PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 140, Number 12, December 2012, Pages 4247–4257 S 0002-9939(2012)11541-2 Article electronically published on April 18, 2012
A VOLTERRA-TYPE INTEGRATION OPERATOR ON FOCK SPACES
OLIVIA CONSTANTIN
(Communicated by Richard Rochberg)
Abstract. We study certain spectral properties and the invariant subspaces for some classes of integration operators of Volterra type on the Fock space.
1. Introduction For analytic functions f,g we consider the Volterra-type integration operator given by z Tgf(z)= f(ζ)g (ζ) dζ. 0 The boundedness and compactness, as well as some spectral properties (such as Schatten class membership) of Tg acting on various spaces of analytic functions of the unit disc D in C have been extensively investigated (see [2, 6] for Hardy spaces, [3, 5, 8, 17, 19] for weighted Bergman spaces, and [9, 10] for Dirichlet spaces, or the surveys [1, 20] and the references therein). Furthermore, the spectrum of Tg on weighted Bergman spaces was recently characterized in [3]. p In this paper we consider the operator Tg acting on the Fock spaces F ,p>0 (here, the symbol g is an entire function with g(0) = 0). Recall that, for p>0, the Fock space F p consists of entire functions f for which 1 | |2 p p − z p f p = f(z)e 2 dA(z) < ∞. 2π C
The boundedness and compactness of Tg follow by classical methods, and we include a sketch of their proofs in Section 3 for the sake of completeness. It turns out that, p q if p ≤ q, Tg : F →F is bounded if and only if the symbol g is a polynomial of degree ≤ 2, while the necessary and sufficient condition for compactness is: g is a polynomial of degree ≤ 1. For p>q, the operator Tg is bounded if and only if it is compact, and these are equivalent to the condition 2p q> and g(z)=az for some a ∈ C. p +2 In particular, this shows that the primitive of a function in F p is in F q for q> 2p/(p + 2), and this result is sharp. In Section 4 we turn to the spectral properties of Tg. Regarding the Schatten class membership of Tg|F 2 , we show that, provided p it is compact, Tg fails to be Hilbert-Schmidt, but it belongs to all classes S for
Received by the editors May 30, 2011. 2010 Mathematics Subject Classification. Primary 30H20, 47B38. Key words and phrases. Fock spaces, integration operator, spectrum, invariant subspaces.
c 2012 American Mathematical Society Reverts to public domain 28 years from publication 4247
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4248 OLIVIA CONSTANTIN
p>2. Moreover, we prove that the spectrum of Tg|F 2 is a closed disc centred at the origin whose radius depends on the symbol g. In the last section of this paper we characterize the invariant subspaces of the Volterra operator z Vf(z)= f(ζ) dζ, 0
which is a particular case of the operator Tg, corresponding to g(z)=z.Acomplete description of the invariant subspaces of V , when acting on various classical spaces of analytic functions on the unit disc (Hardy spaces, weighted Bergman spaces, Dirichlet spaces) was obtained in [4]. We prove the Fock space analogue of these results, showing that the proper invariant subspaces of V : F p →Fp, p>0, are given by F p p { k ≥ } AN = Span z : k N +1 , where N is an arbitrary nonnegative integer.
2. Preliminaries Recall that the point evaluation functionals are bounded on F p, p>0. More precisely, for every f ∈Fp,wehave
|z|2/2 (1) |f(z)|≤e f p,z∈ C. √ Moreover, the monomials zn/ n!,n≥ 0, form an orthonormal basis for F 2,and 2 λz¯ therefore the reproducing kernel of F is given by Kλ(z)=e , i.e. 1 2 f(λ)= f(z) eλz¯ e−|z| dA(z),λ∈ C,f∈F2. π C The following straightforward fact will be used in our further considerations: for an entire function f, the maximum principle and the subharmonicity of |f|p ensure that p p |f(z)| 2 |f(z)| 2 (2) e−p|z| /2dA(z) ∼ e−p|z| /2dA(z) | | p | | p C (1 + z ) |z|>1 (1 + z ) f(z)p 2 ∼ e−p|z| /2dA(z). |z|>1 z Moreover, if f(0) = 0, the above quantities are comparable to p f(z) −p|z|2/2 e dA(z). C z The next proposition can be easily deduced from a result in [7]. We include a sketch of its proof for the sake of completeness. Proposition 1. For an entire function f : C → C and p>0, the following holds: | |p | |p −p|z|2/2 ∼| |p f (z) −p|z|2/2 (3) f(z) e dA(z) f(0) + p e dA(z). C C (1 + |z|) Proof. It is enough to prove the claim for entire functions f with f(0) = 0. Indeed, the result for a general entire function f follows by applying (3) to z → f(z)−f(0).
License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A VOLTERRA-TYPE INTEGRATION OPERATOR ON FOCK SPACES 4249
Suppose now that f(z)=zg(z), where g is entire. Then by Lemmas 5-6 in [7] we have 2 2 |zg(z)|pe−p|z| /2dA(z) ∼|g(0)|p + |g (z)|pe−p|z| /2dA(z). C C This is equivalent to − p | |p −p|z|2/2 ∼| |p zf (z) f(z) −p|z|2/2 (4) f(z) e dA(z) f (0) + 2 e dA(z). C C z The integral on the right side is comparable to − p zf (z) f(z) −p|z|2/2 2 e dA(z). |z|>1 z Using this fact, relation (2) and the obvious inequality (a + b)p ≤ 2p(|a|p + |b|p), we now obtain | |p f (z) −p|z|2/2 | |p −p|z|2/2 p e dA(z) f(z) e dA(z). C (1 + |z|) C Hence one inequality in (3) is proven. We prove the reverse inequality by contra- diction. Assume there exists a sequence of entire functions fn with fn(0) = 0 such | |p −p|z|2/2 that C fn(z) e dA(z)=1and | |p fn(z) −p|z|2/2 ≤ 1 ≥ (5) p e dA(z) ,n 1. C (1 + |z|) n The expression (1+|z|) is bounded above and below on any compact set in C.Then | |p by the subharmonicity of f (z) and relation (5) we deduce that the sequence fn(z) converges to zero uniformly on the compact sets in C and, since fn(0) = 0, the same holds for fn. We now write relation (4) for each fn and we use (2) to get − p p zfn(z) fn(z) −p|z|2/2 1 ∼|f (0)| + e dA(z) n z2 C p p | |p fn(z) −p|z|2/2 fn(z) −p|z|2/2 fn(0) + e dA(z)+ 2 e dA(z) | | z | | z z >1 z >1 p p 1 fn(z) −p|z|2/2 |f (0)| + + e dA(z) n n z2 R>|z|>1 1 | |p −p|z|2/2 + 2 fn(z) e dA(z) R | | z >R p ≤| |p 1 fn(z) −p|z|2/2 1 fn(0) + + 2 e dA(z)+ 2 , n R>|z|>1 z R for any R>1. Letting first n →∞and then R →∞above we obtain a contradic- tion. Thus (3) holds.
3. Boundedness and compactness
The next theorem characterizes the boundedness and the compactness of Tg. p q Theorem 1. (i) For 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4250 OLIVIA CONSTANTIN p (ii) Assume 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A VOLTERRA-TYPE INTEGRATION OPERATOR ON FOCK SPACES 4251 (ii) The proof of the necessity follows a similar approach to the one in [14]. Assume Tg is bounded and let (λi) ⊂ C be a uniformly separated sequence, i.e. infi= j |λi −λj| > 0. Then, for any fixed radius R>0, there exists a positive integer I(R) > 0 such that any disc of radius R contains at most I(R)pointsλi. We also assume that (λi)isε-dense for some ε>0, i.e. every point in C is within ε-distance of some λi. Then, for ε sufficiently small (see Theorem 8.2 in [13] for p ≥ 1and p Theorem 3.1 in [21] for 0 where the ri are the Rademacher functions on [0, 1] (see [14]). Now denote q |g (z)| 2 dμ(z):= e−q|z| /2dA(z) (1 + |z|)q and use Fubini’s theorem together with Proposition 1 to deduce (8) 1 2 q/2 2 q 2 2Re(λ¯iz)−|λi| λ¯iz−|λi| /2 |ci| e dμ(z) ri(t) ci e dμ(z)dt C C i≥1 0 i≥1 1 2 q λ¯iz−|λi| /2 ∼ Tg ri(t) ci e dt q 0 i≥1 1 2 q q λ¯iz−|λi| /2 ≤ Tg ri(t) ci e dt p 0 i≥1 q/p p |ci| . i≥1 Since (λi)isε-dense, it follows that the discs D(λi, 2ε)coverC, and therefore 2 q/2 2 2Re(λ¯iz)−|λi| |ci| e dμ(z) C i≥1 2 −1 q q Re(λ¯iz)−q|λi| /2 ≥I (2ε) |ci| e dμ(z) ≥ D(λi,2ε) i 1 |g (z)|q |c |q dA(z). i (1 + |z|)q i≥1 D(λi,2ε) p Since (ci) ∈ l was arbitrarily chosen, by the previous relation together with (8) we get |g (z)|q (9) dA(z) ∈ lp/(p−q). | | q D(λi,2ε) (1 + z ) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4252 OLIVIA CONSTANTIN By subharmonicity we have |g (z)|q |g (ζ)|q dA(ζ),z∈ D(λ , 3ε/2),i≥ 1, | | q | | q i (1 + z ) D(λi,2ε) (1 + ζ ) with constants depending on ε. We now raise both sides of the above inequality to the power p/(p − q), then we integrate on D(λi, 3ε/2) and sum over i to get g (z) pq/(p−q) g (z) pq/(p−q) dA(z) ≤ dA(z) C (1 + |z|) (1 + |z|) i≥1 D(λi,3ε/2) |g (z)|q p/(p−q) dA(z) < ∞. (1 + |z|)q i≥1 D(λi,2ε) 2p Thus g is constant and q>p+2 . Now let us prove that g constant and q>2p/(p + 2) ensure the compactness of ∞ Tg. As before, let fn be a sequence of entire functions such that supn fn p < and fn → 0 uniformly on compact sets in C. Then, as in (6), for R>0, we have | |q q ∼ fn(z) −q|z|2/2 lim sup Tgfn q lim sup e dA(z) →∞ →∞ (1 + |z|)q n n |z|≤R q |f (z)| 2 + n e−q|z| /2dA(z) | | q |z|>R (1 + z ) 1 (p−q)/p ≤ dA(z) lim sup f q , pq/(p−q) n p |z|>R (1 + |z|) n→∞ ∞ where the last step above follows by H¨older’s inequality. As supn fn p < ,we F q now let R →∞in the above relation to deduce that Tgfn −−→ 0forn →∞.Thus Tg is compact and the proof is complete. 2 4. Schatten class membership and the spectrum of Tg on F 2 Throughout this section, we shall consider the operator Tg acting on F . Recall that an operator T acting on a Hilbert space belongs to the Schatten class Sp, p>0, if the sequence of eigenvalues of (T ∗T )1/2 belongs to lp. p Theorem 2. Suppose Tg is compact. Then Tg belongs to the Schatten class S for all p>2, but it fails to be Hilbert-Schmidt unless g is constant. Proof. By Theorem 1, the operator Tg is compact if and only if g is constant. Let ≡ us first show that Tg fails to be Hilbert-Schmidt if g 0. Given an orthonormal F 2 |z|2 | |2 basis (en)in ,wehaveKz(z)=e = n≥1 en(z) . We use this together with Proposition 1 to infer | |2 −|z|2 2 ∼ en(z) −|z|2 Kz(z)e ∞ Tgen 2 2 e dA(z)= 2 dA(z)= , C (1 + |z|) C (1 + |z|) n≥1 n≥1 which shows that Tg is not Hilbert-Schmidt. Given p>2, a bounded linear operator A on a Hilbert space belongs to Sp if p and only if Aen < ∞ for any orthonormal basis (en) of the Hilbert space n License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A VOLTERRA-TYPE INTEGRATION OPERATOR ON FOCK SPACES 4253 2 (see [11]). For an orthonormal basis (en)ofF we have, as before, | |2 p/2 p ∼ en(z) −|z|2 Tgen 2 2 e dA(z) C (1 + |z|) n≥1 n≥1 2 (p−2)/2 | |2 −|z| ≤ | |2 −|z|2 en(z) e en(z) e dA(z) p dA(z) C C (1 + |z|) n≥1 | |2 −|z|2 en(z) e 1 ∞ = p dA(z)= p dA(z) < , C (1 + |z|) C (1 + |z|) n≥1 by H¨older’s inequality and since (en) is an orthonormal basis. The proof is now complete. 2 2 Let us now turn to the spectrum of Tg : F →F . We assume Tg is bounded, and hence g(z)=az2 + bz,forsomea, b ∈ C. Note first that for λ ∈ C \{0},the equation 1 f − T f = h λ g has the unique analytic solution z g(z)/λ g(z)/λ −g(ζ)/λ (10) f(z)=Rλ,gh(z)=h(0)e + e e h (ζ)dζ, z ∈ C. 0 Thus the resolvent set of Tg consists precisely of those points λ ∈ C for which Rλ,g 2 is a bounded operator on F . Furthermore, the investigation of the spectrum of Tg is related to the behavior of the exponentials eg/λ, λ ∈ C \{0}, as was first pointed out by Pommerenke [18]. Notice that for h ≡ 1 in the last relation above we get g/λ g/λ 2 Rλ,g1=e . Hence e ∈F whenever λ belongs to the resolvent set of Tg.This provides a sufficient condition for a point λ to belong to the spectrum of Tg.A necessary and sufficient condition will be easily obtained once we have proved the next lemma. Lemma 1. Assume g(z)=az2 + bz and |λ| > 2a. Then, for any entire function f, we have | g/λ |2 | g/λ |2 −|z|2 | |2 e f −|z|2 e f e dA(z) f(0) + 2 e dA(z). C C (1 + |z|) Proof. Suppose first that f(0) = f (0) = 0. A simple calculation based on the fact F 2 n that the -norm of an entire function with Taylor series n≥0 anz is given by ( |a |2n!)1/2 shows that n≥0 n f f 2 ≤ . z 2 Replacing f by eg/λf aboveweobtain g/λ g/λ 2az + b f g/λ 2az + b g/λ f e f 2 ≤ e f + ≤ e f + e , λz z 2 λz 2 z 2 and hence | | 2a g/λ g/λ bf g/λ f (11) 1 − e f 2 ≤ e + e . |λ| λz 2 z 2 g/λ g/λ We claim that this implies that e f 2 e f /z 2. Indeed, if this did not hold, there would exist a sequence of entire functions (fn) with fn(0) = fn(0) = 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4254 OLIVIA CONSTANTIN g/λ g/λ ≤ ≥ such that e fn 2 =1and e fn/z 2 1/n for n 1. The last inequality together with (1) show that the sequence fn(z) converges to zero uniformly on the compact sets in C, and since fn(0) = 0, the same holds for fn/z. On the other hand, for R>0, we deduce from (11) that |2a| bf 2 2 1/2 − ≤ g/λ n −|z| 1 | | e e dA(z) λ |z|≤R λz | | 1/2 b | g/λ |2 −|z|2 1 +| | e fn e dA(z) + λ R |z|>R n bf 2 2 1/2 |b| 1 ≤ g/λ n −|z| e e dA(z) + | | + . |z|≤R λz λ R n We first let n →∞, and subsequently R →∞, in the above relation to obtain a contradiction. Thus | g/λ |2 1/2 g/λ g/λ ∼ e f −|z|2 e f 2 e f /z 2 2 e . C (1 + |z| ) The desired result for a general f can be easily deduced by applying the above inequality to z → f(z) − f(0) − f (0)z. 2 Theorem 3. Suppose g(z)=az + bz for some a, b ∈ C. Then the spectrum of Tg is the closed disc centred at the origin and of radius 2|a|. Proof. Let ρ(Tg) denote the resolvent set of Tg. As mentioned before, if λ ∈ ρ(Tg), then eg/λ ∈F2. A straightforward calculation using polar coordinates shows that, if |λ| < 2|a|,theneg/λ ∈F2, and hence D(0, 2|a|) is contained in the spectrum of Tg. Let us now prove that if |λ| > 2|a|, the formal resolvent Rλ,g,givenby(10),is a bounded operator. If |λ| > 2|a|, then clearly eg/λ ∈F2. We first apply Lemma 1, and subsequently Proposition 1, to deduce z g(z)/λ g(z)/λ −g(ζ)/λ Rλ,gf 2 ≤|f(0)| e 2 + e e f (ζ)dζ 2 0 | |2 1/2 f (z) −|z|2 f 2 + 2 e dA(z) f 2, C (1 + |z|) which shows that Rλ,g is bounded. Thus ρ(Tg)={λ ∈ C : |λ| > 2|a|}. 5. Invariant subspaces of the Volterra operator In the particular case g(z)=z, the operator Tg becomes the Volterra operator z Vf(z)= f(ζ) dζ, z ∈ C,f∈Fp,p>0. 0 By Theorem 1, the operator V : F p →Fp is compact. The aim of this section is to characterize the invariant subspaces of V . A closed subspace M in F p is called invariant for V if V M⊂M. We start with the following simple observation. Remark 1. The spectrum of the Volterra operator is the set {0}. Proof. Notice that V has no eigenvalues, since for any f ∈Fp \{0} the function Vf has a zero at z = 0 of one order greater than the corresponding order for f. Since V is compact and has no eigenvalues, its spectrum must be {0}. License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A VOLTERRA-TYPE INTEGRATION OPERATOR ON FOCK SPACES 4255 Remark 2. For x0 = p>0 we consider the sequence defined inductively by 2xn xn+1 = ,n≥ 0. xn +2 Note that (xn) is decreasing and limn→∞ xn = 0. Then, for any nonnegative integer M, it follows by Theorem 1 (ii) that the operator V M is bounded from F p to F q if q>xM . Theorem 4. Assume p>0.ThenM⊂Fp is a proper invariant subspace of V : F p →Fp if and only if there exists a nonnegative integer N such that F p M = {f ∈Fp : f (k)(0) = 0 for 0 ≤ k ≤ N} = Span{zk : k ≥ N +1} . Proof. Clearly, the sets p { ∈Fp (k) ≤ ≤ } AN := f : f (0) = 0 for 0 k N are invariant subspaces for V . Let us now prove that these are all the invariant subspaces of V .Forp =2this follows directly from a result of Nikol ski˘ı (see [15, 16]) about invariant subspaces of weighted shifts. Notice that, with respect to the standard orthonormal basis −1/2 n 2 en(z)=(n!) z , n ≥ 0, of F , the operator V is a weighted forward shift. More precisely, 1 Ve = e ,n≥ 0. n (n +1)1/2 n+1 If the weight sequence of a weighted forward shift belongs to some ls with s<∞, then Nikol ski˘ı’s theorem implies that its only invariant subspaces are the trivial ones, i.e. Nk = Span{en : n ≥ k},k≥ 0. This is certainly the case for our operator; therefore the assertion follows. We are now going to show that the result for p = 2 can be deduced via the case p = 2. Assume first that 0 License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use 4256 OLIVIA CONSTANTIN p ⊆M M p obtain AN+k , and then the choice of k forces = AN+k, so that the proof is done for 0 [1] A. Aleman, A class of integral operators on spaces of analytic functions, Topics in complex analysis and operator theory, 3–30, Univ. Malaga, Malaga, 2007. MR2394654 (2009m:47081) [2] A. Aleman and J. A. Cima, An integral operator on Hp and Hardy’s inequality,J.Anal. Math. 85 (2001), 157–176. MR1869606 (2002k:30068) [3] A. Aleman and O. Constantin, Spectra of integration operators on weighted Bergman spaces, J. Anal. Math. 109 (2009), 199–231. MR2585394 (2011f:47054) [4] A. Aleman and B. Korenblum, Volterra invariant subspaces of Hp, Bull. Sci. Math. 132 (2008), 510–528. MR2445578 (2009h:47014) [5] A. Aleman and A. G. Siskakis, Integration operators on Bergman spaces, Indiana Univ. Math. J. 46 (1997), 337–356. MR1481594 (99b:47039) [6] A. Aleman and A. G. Siskakis, An integral operator on Hp, Complex Variables Theory Appl. 28 (1995), 149–158. MR1700079 (2000d:47050) [7] H.-R. Cho and K. Zhu, Fock-Sobolev spaces and their Carleson measures, preprint available at http://www.albany.edu/∼kzhu/cho-zhu2.pdf. [8] O. Constantin, Carleson embeddings and some classes of operators on weighted Bergman spaces, J. Math. Anal. Appl. 365 (2010), 668–682. MR2587070 (2011c:46051) [9] P. Galanopoulos, D. Girela and J. A. Pelaez, Multipliers and integration operators on Dirich- let spaces, Trans. Amer. Math. Soc. 363 (2011), 1855–1886. MR2746668 (2011m:31013) [10] D. Girela and J.A. Pelaez, Carleson measures, multipliers and integration operators for spaces of Dirichlet type, J. Funct. Anal. 241 (2006), 334–358. MR2264253 (2007m:46034) [11] I. C. Gohberg and M. G. Krein, Introduction to the theory of linear nonselfadjoint opera- tors. Translations of Mathematical Monographs, Amer. Math. Soc., Providence, R.I., 1969. MR0246142 (39:7447) [12] Z. J. Hu, Equivalent norms on Fock spaces with some application to extended Cesaro opera- tors, preprint (May 2011). To appear in Proc. Amer. Math. Soc. [13] S. Janson, J. Peetre and R. Rochberg, Hankel forms and the Fock space, Rev. Mat. Iberoamer- icana 3 (1987), 61–138. MR1008445 (91a:47029) [14] D. Luecking, Embedding theorems for spaces of analytic functions via Khinchine’s inequality, Michigan Math. J. 40 (1993), 333–358. MR1226835 (94e:46046) [15] N. K. Nikolski˘ı, Invariant subspaces of weighted shift operators, Mat. Sb. 74 (1967), 172–190. MR0229081 (37:4659) License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use A VOLTERRA-TYPE INTEGRATION OPERATOR ON FOCK SPACES 4257 [16] N. K. Nikolski˘ı, Unicellularity and non-unicellularity of weighted shift operators, Dokl. Akad. Nauk SSSR 172 (1967), 287–290. MR0217639 (36:728) [17] J. Pau and J. A. Pelaez, Embedding theorems and integration operators on Bergman spaces with rapidly decreasing weights,J.Funct.Anal.259 (2010), 2727–2756 MR2679024 (2011j:46039) [18] Ch. Pommerenke, Schlichte Funktionen und analytische Funktionen von beschr¨ankter mittlerer Oszillation, Comment. Math. Helv. 52 (1977), 591–602. MR0454017 (56:12268) [19] J. R¨atty¨a, Integration operator acting on Hardy and weighted Bergman spaces, Bull. Austral. Math. Soc. 75 (2007), 431–446. MR2331020 (2008f:47056) [20] A. G. Siskakis, Volterra operators on spaces of analytic functions — a survey, Proceedings of the First Advanced Course in Operator Theory and Complex Analysis, 51–68, Univ. Sevilla Secr. Publ., Seville, 2006. MR2290748 (2007k:47052) [21] R. Wallst´en, The Sp-criterion for Hankel forms on the Fock space,0 Faculty of Mathematics, University of Vienna, Norbergstr. 15, 1090 Vienna, Austria E-mail address: [email protected] License or copyright restrictions may apply to redistribution; see https://www.ams.org/journal-terms-of-use