Unbounded Wiener–Hopf Operators and Isomorphic Singular Integral Operators

Some basics of a theory of unbounded Wiener–Hopf operators (WH) are developed. The alternative is shown that the domain of a WH is either zero or dense. The symbols for non-trivial WH are determined explicitly by an integrability property. WH are characterized by shift invariance. We study in detail WH with rational symbols showing that they are densely defined, closed and have finite dimensional kernels and deficiency spaces. The latter spaces as well as the domains, ranges, spectral and Fredholm points are explicitly determined. Another topic concerns semibounded WH. There is a canonical representation of a semibounded WH using a product of a closable operator and its adjoint. The Friedrichs extension is obtained replacing the operator by its closure. The polar decomposition gives rise to a Hilbert space isomorphism relating a semibounded WH to a singular integral operator of Hilbert transformation type. This remarkable relationship, which allows to transfer results and methods reciprocally, is new also in the thoroughly studied case of bounded WH.


Introduction
There is an increasing interest in unbounded Toeplitz and Toeplitz-like operators (see 2.2), which will concern also the related Wiener-Hopf operators (WH). So far, results on WH W κ with unbounded symbol κ are scarce and probably in the literature there exists no introduction to this subject. So Sect. 2 deals with preliminaries and basics regarding unbounded WH. In particular we are concerned with conditions on the symbol κ ensuring that the domain of W κ is either the whole space or dense or trivial, and prove that dom W κ is either trivial or dense. The symbols with non-trivial WH, which are called proper, are determined by a useful integrability property. A classical result on the eigenvalues of a WH is shown to remain valid in the unbounded case. It implies that non-trivial symmetric WH have no eigenvalues. A further result characterizes WH by their invariance under unilateral shifts. In Sect. 3 WH with rational symbols are studied. They constitute a welcome source of densely defined closed operators with finite index. An explicit description of the domains, ranges, kernels, deficiency spaces, spectral and Fredholm points is given. The remainder of this article deals in Sects. 4 and 5 with densely defined semibounded WH. A semibounded operator W κ can be expressed by a product of a closable operator A and its adjoint. Replacing A by its closure one obtains quite naturally a self-adjoint extensionW κ . It is proven to coincide with the Friedrichs extension. Inverting the order of the factors one obtains a singular integral operator L φ of Hilbert transformation type. For the operators of that type to be non-trivial there is a necessary condition analogous to that for WH. The self-adjoint extensionsL φ andW κ are isometric, which follows from the polar decomposition of A. ActuallyW κ is Hilbert space isomorphic to the reduction ofL φ on ker(L φ ) ⊥ , and the spectral representations ofL φ andW κ can be achieved in an explicit manner from each other. It is worth noting that this relationship is new also for bounded WH thus contributing to the well-developed theory of the latter. To conclude, this method is illustrated by a non-trivial example diagonalizing Lalescu's operator and the isometrically related singular integral operator. In [24,Sect. 3.3] the spectral representations of W 1 [−1,1] and the finite Hilbert transformation are related to each other by this method.
Notations Let F denote the Fourier transformation on L 2 (R). For measurable E ⊂ R introduce the projection P E : x ∈ E and = 0 otherwise. Note The theory of WH with bounded symbol is well developed. We content ourselves to refer here to the book [1,Chapter 9] and to mention the origins [2]. Obviously in case of a bounded symbol the operators W κ are bounded with dom W κ = L 2 (R + ) and adjoint W * κ = W κ . W κ is the convolution on the real half line with kernel k, i.e., if κ ∈ L ∞ ∩ L 2 and k := (2π) −1/2 Fκ, or if κ = e i(·)y k(y) d y for k ∈ L 1 (R).
For the case of integrable kernel there is the rather exhaustive theory by M. G. Krein [3]. Generally, the tempered distribution k := (2π) −1/2 Fκ, where κ ∈ L ∞ (R) is considered as a regular tempered distribution, satisfies F M(κ)F −1 u = k u for every Schwartz function u in the distributional sense (e.g. [4,Theorem IX.4]). For instance the kernel for W − sgn is the tempered distribution k(x) = − 1 x or that for W − tanh equals k(x) = 2 i sinh(π x/2) −1 . In the literature the generalizations of WH stay mostly within the realm of bounded operators. One deals with the traces (compressions) of bounded bijective operators in a Banach space on a closed subspace [5]. The results concern the solvability of the associated Wiener-Hopf equations.

Unbounded Wiener-Hopf Operators
As put it by [6] results on unbounded WH are practically inexistent. Indeed they are scarce. See [7, 1.3] for some notes. An important result is due to M. Rosenblum [8], [9], obtained for Toeplitz operators and hence valid for the Hilbert space isomorphic WH W κ (see 2.2). So in the case that the symbol κ is real bounded below not almost constant and (1 + x 2 ) −1 κ is integrable, [9] furnishes the spectral representation of the extensionW κ , which is shown to be the Friedrichs extension 4.3, 4.4 and which by [9] is absolutely continuous. -There are investigations on unbounded general WH dealing with conditions for their invertibility [10]. -In [6] real bounded below Wiener-Hopf quadratic forms from distributional kernels k are considered, and it is shown that such a form determines a WH if and only if the form is closable or, equivalently, if and only if √ 2π k is the Fourier transform of a locally integrable bounded below function κ with integrable (1 + x 2 ) −n κ for some n ∈ N. Clearly κ is the symbol and dom W κ ⊃ C ∞ c (R + ) holds. See further [7]. -Furthermore, the that its partial maps φ y (x) := φ(x + i y) for y > 0 satisfy φ y ∈ L 2 (R), { φ y : y > 0} bounded, and φ y → h for y → 0 in the mean and pointwise a.e. Actually φ converges to h non-tangentially a.e., and φ y ↑ h for y ↓ 0 (see e.g. [ with P H + : L 2 (R) → H + the orthogonal projection. Note that P H + F −1 P * + : The latter is Hilbert space isomorphic to its counterpart the so-called analytic Toeplitz operator with symbol ω := κ • C −1 , see 2.2. M + (κ) and M(κ, H + ) coincide for rational symbols κ holomorphic in the upper half-plane 3.1(c), or more general, if κ is the nontangential limit of an outer function on the upper half-plane. Indeed, in this case ln |κ| 1+x 2 is integrable, whence κ is proper by 2.8 since ln |1 + |κ|| ≤ ln 2 + ln |κ|, and [12, 3.3] applies. We do not put forward this topic.
Finally H ∞ + is the set of all measurable bounded α : R → C such that there is a bounded holomorphic A on the upper half-plane with the partial maps A y → α for y → 0 pointwise a.e. Actually A converges to α non-tangentially a.e. (see e.g. [

Remark on Toeplitz operators 2.2
Let the torus T be endowed with the normalized Lebesgue measure. The Hardy space H 2 (T) is the subspace of L 2 (T) with orthonormal basis e n (w) := w n , n ∈ N 0 . Given a measurable ω : T → C then, quite analogous to (2.2), the Toeplitz operator T ω with symbol ω is defined by Obviously by this well-known relationship (see e.g. [1, 9.5(e)], [24, 3.3.2 (13)]) results and methods regarding Toeplitz operators may be transferred for the study of WH and vice versa. For unbounded Toeplitz and Toeplitz-like operators see: [8], [9] for a spectral theory of Toeplitz operators with bounded below integrable symbols, [7] for closable quadratic forms for semibounded Toeplitz operators, [11] for analytic Toeplitz operators, [19] for Toeplitz-like operators with rational symbols.

Results
As to 2.3 one recalls that in general an everywhere defined linear operator in a Hilbert space need not be bounded.
Proof The second implication is obvious. As to the first one, assume that κ is not bounded. Then there exists There follows a first result on the domain of a WH which applies for instance to rational symbols.

Lemma 2.4
If κ qs ∈ L 2 (R), where q is a polynomial and s is the inverse Fourier transform of a Schwartz function with support in [0, ∞[, then qs ∈ dom W κ and κ is proper. -Suppose that κ qs ∈ L 2 (R) for a polynomial q with only real zeros and for every Schwartz function s.
Proof As to the first claim note that s is a Schwartz function in H + , whence qs ∈ H + by 3.1(c). Now let D denote the differential operator i d d x and let u be any Schwartz function with support in R + . Then q(D)u is still such a function, and F −1 u, Indeed, regarding g as a regular distribution in D , one has g q(D)φ = q(− D)g (φ) = 0 for all test functions φ, whence q(− D)g = 0. Thus g is a solution of the differential equation q(− D)F = 0 for F ∈ D . As known all its solutions are regular. Hence g ∈ T , where T denotes the space of linear combinations of functions on R + of the kind x → x k e i λx , k ∈ N 0 and λ ∈ R. Then G ∈ T for G(x) := x One has G(x) → g 2 < ∞ for x → ∞. Write G = n k=0 x k A k with A k a linear combination of periodic functions e i λx . Then A n (x) → 0 for x → ∞. Since A n is almost periodic this implies A n = 0. The result follows.
The following invariance H ∞ + dom M + (κ) ⊂ dom M + (κ) is essential for the density of dom W κ in 2.10.
Since αh ∈ H + , the latter is equivalent to and apply the foregoing results.
We turn to characterizations of proper symbols κ, as in particular by the condition of the integrability of ln(1 + |κ|)/(1 + x 2 ) in 2.8. Proposition 2.6 Let p ∈]0, ∞]. Let j denote a real-valued function such that j 1+x 2 is integrable. Then κ proper ⇔ κ e j ∈ L p (R) for some j Proof Here we prove the case p = 2 and the implication ⇐ for p = ∞ and p = 1. The remainder is shown in the proof of 2.7.
The general case is easily reduced to the claim that, for r > 1 and κ ≥ 0, κ is proper if and only if κ r is proper. So let κ be proper. Let n ∈ N satisfy r ≤ 2 n . By the foregoing result κ 2 n is proper. Then 1 + κ 2 n is proper and κ r ≤ 1 + κ 2 n . Hence κ r is proper. Conversely, if κ r is proper, then 1 + κ r is proper and κ ≤ 1 + κ r , whence κ is proper. Now we complete the proof of 2.6. Consider first the case p ∈]0, ∞[. Let κ e j ∈ L p (R) for some j. Then |κ| p/2 e p 2 j ∈ L 2 (R). Hence |κ| p/2 is proper by 2.6 ( p = 2). The foregoing result applies, whence κ is proper. The converse follows quite similarly.
Now the alternative is shown that a WH is either trivial or densely defined. [17, Chapter 3]). So U converges non-tangentially a.e. to a function u on T satisfying |u| = e j•c ∈ L 2 (T), whence U ∈ H 2 (D). Therefore, as known (see e.g. [21,Sect. 3

]), U H ∞ (D) is dense in H 2 (D). This implies that u H ∞ (T) is dense in H 2 (T).
Put h := u • C. Note |h | = e j = |h|. The above result is transferred to H + by in 2.2. Accordingly, 1 x+i h ∈ H + and 1 Two results on the eigenvalues of WH follow. The proof of 2.11 uses an alternative argument with respect to [22, 2.8] for the case of bounded κ.

Theorem 2.11
Let κ be not almost constant. If λ ∈ C is an eigenvalue of W κ , then λ is not an eigenvalue of W κ . If κ is almost real, then W κ has no eigenvalues and in particular W κ is injective.
The following result from [12, 3.2] obviously is stronger than 2.12: Let λ be an eigenvalue of W κ = λI . Then ln |κ−λ| 1+x 2 is integrable. The converse still does not hold as the case of real-valued symbols shows by 2.11. By 2.13 the symbol of a WH is uniquely determined.
We turn to symmetric WH. Corollary 2.14 Let W κ be densely defined. Then W κ is symmetric if and only if κ is almost real.
e. by 2.13. The converse is obvious.
Hence due to 2.11 densely defined symmetric WH have no eigenvalues. -Now the numerical range of densely defined semibounded WH is determined.

Theorem 2.15 Let W κ be densely defined symmetric. Then W κ is bounded below if and only if κ is real essentially bounded below, and the maximal lower bound of W κ equals the maximal essential lower bound of κ. If W κ is bounded below and not bounded, then the numerical range
Proof By 2.14 let κ be real. First suppose that κ is bounded below with maximal lower bound a.
for λ sufficiently small. Therefore B is a null set, whence κ ≥ 0 a.e. and α is an essential lower bound of κ. -This proves a = α.
Since M + (κ) is not bounded above by assumption, the numerical range For bounded WH, 2.15 implies the following well-known result.
. It shows again the invariance 2.5(c)

Rational Symbols
WH for rational symbols κ = P Q R with polynomials P = 0, Q = 0 permit some more general analysis. By 2.4 they are densely defined. In 3.2 we show that they are closed and we determine their domains, ranges, and kernels and deficiency spaces, which are finite dimensional, and their spectral and Fredholm points. In particular, in the symmetric case, i.e., for a real rational symbol the deficiency spaces and indices are explicitly available 3.4.
Recall that a densely defined closed operator between Banach spaces with finite dimensional kernel and cokernel is called a Fredholm operator if its range is closed (cf. [25]).
Mostly we will omit | R indicating the restriction on R. A polynomial with a negative degree is the null function. For convenience we will deal with M + (κ) (2.2) in place of W κ . We are indebted to the reviewer for a significantly simplified proof of (b) and (c) of the following lemma 3.1.  Proof (a) The result is known and we omit the proof. [22,Theorem 3.1]. Therefore h = R/P and k = R/Q. Now, h ∈ H + implies by (a) that deg R < deg P and all zeros of P in the closed upper half-plane are zeros of R with at least the same multiplicities. The analogous result regarding Q follows from k ∈ H − . (c) For g := P Q h ∈ L 2 (R) write g = g + + g − with g + ∈ H + , g − ∈ H − . Then, using n, α, β from the proof of (b) and γ := (x + i) −n Q, it follows , since Q has no zeros in the upper half-plane by assumption. (d) Assume first that Q has no real zeros. Let φ and ψ be the holomorphic functions on the upper half-plane related to h and P Q h, respectively. Then P Q φ is meromorphic on the upper half-plane and converges non-tangentially to P Q h a.e. Since ψ does the same, according to [16,IV 2.5], ψ = P Q φ holds. Hence φ/Q is holomorphic on the upper half-plane with (φ/Q) y → h/Q for 0 < y → 0 a.e. Let 0 < δ < c such that C := [−c, c] × i[δ, c] is a neighborhood of the zeros of Q in the upper half-plane. Then |1/Q| is bounded by some constant L on {z : Im z ≥ 0} \ C, and |φ/Q| is bounded on C by some M. Recall that φ y 2 2 is bounded for y > 0 by some K . Then Now we turn to the general case. Note h/Q ∈ L 2 (R), cf. (v). Write Q = q Q 0 , where q has only real zeros and Q 0 has no real zeros. Let q (z) := q(z + i ) for > 0 and put Q := q Q 0 . Note |q(z)/q (z)| < 1 on the upper half-plane. Therefore P Q h = q q P Q h ∈ H + , where Q has no real zeros. Moreover, for > 0 small enough, P and Q have no common zeros. Hence the foregoing result applies so that h/Q ∈ H + . Now h/Q = q q h/Q → h/Q for → 0 in the mean implying h/Q ∈ H + .
Let κ = P Q be a rational function, where the polynomials P and Q have no common zeros. Then let q be the polynomial, the zeros of which are the real zeros of Q with the same multiplicities. Put ς := max{deg q, deg P − deg Q + deg q}. Moreover put P = P < P ≥ , where the zeros of P < (P > ) and P ≥ are exactly the zeros of P in the lower(upper) half-plane and in the closed upper half-plane, respectively.P denotes the polynomial whose coefficients are the complex conjugates of P. Analogous notations concern Q. Theorem 3.2 Let κ = P Q be a rational function, where the polynomials P and Q have no common zeros. Then M + (κ) is densely defined and closed and Proof For the closeness of M + (κ) write P Q in the form P Q = P 0 Q 0 + p q with polynomials P 0 , Q 0 , p, q such that Q = Q 0 q, Q 0 has no real zeros, q has only real zeros, deg P 0 < deg Q 0 , and p and q have no common zeros and satisfy ς = max{deg p, deg q}. Since κh 0 ∈ L 2 (R), and κh 0 ∈ H − . According to 3.1(b) this means h 0 = R/P, where R is a polynomial with deg R < min{deg P, deg Q} and R = P ≥ Q ≤ r , whence the claim. (c) Using (a) one has h 0 ∈ ran M + (κ) Now assume that P H + P R H + is closed. Then the above considerations imply that To complete the proof of the implication (2)⇒(3) it remains to treat the case that deg P ≥ deg Q and P has a real zero. Proceeding as in the forgoing case, here one has deg R = deg P. Hence, assuming that P H + P R H + is closed, one has m ≥ 1 and l ≤ −1 and the same contradiction follows.

Corollary 3.3
For λ ∈ C put P λ := P + λQ. Then referring to M + (κ), λ is The characterization of the Fredholm points of M + (κ) and the fact that at a Fredholm point either the kernel or the deficiency space is trivial, are familiar from Krein's theory [3] for the case of integrable kernel. λ → deg P λ > is locally non-decreasing due to the continuity of the roots of a polynomial on its coefficients [26]. On C \ κ(R) it is even locally constant, since there P λ > = P λ ≥ . Hence, besides ind(M + (κ) − λI ), also dim ker(M + (κ) − λI ) and dim ran(M + (κ) − λI ) ⊥ are constant on the components of C \ κ(R).    (1, 1). -This section is concluded by a much needed Example 3.5 There are WH, even essentially self-adjoint semibounded ones, which are not closed, as for instance W |x| (or W 1/|x| ).

Semibounded Wiener-Hopf operators
In 4.3, 4.4 a densely defined semibounded WH W κ is expressed in a canonical way by the product of a closable operator and its adjoint. Replacing the operator by its closure one obtains a self-adjoint extensionW κ of W κ , which is semibounded by the same bound. The bound is not an eigenvalue of the extension.W κ is shown to be the Friedrichs extension of W κ . We start with a preparatory lemma. The condition γ = 0 not a.e. means that γ −1 ({0}) is not a null set.
The foregoing lemma is needed only in Sect. 5. The main result of this section follows. Proof (a) Apply 4.1 for γ := √ κ. Accordingly, W κ = A A * and, if κ is proper, W κ is densely defined and symmetric nonnegative by 2.14, 2.15, and A * is densely defined. Then A = A * * and by [27, 13.13(a)] A A * is self-adjoint. Clearly A A * is nonnegative. Check that A A * is injective as A * is injective by 4.1 for κ not almost zero.
(b), (c) According to [4,Theorem X.23],W κ is the Friedrichs extension only if domW κ ⊂ H W κ , where H W κ is the completion of dom W κ with respect to the sesquilinear form g, g W κ := g, g + g, W κ g .
Endow dom A * with the inner product g, g A * := g, g + A * g, A * g , by which dom A * becomes a Hilbert space K since A * is closed. Then the subspace dom A A * is dense in K since dom A A * is a core for A * , see [27, 13.13(b)]. One easily checks that H W κ is the closure of dom W κ in K. Therefore dom A A * ⊂ H W κ if and only if H W κ = K, which means that dom W κ is a core of A * . A short computation shows also that H W κ = K is equivalent to dom A * ∩ ran(I + W κ ) ⊥ = {0}.
For κ ≥ 0 and W κ densely defined recall that the deficiency subspace ran(I + W κ ) ⊥ of W κ at −1 is trivial if and only if W κ is essentially self-adjoint.

Isomorphic Singular Integral Operators
This section is concerned with the symmetric singular integral operator in L 2 (E) for y − x f (y) d y (5.1) (in the sense of the principal value at x) where φ : E → R is measurable positive. L φ will turn out to be closely related to W κ , where κ extends φ on R by zero. L φ is rather general. Indeed, it belongs to the studied class of singular integral operators in L 2 (E) of Hilbert transformation type (L(a, b) where a, b are measurable functions on E with a real and b = 0 a.e. There is the obvious unitary equivalence for φ = 2|b| 2 and α = a − |b| 2 , where U is the multiplication operator by b/|b| and M(α) the multiplication operator by α in L 2 (E). So we are concerned with the case a = |b| 2 .
The operator L(a, b) for bounded b and bounded below a is treated by Rosenblum in [28]. It is shown to be self-adjoint on dom L(a, b) = dom M(a), and its diagonalization is achieved. See also [29] and the literature cited in [28,29]. The really unbounded case however is there when b is unbounded. [30] is concerned with this case replacing L(a, b) by the limit of truncated L(a n , b n ) which are bounded. Our analysis of L φ will show 5.3 that L(a, b) ⊂ M(α) if the extension of b on R by zero is not proper. Hence for L(a, b) in (5.2) being not trivial it is necessary that the extension of b is proper. In this case L φ in (5.3) has a self-adjoint extension 5.5.
The Hilbert transformation H on L 2 (R) is defined by the singular integral the trace on L 2 (E) of the Hilbert transformation H . Its spectrum is determined in [32]. For E = [a, b], −∞ ≤ a < b ≤ ∞, H E is called finite and semi-finite Hilbert transformation if E is bounded and semi-bounded, respectively. Its spectral representation is achieved in [33].

Isometry relatingL toW Ä
In what follows we use the polar decomposition C = S|C| of a closed densely defined operator C from a Hilbert space H into another H (see e.g. [4,VIII.9]). |C| denotes the square root of the self-adjoint nonnegative operator C * C in H. One has dom |C| = dom C and dom C * C is a core for C. S is a partial isometry from H into H . Its initial space (ker S) ⊥ equals ran C * = ran |C| = ran |C| 2 . Similarly, its final space ran S equals ran C = ran |C * | = ran |C * | 2 . The partial isometry S * satisfies (ker S * ) ⊥ = ran S and ran S * = (ker S) ⊥ . Note the important relation which means that the reductions of CC * and C * C on the orthogonal complements of their respective null spaces are Hilbert space isomorphic by the restriction of the partial isometry S to its initial state (ker S) ⊥ and its final state ran S. Throughout this section κ ≥ 0 and φ > 0 are related to each other by suppose that κ is proper. Let A = T |A| be the polar decomposition of A. Then the partial isometry T : L 2 (E) → L 2 (R + ) is surjective, its adjoint T * is injective, and As a first result we note that L φ in (5.1) can be trivial:  L(a, b) L(a, b) The latter equals ker L φ . Finally, For example, L e |x| ⊂ 0 with dense dom L e |x| = e −|x|/2 H − . The main outcome of this section is the following: LetL φ and T denote the reduction ofL φ and the restriction of T on the orthogonal complement of the null space ofL φ , respectively. Then T is a Hilbert space isomorphism onto L 2 (R + ) satisfyingL φ = T −1W κ T .
Of course one hasW κ = TL φ T * as well. It allows to studyW κ starting fromL φ . For the still remarkable bounded case see (5.9). By 4.4 it is easy to extend 5.5 to semibounded symbols.
-We remind that forL φ being injective it is necessary that E is proper or that 1/κ is not proper.

Corollary 5.7
Let κ ≥ 0 be proper. Suppose that E is proper or 1/κ is not proper. Theñ L φ is injective and T : Hence {0} = kerL φ = ker T and T is an isomorphism. Apply 5.5.
So it is worth noting thatL φ is absolutely continuous, if κ 1+x 2 is integrable and if E is proper or 1/κ is not proper.
Let κ be bounded. Then L φ , W κ are bounded,L φ = L φ ,W κ = W κ , and Moreover, if E is proper or 1/κ is not proper, then even L φ = L φ by 5.7, whence L φ = T −1 W κ T . An example for the latter case is L e −|x| , which is injective. Generally, if κ > 0 does not decrease too rapidly (so that 1/κ is proper) the kernel of L κ is not trivial. For an instructive example see Sect. 5.2. The trivial example here is κ = 1 R . Note that W (1 R ) = I L 2 (R + ) and L 1 R is the orthogonal projection on H + , and T : H + → L 2 (R + ), The case of the finite Hilbert transformation H [−1,1] is studied in detail in [24,Sect. 3.3], [24, (3.20)]. In conclusion we remark on the spectral representations ofW κ andL φ in 5.5. By the spectral theorem in the multiplication operator version,W κ is Hilbert space isomorphic to the multiplication operator M(ϕ) on L 2 μ (R) for some Borel-measurable positive ϕ : R → R and finite Borel measure μ. Let V : L 2 μ (R) → L 2 (R + ) be an isomorphism satisfyingW The spectral measure forW κ is given by EW κ ( ) = V M(1 ϕ −1 ( ) )V −1 for measurable ⊂ R.
Since T * is bounded, W 0 is closable and its closure W equals T * V . The remainder is obvious.

Example: Lalescu's Operator
Supposing κ > 0, κ ∈ L 2 , and kernel k ∈ L 1 , in [34] a spectral theory of W κ is proposed by a reduction to a previously developed theory for singular integral operators. Its application in [34] to Lalescu's operator W λ with symbol however does not produce the right normalization (5.14) of the generalized eigenfunctions (5.13). In establishing (5.14) we get also the result 5.10 on orthogonal polynomials. The diagonalization of Lalescu's operator W λ is achieved in Sect. 5.2.1 and that of the associated singular integral operator L λ is derived in Sect. 5.2.2. The generalized eigenfunctions (5.18) of the latter are no longer regular distributions.
Hence the reduction L λ of L λ on the orthogonal complement of its kernel is self-adjoint bounded on M(u)H + with spectrum [0, 2]. The following computations are valid in a distributional sense. So generalized eigenfunctions Q s for L λ are given by A * q s = M( √ λ)F −1 P * + q s since L λ A * q s = A * A A * q s = s A * q s . Recall The claim is that n(s) √ s is the unique positive normalization constant such that Q(s, x) := Q s (x) is the kernel for an isometry W on L 2 (0, 2) in L 2 (R) satisfying for test functions h. The additional factor 1 √ s regarding the normalization constant of Q s corresponds to the factor 1 √ ϕ for W 0 in 5.8 and is suggested heuristically by A * q s , A * q s = q s , A A * q s = s q s , q s . For the proof recall the isomorphism (5.16) and the Hilbert transformation H (5.5). For g : R + → C let g oe denote the odd extension g oe (x) = −g(−x) for x ≤ 0 of g.

Theorem 5.12
The integral operator (5.19) determines a Hilbert space isometry W : L 2 (0, 2) → L 2 (R) with ran W = M(u)H + . It satisfies W h = M(u) 1 2 I + H ( h) oe for h ∈ L 2 (0, 2) and yields the representation L λ = W M(id [0,2] )W * Proof By the change of variable t = 2/(1 + s 2 ) for the integration in (5.19) one obtains The integral is easily done yielding ( ) W h = M(u) 1 [0,2] )h = L λ W h for test functions h. The latter is shown along the lines of the proof of 5.11. We omit the technicalities.
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