range of iπν p , 1 < p < ∞ is plotted in the next picture and is of drop-like form. Noting that for all ξ ∈ R, and we have range(πiν p ) = range(πiν q ), although the orientation of these curves is different.Therefore, if the range of πiν p is known, then so is the range πiν q , 1/p + 1/q = 1. Proof. Identifying H 1 with the operator P bJ, we note that b L ∞ = π and P = 1/ sin(π/p), cf. [11]. Therefore, On the other hand, the point π/sin(π/p) belongs to range(πiν p ) = ess sp H 1 , and the claim follows.

The Point Spectrum of H λ on H p
Recall that a complex number ω belongs to the point spectrum of a linear operator A defined on a Banach space X if Let W p denote the bounded closed set in C, the boundary of which coincides with the range of the function πiν p and let W 0 p be the interior of W p . In particular, we have W 2 = [0, π] and W 0 2 = ∅.
(c) If β / ∈ W p , 1 < p < ∞, then the operator βI − H λ acting on H p or l p is Fredholm with index zero. If λ > 1/2 is real, then βI − H λ is invertible on both spaces H p and l p .
Proof. Since H λ differs from H 1 by a compact operator and compact perturbations do not influence Fredholmness and index, it suffices to prove the assertions (a) and (b) for λ = 1.
As was already mentioned, the natural orientation of R induces an orientation of πiν s . The observations above suffice to describe the spectrum of H 1 on H p and l p . However, we postpone it because we want to consider a more general situation.
A more complete study of the point spectrum of H λ on H p for λ ∈ C \ Z was presented by Aleman et al. [1]. The restriction λ ∈ C \ Z is caused by the fact that in [1], the matrix H λ is represented as an integral operator acting on the space H p , viz. ( where κ = e 2πiλ − 1 and γ is the positively oriented boundary of the Stolz angle It is clear that the representation (4.1) is valid only for λ ∈ C \ Z, since e 2πin −1 = 0 for n ∈ Z + . The key observation in this paper is that H λ "almost commutes" with two specific second order linear differential operators having polynomial coefficients, namely with the operators defined on the set of all functions f analytic on D. More precisely, the following theorem is true. . For any f ∈ H p and λ ∈ C \ Z, the operators D 1,λ and D 2,λ satisfy the equations We also note that the functional Φ λ defined by plays an important role. It is clear that Φ λ (f ) is equal to (H λ−1 f )(0). The Eqs. (4.2) and (4.3) are corner stones in the study of the point spectrum of the operator H λ , λ ∈ C \ Z on H p . One reason for that is because the eigenvalue problem Notice that we can always define a branch of the first factor on the right-hand side on C \ [1, ∞) and a branch of the second factor on C \ (−∞, 1]. The eigenvalue problem (4.5) is more involved. In order to formulate the related results, we have to consider the hypergeometric series where the parameters α, β and γ can take any real or complex values provided that γ = Z − and the symbol (τ ) k , k ∈ Z denotes the numbers (τ ) 0 = 1 and (τ ) k = τ (τ + 1) . . . (τ + k − 1), k = 0. The series (4.6) becomes a polynomial if either α or β belongs to Z − . For all other choices of α and β, the radius of convergence of (4.6) equals one.
In what follows, the solutions a and a of the quadratic equation play an important role. We point out that for any a ∈ C there is a ν ∈ C such that a is a solution of Eq. (4.7).
where M is a constant, which does not depend on x. Moreover, f a ∈ H p , belongs to H p , 1 < p < 2 and −1/p < Re (−λ) − n, but not to H 2 .
A few comments are in order. The fact that the solution of equation D 2,λ f = νf is independent of the choice of the root of the equation (4.7) is known as Euler's formula (see [14, p. 248, formula(9.5.3)]).
Assertion (i) is a consequence of the well-known Gauss' formula [14,Section 9.3]). The last claim in (i) is a consequence of the Fejer-Riesz inequality.
The estimate (4.11) is a consequence of the representation where z 1 and z 2 are arbitrary complex constants such that Γ(n + z 1 ) and Γ(n + z 2 ) are defined (see [14,  λ + a / ∈ Z − insures that Γ(a + λ + 1 + n) and Γ(a + λ + n) are defined. If we expand 1 + log(1/ (1 − x)) in a power series, we get and arrive finally at so that (4.9) follows. The last claim in (iii) is a consequence of the Fejer-Riesz inequality. The proof of (iv) can be carried out similar to the proof of (d), Theorem 4.11. Theorem 4.8 is one of the corner stones in description of the point spectrum of the operator H λ : H p → H p . Another key result is the following lemma. We also need several times the following lemma.  (4.1).
is in the point spectrum of H λ : l 2 → l 2 .
Each eigenvalue −π/(sin(πa)), a = −λ, −λ − 1 has multiplicity one and the corresponding eigenspace is spanned by with the eigenvalue (−1) n π sin(πλ) . (4.13) The proof of this theorem given in [1] is relatively complex. So we restrict ourselves to comments concerning the claim (c) and provide independent proofs for the claims (a), (b) and (d).
Starting with the case p < 2, i.e. with the assertion (c), we note that Ω λ contains at most two elements. Now let −1/p < Re (a) < −1/2 and a + 1 − λ = −a − λ / ∈ Z + . By Theorem 4.8(i), the function f a belongs to H p only if −1/p < Re a. Since the restriction of f a to [0, 1] belongs to L p [0, 1], then H λ f a belongs to H p by Lemma 4.10. Let ξ ∈ C be such that a, Re (a) < −1/2 is a root of the Eq. (4.7). Taking into account the Eq. (4.3), we write so that H λ f a belongs to the space ker(D 2,λ − ξI), which has dimension one by Theorem 4.8. Thus for a μ ∈ C, and the equation f a (0) = 1 yields It is shown in [1] that if Re (y) > 0 and Γ(x), Γ(y) and Γ(x + y) are defined, then 1 and (H λ f a (0) = Γ(a + 1)Γ(−a) = − π sin(πa) .
An independent proof of this result can be given using Corollary 4.2(a). Finally, we assume that Re (a) < −1/2 and λ − a − 1 = a + λ = −n, n ∈ Z + . Notice that we do not suppose that Re (a) > −1. Since H p ⊃ H 2 for p < 2, the set Ω λ is included in the point spectrum of H λ . However, it is worth noting that if Re (λ) > 1/2, then Ω λ can be an empty set and if Re (λ) ≤ 1/2, it can contain one or two elements. Now we consider case (a). If the condition Re (a) < −1/2, λ − a − 1 = a + λ = −n, n ∈ Z + is fulfilled and N ≥ 0 is the largest integer such that where Q n (z) is a polynomial of degree n. More generally, if λ is fixed, then for 0 ≤ k ≤ N the function belongs to H p and Thus for H λ : H p → H p , p ≥ 2, 0 ≤ k ≤ N , and Λ := π/ sin(πλ), we have that are linearly independent (see Remark 4.16). We note that the systems U e , U o differ from the system (4.12) used in [1]. Lemma 4.14 below may serve as a hint, why that system is of relevance. We like to present a transparent and relatively simple proof of the assertion. It is based on the following lemma used in [1], but for different aims. Thus we introduce the forward shift V : V is the multiplication operator by z.
where ν is not specified. Notice that we only have to look for solutions which are eigenfunctions of H λ , more precisely, belong to ker(H λ − μI). It is known that the functions belonging to ker(H λ − μI) must extend analytically in C \ [1, ∞). Using that ker(D 1,λ − v i I) is spanned by and has to belong to ker(H λ − μI), we get that (−λ − v i )/2 necessarily belongs to Z + . If (−λ + v i )/2 = n i ∈ Z + , then (−λ + v i )/2 = −λ − n i and (1 − z) −λ−ni ∈ H p . Now let N is the largest integer such that (1 − z) −λ−N ∈ H p . Choose an integer n such that 0 ≤ n ≤ N and set Obviously, and this system coincides with (4.12). As a consequence we get that μ can take on only 2 values, namely ±Λ. Let V e and V o be the systems defined in Then k e < l e , and the characteristic polynomial of D 1,λ : M Λ → M Λ has degree larger than k e . Thus there is a further eigenvector, say ξ of D 1,λ such that ξ = 0 and ξ / ∈ span V e . Since, ξ ∈ ker(H λ − ΛI), the above considerations show that a scalar multiple of ξ belongs to V e , and we get a contradiction. To prove the last claim in (d), we have to show that the polynomial F (−λ − n + 1, −n; λ; z) = F (−n, −λ − n + 1; λ; z) does not vanish at the point z = 1. The proof is based on the following assertion from [14], formulated as a lemma here.
The proof of Theorem 4.11 is complete if one takes into account the following two lemmas. Proof. Obviously, it suffices to prove the claim for 1 < p < 2. Suppose that a function f = 0 belongs to the space H p and H λ f = 0. By the Eq. (4.3) in Theorem 4.7, we have H λ D 2,λ f = D 2,λ H λ f = 0, and this implies that f and D 2,λ f both belong to ker H λ . By Theorem 4.5, part (II), we obtain that dim ker H λ = 1. But this result implies that D 2,λ f = τ f, where τ is a complex number. Theorem 4.11 entails that the solutions of the eigenvalue problem D 2,λ f = τ f, f is analytic in D, form a one-dimensional space spanned by f a = f a , the functions f a and f a given by (4.8), where a and a are the solutions of the Eq. (4.7) with ν replaced by τ . The theory of the mentioned eigenvalue problem is explained in Theorem 4.8, and Theorem 4.11 together with its proof shows the importance of Theorem 4.8 in the study of the eigenvalue problem of H λ − ωI in the spaces H p . Due to the restriction 1 < p < 2, an inspection of Theorem 4.11 shows that f ∈ ker(H λ − ωI) for an ω = 0. But this contradicts H λ f = 0.
So far, all results in this section are proved under the assumption that λ / ∈ C \ Z. In the next section we show that they are in force also for λ ∈ N.