INVARIANT SUBSPACES AND COMMUTANT FOR THE GAUSSIAN HILBERT TRANSFORM ´ AND WILFREDO URBINA MARIA D. MORAN
Abstract. In this note we study some properties, from the point of view of Operator Theory, of the Gaussian Hilbert Transform. More specifically we will prove that the Gaussian Hilbert Transform is unitary equivalent to the adjoint of the unilateral shift operator acting on H 2 (D) and thus we can completely characterize the invariant subspaces and commutant of the Gaussian Hilbert Transform.
1. Introduction In 1969, B. Muckenhoupt [4] defined the notion of Gaussian Hilbert Transform for the Hermite expansions, in one dimension, by analogy with the classical case of the Hilbert Transform, in Lp of the circumference (see [8]). 2 1 e−x dx, which Let γ1 be the Gaussian measure, given by γ1 (dx) = π1/2 is a probability measure in R. For f ∈ L1 (γ1 ), its Poisson-Hermite integral is defined as u(x, t) =
Z ∞ −∞
R(t, x, y)f (y) γ1 (dy),
t > 0, where 2
2 2
t −r x +2rxy−r 1 Z 1 t exp( 2 log r ) exp( 1−r2 R(t, x, y) = √ 3/2 (1 − r2 )1/2 π 2 0 r(− log r) It is easy to see that u(x, t) satisfies:
2 y2
)
dr.
∂ 2 u(x, t) ∂ 2 u(x, t) ∂u(x, t) + − 2x = 0, 2 2 ∂t ∂x ∂x which is equivalent to ∂ 2 u(x, t) x2 ∂ −x2 ∂u(x, t) + e (e ) = 0. ∂t2 ∂x ∂x 1991 Mathematics Subject Classification. 42B25,47D03,42C10 , Secondary 60H99,42A99. Key words and phrases. Gaussian Measure, Fourier Analysis, Operator Theory, Singular Integrals, Hermite expansions, Shift operator. 1
´ AND W. URBINA M. MORAN
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Then we consider a L-harmonic conjugated v, given by the Gaussian Cauchy-Riemann equations: ∂u(x, t) ∂v(x, t) = − ∂x ∂t ∂u(x, t) ∂ 2 2 ∂v(x, t) = ex (e−x ), ∂t ∂x ∂x and it can be proved that v can be written as v(x, t) =
Z ∞ −∞
Q(t, x, y)f (y) γ1 (dy),
t > 0, where √ Z t2 −r2 x2 +2rxy−r2 y 2 ) 2 1 (y − rx) exp( 2 log r ) exp( 1−r2 dr. Q(t, x, y) = π 0 (− log r)1/2 (1 − r2 )3/2 Muckenhoupt [4] proved that v is Lp (γ1 ) bounded for p > 1, and for f ∈ Lp (γ1 ) , p > 1, v(x, t) tends to √ Z Z −r2 x2 +2rxy−r2 y 2 ) 2 ∞ 1 (y − rx) exp( 1−r2 Hf (x) = drf (y) γ1 (dy), π −∞ 0 (− log r)1/2 (1 − r2 )3/2 as t → 0,in Lp (γ1 )-norm and also almost everywhere. We shall call H the Gaussian Hilbert Transform. Let D be the open unit disk, T the circumference, consider the square integrable functions in T 2
L (T) = {f : T → C :
Z π −π
|f(i≈ )|2 ≈ < ∞},
let {en }n∈Z be the trigonometric system, en (ξ) = ξ n , ξ ∈ T, which is a complete orthonormal system in L2 (T), and finally set S : L2 (T) → L2 (T) be shift operator given by (Sf )(ξ) = ξf (ξ), for all ξ ∈ T. For more details on the shift operator we refer to Nikol’skii [5]. If we consider H 2 (D) = {f : D → C : f =
∞ X n=0
an zn and
∞ X
|an |2 < ∞},
n=0
then, we can identify H 2 (D) with the subspace of L2 (T) consisting of functions such that hf, en i = 0, ∀n < 0.
INVARIANT SUBSPACES AND COMMUTANT
3
The restriction of the bilateral forward shift operator to H 2 (D) , which, abusing the notation, we shall also call S, is the it unilateral forward shift, which leaves the space H 2 (D) invariant and S(
∞ X n=0
an z n ) =
∞ X
an z n+1 =
n=0
∞ X
an−1 z n .
n=1
On the other hand, consider on R the Gaussian measure γ1 and the space of the square integrable functions on R with respect to γ1 L2 (γ1 ) = {f : R → C :
Z ∞ −∞
|f(≈)|2 γ1 (≈) < ∞}.
Let {Hn } be the Hermite polynomials, which can be defined as: H0 (x) = 1 and for n > 1
dn −x2 (e ). dxn They form a complete orthogonal system in L2 (γ1 ). Let us denote with hn the normalized Hermite polynomial of order n, Hn (x) = (−1)n ex
hn (x) =
2
Hn (x) . (2n n!)1/2
It can be seen that if f ∈ L2 (γ1 ) and f = ∞ n=0 hf, Hn iHn then the Gaussian Hilbert transform is given by ∞ √ X Hf = 2nhf, Hn iHn−1 . P
n=1
Observe that, in particular, we have √ Hn Hn−1 Hhn = H( n 1/2 ) = 2n n 1/2 = hn−1 . (2 n!) (2 n!) In this paper we will prove that the Gaussian Hilbert Transform is unitary equivalent to the adjoint of the unilateral shift operator acting on H 2 (D) and thus we are able to completely characterize the invariant subspaces and commutant of the Gaussian Hilbert Transform. More explicitly the main results are the following: Theorem 1.1. The Gaussian Hilbert Transform H as an operator on L2 (γ1 ) is unitary equivalent to the adjoint of the restriction of the shift operator on H 2 (D). With this result we can obtain the following results. The first one completely characterizes the invariant subspaces of the Gaussian Hilbert Transform.
´ AND W. URBINA M. MORAN
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Theorem 1.2. Given the Gaussian Hilbert Transform H and A a proper and closed subspace of L2 (γ1 ) then H(A) ⊂ A if and only if there P n exist a sequence of complex numbers {an } such that | ∞ n=0 an z | = 1 almost everywhere in T and A = {f =
∞ X
hf, hn ihn ∈ L2 (γ1 ) :
n=0
∞ X
hf, hn ian−k = 0, ∀k ≥ 0}
n≥k
The next result characterizes the commutant of the Gaussian Hilbert Transform. Theorem 1.3. Let F be a linear operator on L2 (γ1 ). If FH = HF then there exist g ∈ H ∞ (D) such that Ff = F(
∞ X
hf, hn iL2 (γ1 ) hn )
n=0 ∞ X ∞ X
=
hf, hn iL2 (γ1 ) hg, en−k iH 2 (D) hk .
k=0 n≥k
Conversely if this relation holds and P0 f = hf, h0 iL2 (γ1 ) h0 , then FH = HF(I − P0 ). Finally we have: Theorem 1.4. Let A be a (closed) subspace of L2 (γ1 ) that is invariant for H and PA is the orthogonal projection of L2 (γ1 ) onto A. If F is a linear operator on A such that F(PA H∗ ) = (PA H∗ )F, then there exists F1 a linear operator acting on L2 (γ1 ) such that F∞ H = HF∞ and F = PA F∞ ∗ |A . We want to express our gratitude to the referee whose detailed observations improve substantially the presentation of this article. 2. Proofs Proof of Theorem 1.1: Let us consider Ω : L2 (γ1 ) → H 2 (D) defined by Ω(
∞ X
hf, hn ihn ) =
n=0
∞ X
hf, hn ien .
n=0
INVARIANT SUBSPACES AND COMMUTANT
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It is easy to see, by Parseval’s identity, that Ω is a well defined operator and unitary, also Ω intertwining H and S ∗ i.e. ΩH = S ∗ Ω : ∞ ∞ √ X X ΩH( hf, hn ihn ) = ΩH( ( 2n n!)−2 hf, Hn iHn ) n=0
= Ω( = Ω(
n=0 ∞ X √
√ ( 2n n!)−2 2nhf, Hn iHn−1 )
n=1 ∞ X
hf, hn ihn−1 ) =
n=1 ∞ X
= S ∗(
∞ X
hf, hn ien−1
n=1
hf, hn ien ) = S ∗ Ω(
n=0
∞ X
hf, hn ihn ).
n=0
Proof of Theorem 1.2: We shall prove the condition is necessary. Let Ω intertwining H and S ∗ as in Theorem 1.1. It is clear that H(A) ⊂ A if and only if S ∗ (ΩA) ⊂ ΩA, and this is equivalent to S(H 2 (D) ΩA) ⊂ H2 (D) ΩA. Now H 2 (D) ΩA = Ω(L2 (γ1 ) A) 6= 0 by hypothesis, then H 2 (D) ΩA is a non trivial, closed subspace of H 2 (D), then ( see [1] and [2] ) there exists θ ∈ H 2 (D) with |θ(ξ)| = 1 for almost all ξ ∈ T such that H 2 (D) ΩA = θH2 (D), or equivalently A = Ω−1 [H 2 (D) θH2 (D)] = L2 (γ1 ) Ω−1 θH2 (D). Let θ(z) =
P∞
n=0
an z n , then f =
P∞
n=0 hf, hn ihn
∈ A if and only if
hf, Ω−1 θui = 0, for all u ∈ H 2 (D). Given k ≥ 0 let us take u = ek , then 0 = hf, Ω−1 θek i = hΩf, θek i =
∞ X
hf, hn ihen , θek i,
n=0
but we have that (
hen , θek i =
an−k , if k ≥ n, 0 if k < n,
then for for all k ≥ 0, ∞ n=0 hf, hn ian−k = 0. P n We shall now prove the sufficiency. Let θ(z) = ∞ n=0 an z then given f ∈ L2 (γ1 ), P
f ∈ A if and only if for all k ≥ 0 hΩf, θek i = 0.
´ AND W. URBINA M. MORAN
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Thus, if u ∈ H 2 (D), then we have |hΩf, θui| ≤ |hΩf, θ(u −
n X
hu, ek iek )i|
k=0
≤ ||Ωf ||H 2 ||θ(u − ≤ ||f ||L2 (γ1 ) ||u −
n X
hu, ek iek )||H 2
k=0 n X
hu, ek iek ||H 2 ,
k=0
but lim ||u −
n→∞
n X
hu, ek iek ||H 2 = 0.
k=0
Therefore hΩf, θui = 0 for all f ∈ A, u ∈ H 2 (D) thus ΩA = H 2 θH 2 (D), and, then HA = Ω−1 ΩHA = Ω−1 S ∗ ΩA = Ω−1 S ∗ (H 2 θH 2 (D)) ⊂ Ω−1 (H2 θH2 (D)) = A. Proof of Theorem 1.3: Let G = ΩFΩ−1 . It is clear that G ∈ L(H 2 (D)) and S ∗ G = S ∗ ΩFΩ−1 = ΩHFΩ−1 = ΩFHΩ−1 = ΩFΩ−1 S ∗ = GS ∗ , thus S ∗ G = GS ∗ , and also G∗ S = SG∗ . Let g = G∗ e0 , then it is easy to check that g ∈ H ∞ (D), G∗ u = gu for all u ∈ D and Ff = F(
∞ X
hf, hn iL2 (γ1 ) hn ) =
= =
∞ X
hf, hn iL2 (γ1 ) Fhn
n=0
n=0
=
∞ X
hf, hn iL2 (γ1 ) Ω−1 FΩhn =
n=0 ∞ X
∞ X
hf, hn iL2 (γ1 ) Ω−1 (
n=0
hf, hn iL2 (γ1 ) Ω−1 (
n=0 ∞ X n X
∞ X
∞ X
hGen , ek iek )
k=0
hen , gek iek ) =
∞ X
hf, hn iL2 (γ1 ) Ω−1 (
n=0
k=0
hf, hn iL2 (γ1 ) hg, en−k ihk =
n=0 k=0
∞ X X
(
n X
hg, en−k iek )
k=0
hf, hn iL2 (γ1 ) hg, en−k ihk ).
k=0 k≥n
Conversely, FHf = FH(
∞ X
hf, hn iL2 (γ1 ) hn ) = F(
∞ X
hf, hn iL2 (γ1 ) hn−1 )
n=0 ∞ X
∞ X X
n=0
k=0 n≥k
= F(
hf, hn+1 iL2 (γ1 ) hn ) =
n=1
hf, hn+1 iL2 (γ1 ) hg, en−k ihk
INVARIANT SUBSPACES AND COMMUTANT
7
and HF(I − P0 )f = HF( = H(
∞ X
hf, hn iL2 (γ1 ) hn ) = H(
n=1 ∞ XX
∞ X X
hf, hn iL2 (γ1 ) hg, en−k ihk ) =
k=0 n≥k
=
∞ X X
hf, hn+1 iL2 (γ1 ) hg, en−k ihk )
k=0 n≥k ∞ X X
hf, hn iL2 (γ1 ) hg, en−k ihk−1
k=1 n≥k ∞ X X
hf, hn iL2 (γ1 ) hg, en−(k+1) ihk =
k=0 n≥k+1
hf, hn+1 iL2 (γ1 ) hg, en−k ihk
k=0 n≥k
thus FH = HF(I − P0 ) . Proof of Theorem 1.4: Recall that H(A) ⊂ A implies that H(A⊥ ) ⊂ A⊥ . Set B = ΩA, then S(H 2 (D) B) ⊂ H2 (D) B. Let T a linear operator in B given by T = ΩFΩ−1 , then T PB S|B = ΩFΩ−1 PB S|B = ΩFPA Ω−1 S|B = ΩFPA H∗ Ω−1 |B = ΩPA H∗ FΩ−1 |B = PB ΩH∗ Ω−1 T |B = PB ST |B . Thus, by Sarason’s Generalized Interpolation Theorem [6], there exist g ∈ H∞ (D) such that T = PB Mg |B ( i.e. T f = PB (f g) ∀f ∈ B), and if F1 = ΩMg Ω−1 , then PA F1 |A = PA ΩMg Ω−1 |A = Ω−1 PB ΩΩ−1 Mg Ω|A = Ω−1 PB Mg Ω|A = Ω−1 T Ω|A = F, and F1 H∗ = Ω−1 Mg ΩH∗ = Ω−1 Mg SΩ = Ω−1 SMg Ω = Ω−1 SΩΩ−1 Mg Ω = Ω−1 ΩH∗ F1 = H∗ F1 . References [1] Beurling, A. On two problems concerning linear operators in Hilbert space. Acta Math. 81 (1949) [2] Helson, H. Lectures on invariant subespaces. New York. Academic Spaces (1964). [3] Meyer, P.A. Transformations de Riesz pour les lois Gaussians. Lectures Notes in Math. 1059 (1984) Springer-Verlag. Berlin. 179-193 . [4] Muckenhoupt, B. Hermite conjugated expansions. Trans.Amer. Math. Soc. 139 (1969) 243-260. [5] Nikol’skii, N.K. Treatise on the Shift Operator: Spectral Function Theory. Springer-Verlag.Berlin (1986). [6] Sarason, D. Generalized interpolation in H ∞ . Trans.Amer. Math. Soc. 127 (1967) 179-203. [7] Stein, E. Singular Integrals and differenciability properties of functions. Princeton Univ. Press. Princeton (1970) .
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´ AND W. URBINA M. MORAN
[8] Zygmund, A. Trigonometric Series. 2nd. Ed. Cambridge Univ. Press. Cambridge (1959). ´ tica, Facultad de Ciencias UCV, Apt. 20513 Departamento de Matema Caracas 1020-A Venezuela E-mail address:
[email protected] ´ tica, Facultad de Ciencias UCV Apt. 47195 Departamento de Matema Los Chaguaramos, Caracas 1041-A Venezuela E-mail address:
[email protected]
