Unbounded Wiener-Hopf Operators and Isomorphic Singular Integral Operators

Some preliminaries and basic facts regarding unbounded Wiener-Hopf operators (WH) are provided. WH with rational symbols are studied in detail showing that they are densely defined closed and have finite dimensional kernels and deficiency spaces. The later spaces as well as the domains and ranges are explicitly determined. A further topic concerns semibounded WH. Expressing a semibounded WH by a product of a closable operator and its adjoint this representation allows for a natural self-adjoint extension. It is shown that it coincides with the Friedrichs extension. Polar decomposition gives rise to a Hilbert space isomorphism relating semibounded WH to singular integral operators of a well-studied type based on the Hilbert transformation.


Introduction
Results on Wiener-Hopf operators (WH) W κ with unbounded symbol κ are scarce and probably in the literature there exists no introduction to this subject. So sec. 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. 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 sec. 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 sec. 4 and sec. 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 type Hilbert transformation. As shown, for the operators of the mention type being not 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. To conclude, this method is illustrated by a non-trivial example diagonalizing Lalescu's operator and the isometrically related singular integral operator. In [22, sec. 3.3] the spectral representations of W 1 [−1,1] and the finite Hilbert transformation were 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 : L 2 (R) → L 2 (E), (P E f )(x) := f (x). (For convenience define L 2 (E) = {0}, P E = 0 if E = ∅.) Its adjoint P * E is the injection (P * E f )(x) = f (x) for x ∈ E and = 0 otherwise. Note with M (1 E ) the multiplication by the indicator function 1 E for E. We call E proper if neither E nor the complement R \ E is a null set. Put R + :=]0, ∞[ and P + := P R+ . Analogously define P − . Throughout let κ : R → C denote a measurable function and M (κ) the multiplication operator by (1) Definition. The operator in L 2 (R + ) W κ := P + F M (κ)F −1 P * + is called the Wiener-Hopf operator (WH) with symbol κ. Occasionally we write W (κ) instead of W κ . Clearly, W κ = W κ ′ if κ = κ ′ a.e. Often we shall refer to this tacitly. The symbol κ is called proper if dom W κ = {0}.
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 F κ, 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 F κ, 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(πx/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 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 (3)). 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 (14), (15) 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 √ 2πk 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 methods applied for the study of unbounded analytic Toeplitz operators [11] can also produce results on unbounded WH, as (4)(i), (13), (14)(c).
Starting the preliminary remarks note that dom W κ = dom M (κ)F −1 P * + . Therefore κ is proper if and only if κh is square-integrable for some Hardy function (2) But unbounded WH may and may not be closed (8), (11). If W κ is densely defined then W κ is symmetric, i.e., W κ ⊂ W * κ , if and only if κ is almost real (4)(n). If W κ is densely defined symmetric, then W κ is bounded below if and only if κ is essentially bounded below (4)(o). Recall that the numerical range { g, W κ g : g ∈ dom W κ , ||g|| = 1} of W κ is convex. (Indeed, the numerical range of every operator in Hilbert space is convex [12].) It is determined in (4)(o),(p).
If the symbol κ is unbounded, then dom W κ = L 2 (R + ) (4)(a). The alternative holds that either dom W κ is trivial or dom W κ is dense. In other words, as shown in (4)(i), if κ is proper, then W κ is densely defined. In (4)(e) and (4)(g) explicit characterizations of proper symbols are given. There is also the useful criterion in (4)(b) for κ to be proper. So proper symbols may have polynomial growth and countably many singularities with integrable logarithm like as exp |x| α , −1 < α < 0. It is easy to give examples of non-proper symbols (4)(h).
(2) Hardy spaces. Recall the Hardy spaces H ± := ran( . We tacitly refer to the well-known Paley-Wiener Theorem characterizing the Fourier transforms of L 2 -functions vanishing on a half-axis, see e.g. [13,Theorem 95]. In particular h ∈ H + if and only if there is a φ holomorphic on the upper half-plane such that its partial maps φ y (x) := φ(x + i y) for y > 0 satisfies 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. [14, III 3.3, II 2.6]). Moreover, every h ∈ H ± \ {0} vanishes only on a null set. Indeed, according to a Luzin-Privalov Theorem [15, IV 2.5] a meromorphic function on the upper or lower half-plane which takes non-tangential boundary values zero on a set of positive Lebesgue measure is zero. The former property is also an immediate consequence of the following result on the modulus of a Hardy function: with P H+ f the orthogonal projection of f ∈ L 2 (R) on H + . Note that P H+ F −1 P * + : L 2 (R + ) → H + is a Hilbert space isomorphism with its inverse P + F P * H+ , by which W κ is Hilbert space isomorphic to M + (κ). Often it is convenient to deal with M + (κ) in place of W κ .
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. [14,III 3.3, II 2.6]).
(3) Remark on Toeplitz Operators. 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 . Let Γ : is the Toeplitz operator with symbol ω. Explicitly one has the formula T ω = P H 2 (T) M (ω)P * H 2 (T) quite analogous to (2.2). Obviously by this relationship results and methods regarding Toeplitz operators may be transferred for the study of WH.
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.
otherwise. Then one has the translational invariance 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 (f) κ is proper if and only if κ 2 is proper. More generally, let r > 0 and let κ 1 , κ 2 be two symbols satisfying |κ 1 | = |κ 2 | r . Then κ 1 is proper if and only if κ 2 is proper. Finally, if κ 1 and κ 2 are proper symbols then so are κ 1 κ 2 and κ 1 + κ 2 .
x+i h ′ is dense and contained in dom M + (κ). (k) 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.
(n) Let W κ be densely defined. Then W κ is symmetric if and only if κ is almost real.
(o) 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 { g, W κ g : g ∈ dom W κ , ||g|| = 1} equals ]α, ∞[ with α the maximal lower bound.
(p) Let W κ be bounded symmetric and not a multiple of I. Then the numerical range equals ]a, b[ with a and b the minimum and maximum, respectively, of the essential range of κ.
(b) As to the first claim note that s is a Schwartz function in H + , whence qs ∈ H + by (7)(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, The claim is g = 0. 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.
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 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 (e). 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 as shown in (e)(p = 2). The foregoing result applies, whence κ is proper. The converse follows in the same way due to (e)(⇒ for p = 2). -Now let p = ∞ and let κ be proper. As just shown, κe j is integrable for some real-valued j with integrable j(1 + x 2 ) −1 .
Since h − vanishes on the non-null set E, h − = 0 follows. Hence h + vanishes on the non-null set R \ E implying h + = 0 and hence g = 0.
e. by (m). The converse is obvious.
Therefore B is a null set, whence κ ≥ 0 a.e. and α is an essential lower bound of κ. This proves a = α.
M + (κ) not being bounded above by assumption, the numerical range R of M + (κ) is not bounded (p) follows readily from (o).
As mentioned section 4 is concerned with the case that κ is proper real and semibounded. This is the general case that W κ is densely defined symmetric semibounded (4)(n),(o). The natural self-adjoint extensionW κ is studied in (14), (15).
If κ is proper real and even (i.e. κ(−x) = κ(x)) then W κ is densely defined symmetric and has a self-adjoint extension. This holds true since L 2 (R + ) → L 2 (R + ), g → g is a conjugation, which leaves dom W κ invariant and satisfies W κ g = W κ g (see [4,Theorem X.3]). If κ is odd instead of even then in general W κ has no self-adjoint extension. Examples are furnished by real rational symbols as e.g. κ(x) = x. In (10) an explicit description of the deficiency spaces of W κ for real rational κ are given yielding further examples of densely defined symmetric WH with self-adjoint extensions.
Concluding this section we deal with the unilateral translation invariance (4)(d) of WH. We are inspired by [6, sec. 2.3)] which treats the bounded case (6). Observe the easily verifiable relation Indeed, let f ∈ L 2 (R) and ε > 0. Since f a : For the translation invariance of D it suffices to show T * c P * We are going to show that lim b→∞ f b exists.
Hence C * 1 is densely defined and the closure C := C 1 exists. Clearly for some measurable function κ. Hence P + CP * + = W κ . Finally, for g ∈ dom A one has P * + g ∈ D and W κ g = P + C 1 P * For (6) see also [6, (2.10)], where the existence of lim b→∞ T * b P * + AP + T b f (see (ii) of the proof of (5)) is not proven.

Rational Symbols
WH for rational symbols κ = P Q R with polynomials P = 0, Q = 0 permit some more general analysis. According to (4)(b) they are densely defined. In (8) 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 (10).
Mostly we will omit | R indicating the restriction on R. A polynomial with a negative degree is the null function.
Then there is a polynomial R with deg R < min{deg P, deg Q} such that all zeros of P in the closed upper half-plane as well as all zeros of Q in the closed lower halfplane are zeros of R and such that h = R P . Conversely, it is obvious that h + := R P and h − := R Q satisfy h ± ∈ H ± and P Q h + = h − . (c) Let Q have no zeros in the upper half-plane. If h ∈ H + and P Q h ∈ L 2 (R), then P Q h ∈ H + . (d) Let P and Q have no common zeros. Suppose that h ∈ H + and P Q h ∈ H + . Then h/Q ∈ H + .
Proof. (i) As to (a) put h := P Q R . Obviously h ∈ L 2 (R) if and only if deg P < deg Q and Q has no real zeros. Moreover, if all zeros of Q are in the lower half-plane, then ψ := P Q is holomorphic in the upper half-plane with bounded { ψ y 2 : y > 0}, whence h ∈ H + . Conversely, let φ be the holomorphic function on the upper half-plane associated with h ∈ H + . Then φ converges to h non-tangentially a.e., and ψ is meromorphic without real poles, whence ψ(z) → h(x) for z → x ∈ R. Hence φ and ψ coincide on the upper half-plane by [15,IV 2.5] so that ψ has no poles there.
(ii) As to (b) we prove h = R/P for some polynomial R supposing that Q has no real zeros and deg Q ≥ deg P .
Let φ and ψ be the holomorphic functions on the upper half-plane and lower half-plane associated with h and P Q h, respectively. There is δ > 0 such that B := P/Q is holomorphic and bounded in the strip {z : −δ < Im z < δ}, and hence Bφ is holomorphic on {0 < Im z < δ}. Since B is bounded, one easily infers J |(Bφ)(x + i y) − ψ(x − i y)| d x → 0, 0 < y → 0 for any bounded interval J. Then by [21,Theorem II] there is a holomorphic function χ on {z : Im z < δ} extending Bφ and ψ. So Qχ is still holomorphic on {z : Im z < δ} coinciding with P φ on {z : 0 < Im z < δ} and with Qψ on the lower half-plane. Hence there is an entire function R extending P φ on the upper and Qψ on the lower half-plane. Introduce S being equal to P on the upper half-plane and equal to Q on the lower half-plane. Analogously define Σ with respect to φ and ψ.
Fix z ∈ C, |z| > 1. We use the representation R(z) = 1 π D R(z + w) d 2 w were D denotes the disc with center 0 and radius 1. Then The first integral is easily estimated ≤ constant |z| 2n with n := deg Q. The double integral is bounded Let a ∈ C be a zero of P and write P = (x− a)P ′ . Note that (x− a)h ∈ L 2 (R) since |x− a| ≤ c|P (x)|, x ∈ R \ J for some bounded interval J and constant c. Hence it suffices to prove (x − a)h ∈ H + and proceed with P ′ in place of P . Actually the claim is xh ∈ H + and hence equivalently ( where the zeros of q are exactly the real zeros of Q. Let q ǫ (z) := q(z + i ǫ) for ǫ > 0 and put Q ǫ := q ǫ Q < . Note |q(x)/q ǫ (x)| < 1, x ∈ R. Hence h/Q ǫ ∈ L 2 (R). Since 1/Q ǫ is bounded on the upper half-plane, h/Q ǫ ∈ H + . Moreover, h/Q ǫ → h/Q for ǫ → 0 pointwise and in the mean, whence h/Q ∈ H + .
(v) It follows the proof of (c). Without restriction let P and Q be without common zeros. Then P Q h ∈ L 2 (R) implies 1 Q h ∈ L 2 (R). Indeed, let K be a compact neighborhood of the real zeros of Q containing no real zero of P . Then |h/Q| is bounded by C|P h/Q| on K and by c|h| on R \ K for some finite constants C, c, whence 1 Q h ∈ L 2 (R). -Now 1 Q h ∈ H + by (iv) and hence P 1 Q h ∈ H + by (iii). (vi) We proceed with the proof of (b). Without restriction let P and Q be without common zeros. Write Q = q Q 0 , where the zeros of q are exactly the real zeros of Q. Then P Q h ∈ L 2 (R) implies 1 q h ∈ L 2 (R), cf. (v). Hence h ′ := h/q ∈ H + by (iv). Next let P = p P 0 with deg P 0 = deg Q 0 if deg P > deg Q 0 and p = 1 otherwise. Then h ′′ := ph ′ ∈ L 2 (R), since h ′′ = Q0 P0 P Q h, where in the case p = 1 the factor Q0 P0 is bounded outside a bounded interval. By (iii) this implies h ′′ ∈ H + . By assumption P0 Q0 h ′′ ∈ H − . Therefore h ′′ = R 0 /P 0 by (ii) for some polynomial R 0 . It follows h = R/P ∈ H + for R := qR 0 and hence R/Q ∈ H − . The proof is accomplished applying (a) proved in (i).
(vii) As to the proof of (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 [15,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 the zeros of q are exactly the real zeros of Q. Let q ǫ (z) := q(z + i ǫ) for ǫ > 0 and put Q ǫ := q ǫ Q 0 . Note |q(z)/q ǫ (z)| < 1 on the upper half-plane. Therefore P Qǫ h = q qǫ P Q h ∈ H + , where Q ǫ has no real zeros. Moreover, for ǫ > 0 small enough, P and Q ǫ have no common zeros. Hence the foregoing result applies so that h/Q ǫ ∈ H + . Now h/Q ǫ = q qǫ h/Q → h/Q for ǫ → 0 in the mean implying h/Q ∈ H + .
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. [23]).  Here the zeros of the polynomial q are the real zeros of Q, ς := max{deg q, deg P − deg Q + deg q}. Moreover P = P < P ≥ , where the zeros of P < and P ≥ are exactly the zeros of P in the lower 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. Proof. For the closeness of M + (κ) write P Q in the form P Q = P0 Q0 + 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 and it remains to show that M + ( p q ) is closed. Let h n ∈ dom M + ( p q ) such that (h n ) converges to some h ∈ H + and M + ( p q )h n converges to some k ∈ H + . By (7)(c), p q h n ∈ H + . Hence one has h n → h and p q h n → k in L 2 (R). Since M ( p q ) is closed, h ∈ dom M + (κ) and k = M + ( p q )h follows. (a) dom M + (κ) is dense by (4)(b). Arguing as above it remains to show dom M + ( p q ) = q (x+i) ς H + . Let h ∈ H + . Then, by (7)(c), q (x+i) ς h ∈ H + and p q q (x+i) ς h ∈ H + implying q (x+i) ς H + ⊂ dom M + ( p q ). For the converse inclusion argue g ∈ dom M + ( p q ) ⇒ g ∈ H + , p q g ∈ L 2 ⇒ h := p q g ∈ H + by (7)(c). Hence g = q p h, whence 1 p h ∈ H + by (7)(d). Since ς = max{deg p, deg q} one infers k := (x+i) ς p h ∈ L 2 , whence k ∈ H + applying (7)(c) to 1 p h ∈ H + . This shows g = q (x+i) ς k ∈ q (x+i) ς H + . -Now the claim about ran M + (κ) is obvious.
(c) Using (a) one has h 0 ∈ ran M + (κ) Hence M (R) is a homeomorphism on L 2 (R), whence RH + is closed. Since (Q > ) ∼ =Q < it follows by (7)(c) that RH + ⊂ H + . Hence RH + = ran M + (κ ′ ) for κ ′ := P Q ′ with Q ′ :=Q < Q < q, whence dim(RH + ) ⊥ < ∞ by (c). So it suffices to show that RH + ⊂ ran M + (κ). By (7)   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.
From (8) one immediately obtains (9) Corollary. Let λ ∈ C and 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 [24]. 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).
Proof. For dom M + (κ) see (8)(a). Adopting the notation of (8) one has M + (κ) − i I = M + P Q with P := p − i q and Q = q. Obviously P and Q have no common zeros. In view of (8)(c) noteQ < = q < , An interesting property of polynomials follows immediately from the foregoing considerations. Let R be a polynomial without real zeros, and let n > and n < denote the numbers of its zeros in the upper and lower half-plane, respectively. Write R = p + i q with real polynomials p and q, and let n p and n q denote half the numbers of the non-real zeros of p and q, respectively. Then max{n p , n q } ≤ min{n > , n < }. To illustrate this consider the case that all zeros of R lie in one half-plane. Then all zeros of p and q are real. However the converse does not hold as for instance R = z 2 + i shows.
W κ for κ in (10)  This section is concluded by a much needed (11) Example. W |x| is essentially self-adjoint but not closed. (The same holds true for W 1/|x| .) So there are WH (also essentially self-adjoint semibounded ones), which are not closed.

Semibounded Wiener-Hopf operators
In (14), (15) a semibounded densely defined WH W κ is expressed in a natural way by the product of a closable operator and its adjoint. Replacing the operator by its closure one obtains a self-adjoint extensioñ W 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 .
In view of (12)(⋆) recall the results on the domain of a WH in (4). If A is closable it need not be closed, even if AA * is closed. Indeed, W x 2 = AA * is closed and =W x 2 = AA * by (10), (14). Recall that ran A * is not dense if A is not injective. A is not injective for γ = 0 a.e. if for instance q γ F −1 s ∈ L 2 (R) with q a polynomial and s a Schwartz function with support in ] − ∞, 0]. (Indeed, qF −1 s is a Schwartz function in H − , whence q γ F −1 s ∈ ker A.) Recall that A is injective if and only if γ = 0 not a.e. or γ = 0 a.e. and 1 γ non-proper.
(13) Lemma. Let γ be proper. Suppose that γ = 0 not a.e. or that γ = 0 a.e. and 1 γ is not proper. Then A is closable and A is injective.
The foregoing lemma is needed only in sec. 5. The main result of this section follows.  a) W κ is densely defined symmetric nonnegative andW κ is an injective nonnegative self-adjoint extension of W κ .
(b) dom W κ is a core of A * and dom A * ∩ ran(I + W κ ) ⊥ = ∅ holds.
(c)W κ is the Friedrichs extension of W κ .
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 AA * is dense in K since dom AA * is a core for A * , see [25, 13.13(b)]. One easily checks that H Wκ is the closure of dom W κ in K. Therefore dom AA * ⊂ H Wκ if and only if H Wκ = K, which means that dom W κ is a core of A * . A short computation shows also that H Wκ = K is equivalent to dom A * ∩ ran(I + W κ ) ⊥ = ∅.
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 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 turns out to be closely related to W κ , where κ extends φ on R by zero. L φ belongs to the studied class of singular integral operators in L 2 (E) of type Hilbert transformation 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 [26]. It is shown to be self-adjoint on dom L(a, b) = dom M (a), and is diagonalization is achieved. See also [27] and the literature cited in [26], [27]. The really unbounded case however is there when b is unbounded. [28] 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 (18) 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 (20).
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 [30]. 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 [31].

Isometry relatingL φ toW κ
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) ⊥ . Important is the 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.
Now 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 (c)W κ := AA * is a self-adjoint extension of W κ .
(d)L φ := A * A is a self-adjoint extension of L φ .
As a first result we note that L φ in (5.1) can be trivial.
For example κ = e x is positive and, by (4)(h), κ and 1/κ are not proper. Hence dom L e x = {0}. Now (17) and (5.3) have the following important outcome on L(a, b). Accordingly, for L(a, b) in (5.2) being not trivial it is necessary that the extension of b on R by zero be proper. is not proper (19) Corollary. Let E = R and suppose that 1 φ is bounded.. Then L φ is densely defined and If φ is proper then kerL φ = ker L φ = L 2 (R). If φ is not proper then L φ = 0.
The main outcome of this section is (20) Theorem. Let κ ≥ 0 be proper. ThenL 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 + ) satisfying Proof. (5.8) follows immediately from polar decomposition A = T |A| (cf. (5.7)). By (16) T is surjective. Hence T ′ is a Hilbert space isomorphism by (16)(e). Finally recall (5.8).
The special case of (21) that φ is bounded is treated in [26, sec. 3] and [27,Theorem]. -We remind that forL φ being injective it is necessary that E is proper or that 1/κ is not proper.
(22) Corollary. Let κ ≥ 0 be proper. Suppose that E is proper or 1/κ is not proper. ThenL φ is injective and T : L 2 (E) → L 2 (R + ) is a Hilbert space isomorphism with
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 φ be bounded. Then L φ , W κ are bounded andL φ = L φ ,W κ = W κ , and L φ = T * W κ T . Moreover, if E is proper or 1/κ is not proper, then even L ′ φ = L φ by (22), whence 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 sec. 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 + ), T ′ h := P + F h. Examples for proper E are the isomorphic pairs W (1 E ) ≃ 1 2 (I + H E ) with H E in (5.6), which we like to write as The case of the finite Hilbert transformation H [−1,1] is studied in detail in [22, sec. 3.3], [22, (3.20)].
In conclusion we make a remark on the spectral representations ofW κ andL φ in (20). 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 satisfying The spectral measure forW κ is given by EW κ (∆) = V M (1 ϕ −1 (∆) )V −1 for measurable ∆ ⊂ R. For the last equality recall A * = T * |A * |. Further note that dom(|A * | −1 ) = ran(|A * |) is dense as |A * | = W κ 1/2 is self-adjoint injective. 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 [32] a spectral theory of W κ is proposed by a reduction to a previously developed theory for singular integral operators. Its application in [32] to Lalescu's operator W λ with symbol λ(x) := 2 1 + x 2 (5.11) however does not produce the right normalization (5.13) of the generalized eigenfunctions (5.12). In establishing (5.13) we get also the result (25) on orthogonal polynomials.
The diagonalization of Lalescu's operator W λ is achieved in sec. 5.2.1 and that of the associated singular integral operator L λ is derived in sec. 5.2.2. The generalized eigenfunctions (5.17) of the latter are no longer regular distributions.

Spectral representation of W λ
Clearly the spectrum of the WH W λ lies in [0, 2]. The kernel for W λ is e −|x| . The obvious ansatz a e αx +b e βx for u(x), x > 0 in s u(x) − 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 (23) and is suggested heuristically by A * q s , A * q s = q s , AA * q s = s q s , q s . For the proof recall the isomorphism Γ (5.15) 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.  (Γh) oe 2 = h 2 . So W is an isometry, and (⋆) extends to all h ∈ L 2 (0, 2). Obviously W h ∈ M (u)H + . Moreover, if k ∈ H + then k = 1 2 I + H (k −ǩ) asǩ ∈ H − . Since k −ǩ is odd, this implies ran W = M (u)H + . The representation of L λ follows from W M (id [0,2] )h = L λ W h for test functions h. The latter is shown along the lines of the proof of (26). We omit the technicalities.