Spectra of integration operators on Bergman and Hardy spaces. Alexandru Aleman

Transcription

1 Spectra of integration operators on Bergman and Hardy spaces Alexandru Aleman Let g be a fixed analytic function in the unit disc and consider the integral operator T g by T g f(z) = where f is analytic in. z f(t)g (t)dt, 1

2 1 Spaces of functions All spaces considered here are on the unit disc = {z C : z < 1}, and T = {z C : z = 1}. Hardy spaces. For < p < H p consists of those analytic functions f with the property that f p p = lim r 1 2π f(re it ) p dt 2π = H = bounded functions in. 2π f(e it ) p dt 2π. Weighted Bergman spaces If A denotes the area measure on and w is a nonnegative integrable function on, for < p < L p a(w) = L p (wda) Hol(), provided that the right hand side is closed in L p (wda). For example, this happens when w is continuous and positive. The weights w(z) = (1 z 2 ) α, α > 1 are called standard weights and the corresponding Bergman spaces are denoted by L p,α a. Mean oscillation BMOA = {Cauchy transforms of bounded functions on T}, V MOA = {Cauchy transforms of continuous functions on T}. 2

3 The Bloch space. This is denoted by B and consists of analytic functions f with f (z) = O((1 z ) 1 ), when, z 1. The little Bloch space B is defined by the fact that f (z) = o((1 z ) 1 ), when, z 1. Analytic Lipschitz classes. These are denoted by Λ α, < α 1 and consist of those analytic functions f with f(z) f(w) C z w α. Equivalently, f (z) = O((1 z ) α 1 ), when, z 1. The little oh version of Λ α is denoted λ α and can be defined by the condition f (z) = o((1 z ) α 1 ), when, z 1. 3

4 4) A theorem of Hardy and Littlewood. If < p < 1 f H p = f H p 1 p. In other words for < p < 1 T z is a bounded operator from H p into H p 1 p. Bergman space analogue If < p < 2 f L p a = f L 2p 2 p a. 4

5 The John-Nirenberg inequality for analytic functions There exist A, B > such that for all g BMOA with g 1 we have exp(a(g g())) H 2 B. A proof by Pommerenke 77 Step 1 If g BMOA then T g is a bounded operator from H 2 into itself. Indeed, for f H 2 (T g f) = fg and T g f 2 2 = 2 π (Littlewood-Paley formula) f 2 g 2 log 1 z da C g 2 f 2 2 Step 2 If g() = we have λt g e λg = e λg 1. If λ C g < 1 then I λt g is invertible with e λg = (I λt g ) 1 1 H 2 and e λg (1 λ C g ) 1. 5

6 2 Boundedness, Compactness, etc. Get rid of the integral! If the norm on a space Y can be expressed in terms of the derivative, f Y E(f ), f Y, f() = then T g maps X into Y if and only if E(fg ) C f X for all f X. This is essentially an embedding problem. Example In L p,α a we have f p p,α f p (1 z ) α+p da, f() = so that T g maps L p,α a into itself whenever fg p (1 z ) α+p da f p (1 z ) α da 6

7 Theorem 2.1. Let p, q >. Then (i) For p > q T g maps L p,α a g L s,α a, where 1 s = 1 q 1 p, into L q,α a if and only if (ii) T g maps L p,α a into itself if and only if g B. (iii) For p < q and 1 p 1 q 1/(α + 2), T g maps L p,α a if and only if L q,α a g (z) = O ( (1 z ) (α+2)( 1 p 1 q ) 1), into when z 1. Corollary 2.1. Let p, q >. Then: (i) For p > q and g H s, 1 s = 1 q 1 p, T g : L p,α a compact. L q,α a is (ii) T g : L p,α a L p,α a is compact if and only if g B, (iii) For p < q and 1 p 1 q 1/(α + 2), T g : L p,α a compact if and only if ( g (z) = o (1 z ) (α+2)( 1 p 1) 1) q, when z 1. L q,α a is 7

8 Part (ii) of Theorem 2.1 and Corollary 2.1 have been proved by A. and Siskakis 96 for more general radially weighted Bergman spaces. Hu 23 has extended the part (ii) of Theorem 2.1 to several variables. Theorem 2.1 and Corollary 2.1 actually hold for all radial weights w which are normal (Zinner, Thesis, Univ. of Hagen, Germany). There is an important class of of non-radial weights where an analogous result holds (A.- O. Constantin, 9), namely the so-called Bekollé-Bonami weights. 8

9 On the Hardy spaces H p, p 2 the situation is a little more subtle. (ideas from Aleksandrov-Peller 96) For an analytic function f in let ( 1 G(f)(e it ) = f (re it ) 2 (1 r)dr ) 1/2 Fefferman and Stein 72 proved that for < p < f p p 2π G p (f)(e it )dt, f() =. Write to deduce that ( 1 G(T g f) = ( 1 = fg = (fg) gf ) 1/2 fg (re it ) 2 (1 r)dr ) 1/2 [(fg) gf ](re it ) 2 (1 r)dr G(gf) + g G(f), g = radial max-function of g. This shows that If g H s then T g maps H p into H q where 1 s = 1 q 1 p, or T gh p H p when g H. What about the case when g BMOA? If Reg is bounded then e g is bounded and T g f(z) = z e g(t) f(t)e g(t) g (t)dt = = T e ge g f(z) z e g(t) f(t)(e g ) (t)dt 9

10 The following results were proved by A.-Siskakis, 95, Aleksandrov- Peller, 96 and A.-Cima, 21. Theorem 2.2. Let p, q >. Then (i) For p > q T g maps H p into H q if and only if g H s, where 1 s = 1 q 1 p, (ii) T g maps H p into itself if and only if g BMOA. (iii) For p < q and 1 p 1 q 1, T g maps H p into H q if and only if g Λ 1 p 1 q, (iv) If 1 p 1 q constant. > 1, and T g maps H p into H q then g is 1

11 Corollary 2.2. Let p, q >. Then: (i) For p > q and g H s, 1 s = 1 q 1 p, T g : H p H q is compact. (ii) T g : H p H p is compact if and only if g V MOA, (iii) For p < q and 1 p 1 q < 1, T g : H p H q is compact if and only if g λ 1 p 1 q. Theorem 2.3. For 1 < p < the operator T g H 2 or T g L 2,α a belongs to the Schatten-class S p if and only if g belongs to the Besov space B p, i.e. g (z) p (1 z 2 ) p 2 da(z) <. If T g H 2 belongs to the trace class S 1 then g is constant. 11

12 There are cases when the boundedness of T g is mysterious... 1) The weighted irichlet spaces p,β = {f : f L p,β a }, f p p,β = f() p + For a fixed analytic function g and f p,β T g f p p,β = f p g p (1 z ) β da, so that T g is bounded on p,β if and only if g p (1 z ) β da is a Carleson measure for p,β. f p (1 z ) β da. The problem is that these measures are not well understood. Stegenga 78, Arcozzi-Rochberg-Sawyer beginning of 2. 12

13 2) The vector-valued H 2 -space, i.e. the space of H-valued power series (H is a fixed separable Hilbert space) f(z) = n o a n z n, with f 2 2 = n a 2 <. Here T g can be defined for an analytic operator-valued function g. The Littlewood-Paley formula works in this context as well so that T g f 2 2 = 2 g f 2 log 1 π z da, and again T g is bounded if and only if the positive operatorvalued measure g g log 1 z da satisfies the Carleson embedding estimate. When H has infinite dimension: a) We do not have a characterization of these measures Nazarov-Treil-Volberg 97. b) We do not even know whether g f 2 log 1 z da f 2 2 for all f H 2 (H). g f 2 log 1 z da f 2 2 c) We do not have an intrinsic characterization for Cauchy integrals of bounded operator-valued functions = operatorvalued BM OA, Nazarov-Pisier-Treil-Volberg 2. 13

14 Relation to Hankel forms There is a partial estimate that has attracted attention. On H 2, L 2,α a T g is bounded if and only if the (little) Hankel operator with symbol g is bounded. In other words for α 1 sup f, h 1 g, fh 2 sup f 1 fg 2 (1 z 2 ) α+2 da, or sup f, h 1 g (fh) (1 z 2 ) α+2 da 2 sup fg 2 (1 z 2 ) α+2 da. f 1 What about the case α < 1? Arcozzi-Rochberg-Sawyer- Wick showed the estimate holds when α = 2 What about the vector-valued version ( Say Hilbert-Schmidt valued functions) : sup f, h 1 2 tr(g (fh) )(1 z 2 ) α+2 da fg 2 S 2 (1 z 2 ) α+2 da? sup f 1 14

15 A more tractable form (f, g) g fh (1 z 2 ) α+2 da Such forms appeared in the work of Peller Rochberg-Wu proved in the 9 s that its boundedness is equivalent with the boundedness of T g and Treil-Volberg proved a Nehari theorem for this form. In general, it is not known whether the boundedness of the 2 Hankel forms presented here is equivalent or not. For Hilbert-Schmidt valued functions (f, g) tr(g fh )(1 z 2 ) α+2 da For α = 1, the boundedness of this form is not equivalent to the boundedness of the previous one (Pisier). The boundedness problem makes sense when p 2 as well. Let p,β (S q ), 1 < p <, be the space of S q -valued holomorphic functions f in with ( 1/p f p,β (S q ) = f() S q + f (z) p S q (1 z 2 ) da(z)) β <. 15

16 Theorem 2.4. (A.- K.-M. Perfekt) Let α, β, γ, p, q with α > 1, p, q > 1, 1 p + 1 p = 1, 1 q + 1 q = 1and β p + γ p = α. Then p sup tr(g fh )(1 z 2 ) α da f p,β (S q), h p,γ (Sq ) 1 sup f p,β (S q) 1 g f p S q (1 z 2 ) β da, 16

17 The spectrum 1) The Cesáro operator is usually defined on sequence spaces by (Cx) n = 1 n x k n + 1 k= If we identify sequences with power series we can easily verify that Cf(z) = 1 z f(t) dt z 1 t = 1 z T gf(z), with g(z) = log(1 z). The Cesáro operator is bounded on the Hardy and weighted Bergman spaces because g(z) = log(1 z) belongs to BMOA Boundedness on H 2 has been noted by Hardy, Polya-Szegö, and on H p, < p < by Siskakis 87, 9 for p 1, Miao 92 for p < 1. On the standard weighted Bergman spaces L p,α a, < p <, α > 1 this was observed by Andersen 96. The spectrum of the Cesáro operator on H p, 1 < p <, (Siskakis 87, Miller, Miller and Smith 98) is the closed disc { z p 2 p }. 2 If p,α = { z } p 2(2 + α) p 2(2 + α) 17

18 then the spectrum of the Cesáro operator on L p,α a, p > 1, α > 1 is (ahlner 23) p,α { 1 n, n N, 1 n }. The point spectrum is the finite set outside p,α. Remark 2.1. Let w(z) = exp 1 1 z, z. On L 2 a(w) the Cesáro operator is compact. Its spectrum consists of the eigenvalues 1 n, n N. The Cesáro operator on the Hilbert space H 2 is subnormal (Kriete and Trutt 71, hyponormality was proved by Brown and Halmos 65), i.e. there is a normal extension of C to a larger Hilbert space. The Cesáro operator on H p, 1 < p < and L p,α a, 1 < p < is subdecomposable (Miller, Miller and Smith 98 resp. ahlner 23), i.e. there is a larger Banach space such that C can be extended to a decomposable operator. This is an operator T : X X with the property that for every open cover {U, V } of C there exist closed invariant subspaces Y, Z with X = Y + Z such that σ(t Y ) U and σ(t Z) V. 18

19 The results on the spectrum of the Cesàro operator generalize to linear combinations of rotations of the Cesàro operator. The following theorem is due to Young 4, Albrecht-Miller-Neumann 5, A.-Persson 1. It refers to the normalized version of T g C g f(z) = 1 z z f(t)g (t)dt. in the case when g is essentially a rational function. Theorem 2.5. Let g H() with g() = such that g has the form n (2.1) g a k (z) = 1 b k z + h(z), k=1 where h H, a 1,..., a n and b 1,..., b n are distinct. Let X be one of the spaces H p, L p,α a, p 1 or even a growth class A β. Then there exists a constant γ = γ(x) > such that: (i) The point spectrum σ p (C g ) is void if g () = and if g () then σ p (C g ) = { g () : k Z +, Re ka j k g () } ka j < γ, (1 ξ) g () X, 1 j n. (ii) σ(c g ) = σ p (C g ) ( n j=1 {λ C : λ a j 2γ a j 2γ } ), (iii) σ e (C g ) = {λ C : λ a j 2γ = a j 2γ } and for λ 19

20 C \ σ e (C g ), the Fredholm index of λi C g is given by n ind(λi C g ) = χ j (λ), j=1 where j = {ζ C : ζ a j 2γ < a j 2γ }, and χ j its characteristic function. denotes (iv) If T (λ) : (λi C g )X X denotes the left inverse of λi C g, λ C \ (σ e (C g ) σ p (C g )) then λ T (λ) is locally integrable on C \ {}. Corollary 2.3. All these operators have decomposable extensions. 2

21 Recall Pommerenke s result: The general case e g λ H 2 (orh p, L p,α a ), p 1 whenever λ is in the resolvent set of T g. The operators in the previous theorem satisfy the converse as well, the resolvent set is characterized by this property. Is this true in general? i.e. e g λ H 2 (orh p, L p,α a ), p 1 if and only if λ is in the resolvent set of T g? This would be a nice sharp version of the John-Nirenberg inequality but it is unfortunately not true. 21

22 There exist functions g B such that e g λ H p, p, λ but σ(t g L 2 a) has nonvoid interior (A.-O. Constantin 9). On the other hand, if g is the Cauchy transform of a finite measure: g 1 (z) = ζ z dµ(ζ), { ζ =1} then λ is in the resolvent set of T g L p,α a if and only if e g λ L p,α a. 22

23 Spectra and weighted norms Formal inverse: If g() = and λ the equation has the solution λf T g f = h (λi T g ) 1 h(z) = 1 λ h()eg(z)/λ + 1 z λ eg(z)/λ e g(t)/λ h (t)dt. This operator will be bounded if and only if e g λ the space and the second term z Rh(z) = e g(z)/λ e g(t)/λ h (t)dt is bounded. In fact we should have Rh h on the set of h with h() =. belongs to If f = e g λh and f() =, we can write Rh = e g λ f. Assume again that we are working on a space X whose norm can be expressed in terms of the derivative, f X E(f ), f X, f() =. Then with the above notation the estimates for the operator R are equivalent to e g λ f X E(e g λ f ) f X, f() =. 23

24 This can be seen as a question about weighted spaces. If we denote exp(pre g(z) λ )(1 z )α = w(z) (weight) then on L p,α a the condition reads: f p wda f p w(1 z ) p, f() = On H p we usually deal with weights w defined on the boundary and in our specific case we set ( w = exp pre g ). λ Since g BMOA, we see that exp(pre g(z) λ ), z are the values of the outer function whose modulus equals w a.e. on the boundary. Thus for a general (log-integrable) weight w the condition reads: 2π 2π ( 1 f(e it ) p w(e it ) ) p/2 f (re it ) 2 W (re it ) 2/p (1 r)dr for functions that vanish at the origin, where W is the outer function whose modulus equals w a.e. on the boundary. In both cases (Bergman and Hardy) the estimate is crucial! When working with harmonic functions the opposite estimate is the most important (Nazarov-Treil 5). 24

25 Bergman spaces Here we need the Bekollé-Bonami condition mentioned before. We say that a weight w on satisfies the condition B p (α), p > 1, α > 1, if w B p (α) ( ) ( sup h p(α+2) wda h,θ S h (θ S h (θ ) p/p w p /p (1 z ) αp da where 1 p + 1 p = 1 and S h (θ) =Carleson box. Bekollé s theorem asserts that the standard weighted projection α + 1 P α f(z) = f(ζ) (1 ζz) α+2(1 ζ 2 ) α da(ζ) is a bounded operator from L p (, wda) into L p a(w) if and only if w satisfies B p (α). <, Theorem 2.6. (A.-O. Constantin) The estimate f p wda f p w(1 z ) p, holds for all f L p a(w) with f() =, if and only if w satisfies the condition B q (β) for some q > 1, β > 1. The complete characterization of the spectrum is as follows. Corollary 2.4. (A.-O. Constantin 9) Let g B. For λ C \ {} the operator λi T g is invertible on L p,α a if and only if the weight function w(z) = (1 z ) α exp(re g(z) λ ) satisfies B q (β) for some q > 1, β > 1. 25

26 Hardy spaces The estimate is related to the Muckenhoupt conditions. These characterize the weights w on the unit circle for which the Riesz projection P + f(z) = 2π is bounded on L p (w), p > 1. f(e it ) 1 e it z dt The famous Hunt-Muckenhoupt-Wheeden theorem states that these weights are characterized by the A p -condition ( ) ( ) p /p sup I p w dz w p/p dz <. I I I Here I denotes an arc on the unit circle and 1 p + 1 p = 1. The limit case when p is of interest here. By Jensen s inequality, A p implies 1 w dz Ce 1 I log w dz I I I for all arcs I. This is called the A -condition. 26

27 Theorem 2.7. (A.- Peláez, preprint) For a weight w with w, log w L 1 (T) and p 1 the following are equivalent: (i) w satisfies A, (ii) For analytic functions f in with f() = 2π 2π ( ) p/2 f(e it ) p w(e it ) f 2 W 2/p da Γ(t) (iii) For analytic functions f in with f() = 2π f(e it ) p w(e it ) W f p log 1 z da Corollary 2.5. Let g BMOA and λ C \ {}. Then λi T g is invertible on H p, p 1 if and only if the weight w = e pre g λ satisfies the A -condition. 27

28 3 Invariant subspaces of Volterra operators In real variables the Volterra operator V f(x) = x f(t)dt, f L 2 ([, 1], and its invariant subspaces have a long and honored history. Gelfand raised the problem of describing the invariant subspaces of V in Agmon essentially solved the problem in showed that these are He M t = {f L 2 ([, 1]) : f = a.e. on [, t]}. Kalisch (1957) extended the result for a large class of integral operators. Sarason (1965) observed that V is unitarily equivalent to the restriction of the backward shift to a very simple invariant subspace and gave a beautiful new proof of the original theorem. 28

29 Volterra operators on the disc For a fixed let V a f(z) = z a f(t)dt, f H 2, or some other space of analytic functions. We are interested in the invariant subspaces of these operators. Note that V = T g with g(z) = z. Moreover, if a on all spaces considered here the invariant subspaces of V a are in one-to-one correspondence with those of T ϕa, where ϕ a (z) = z+a 1+az (the correspondence is implemented by composition with ϕ a. Examples: 1) (z a) N H 2 if a < 1. 2) A theorem of onoghue (1957) about invariant subspaces of weighted shifts implies that all invariant subspaces of V are of the form z N H 2. 3) If a = 1 let S a (z) = exp z+a z a function). Then (atomic singular inner V a S a H p S a H p. 29

30 Shift invariance Let H be a Hilbert space of functions in a set Ω C such that 1. Polynomials are dense in H 2. The shift Sf(z) = zf(z) is bounded on H 3. For some a Ω the operator V a f(z) = z a f(t)dt is bounded with σ(v a ) = {} We want to show that V a -invariance implies S-invariance The Borel transform H of H is defined as the space of functions of the form h(λ) = e λ, h, h H, e λ (z) = e λz, λ C, z Ω. The norm is transported from H, i.e. we set h = h. Then H becomes a Hilbert space of entire functions of exponential type, since by 3) e λ e λ S h(λ) e λ S h. 3

31 Let M H be closed with V a M M and set N = { h : h M }. Proposition 3.1. If f N vanishes at λ C then z f(z) z λ belongs to N. Proof. By 4) and a direct computation we have hence (I λv a ) 1 f(z) = e λ(z a) f(a) + z a e λ(z t) f (t)dt (I λv a ) 1 e ζ = ζ ζ λ e ζ + e ζ(a) λ e λ (a) ζ λ e λ. Now let f N, that is f(ζ) = e ζ, g for some g M, and assume that f(λ) =. Since M is invariant for [(I λv a ) 1 ] the function given by e ζ, [(I λv a ) 1 ] g = (I λv a ) 1 e ζ, g belongs to N. = ζ f(ζ) f(ζ) = f(ζ) + ζ λ ζ λ 31

32 The second step is the use of Hadamard s formula for functions of exponential type. Every f N \ {} can be written as f(z) = z m e az+b f(λ)= λ ( 1 z λ where a, b C, m N {}. Moreover, λ 1 ε < for all ε >. f(λ)= λ ) e z λ, 32

33 Equivalently, f (z) = f(z) a + m z + f(λ)= λ z. λ(z λ) Each term of the series on the right belongs to N by the previous proposition. If this series converges in H then f belongs to N and since f (λ) = Se λ, g = e λ, S g we conclude that if N is differentiation invariant, then M is S -invariant, hence M is S-invariant. 33

34 So what needs to be done is to show the convergence of that series. For example, it suffices to show that for some ε > zf (z λ) = O( λ ε ), when λ and f(λ) =, since we know that λ 1 ε < for all ε >. tricky. f(λ)= λ In specific cases these estimates can be Theorem 3.1. (A. - Korenblum, 7) (i) A proper subspace M of H p, L p,α a, p 1 is V a -invariant, where a < 1 if and only if there exists a positive integer N such that M = (z a) N H 2 (ii) A proper subspace M of H p, L p,α a, p 1 is V a -invariant, where a = 1 if and only if there exists a t > such that M is the closure of polynomial multiples of Sa(z) t = exp(t z+a z a ). 34