Complex Anal. Oper. Theory DOI 10.1007/s11785-015-0481-8

Complex Analysis and Operator Theory

Cesàro-Like Operators on the Hardy and Bergman Spaces of the Half Plane S. Ballamoole1 · J. O. Bonyo1 · T. L. Miller1 · V. G. Miller1

Received: 3 February 2015 / Accepted: 15 June 2015 © Springer Basel 2015

Abstract We construct integral operators associated with strongly continuous groups of invertible isometries on the Hardy spaces and the weighted Bergman spaces of the upper half plane. Specifically, we obtain the spectrum and point spectrum of the generator and represent resolvents as integral operators related to the Cesàro operator investigated by Arvanitidis and Siskakis 1 C1 f (z) := z



z

f (ζ ) dζ

0

on the Hardy Spaces H p (U), p > 1, Arvanitidis and Siskakis (Can Math Bull 153:1–12, 2011). Keywords One-parameter operator group · Spectrum · Resolvent · Compact operator

Communicated by Simeon Reich.

B

T. L. Miller [email protected] S. Ballamoole [email protected] J. O. Bonyo [email protected] V. G. Miller [email protected]

1

Department of Mathematics and Statistics, Mississippi State University, Drawer MA, Mississippi State, MS 39762, USA

S. Ballamoole et al.

Primary 47B38 · 47D03 · 47A10

Mathematics Subject Classification

1 Introduction Let U denote the upper half plane and D the unit disc of the complex plane C. For an open subset  of C, let H() denote the Fréchet space of analytic functions f :  → C endowed with the topology of uniform convergence on compact subsets of . Let Aut() ⊂ H() denote the group of biholomorphic maps f :  → . Let d A denote area measure and for every α > −1 define measures dμα (ω) = (ω)α d A(ω), ω ∈ U, and dm α (z) = (1 − |z|2 )α d A(z) for z ∈ D. For 1 ≤ p < ∞, the corresponding weighted Bergman spaces are defined by p p L a (U, μα ) = L p (μα ) ∩ H(U) and L a (D, m α ) = L p (m α ) ∩ H(D). Also consider the Hardy spaces H p (U) and H p (D), 1 ≤ p < ∞, consisting of analytic functions f ∈ H(U), resp. H(D), such that  f H p (U) := sup

1/ p

∞ −∞

y>0



f H p (D) := sup

0≤r <1

| f (x + i y)| p d x

2π

< ∞, resp.

1/ p | f (r e )| dθ iθ

< ∞.

p

0

Here and throughout, we identify functions in H p with their boundary values; in particular, p f H p (U )

 =

| f (x)| d x and p

R

p f H p (D )

 =

∂D

| f (z)| p dm(z),

where m denotes Lebesgue measure on the unit circle ∂D. As noted in [2] in the case of the disc, the Hardy space H p (U) behaves in many p ways as the limiting case of L a (U, μα ) as α → −1+ . Therefore, we let X denote either p p the Hardy space H (U) or the weighted Bergman space L a (U, μα ), and we associate α+2 with each X , a parameter γ = p , where α = −1 in case that X = H p (U). Let X (D) denote the corresponding space of analytic functions on D. Thus we may formulate the well known growth conditions for Hardy and Bergman spaces simultaneously: for each X there exists a constant K such that, for every f ∈ X and ω ∈ U, | f (ω)| ≤

K f ; (ω)γ

(1)

see, for example, [3, Equation 1.1] and [12, Theorem 4.14]. Similarly, for each X (D) there is a constant K such that |g(z)| ≤ for every g ∈ X (D) and z ∈ D.

K g (1 − |z|2 )γ

Cesàro-Like Operators on the Hardy and Bergman…

x Motivated by work in [6] concerning the Cesàro operator C f (x) := x1 0 f (y) dy for f ∈ L 2 (0, ∞), Arvanitidis and Siskakis [3], introduced a strongly continuous group of weighted composition operators on H p (U) for 1 < p < ∞, and determined the spectral properties of its generator, . They then realized the operator 1 C1 f (z) := z



z

f (ζ ) dζ

0

as a resolvent, C1 = R(1 − 1/ p, ), and applied the theory of strongly continuous semigroups to obtain the norm, spectrum and adjoint of C1 . The first goal of this paper is to obtain similar representations for other resolvents of on H p (U) for p = 1 as well as p > 1 and to extend these results to the setting of the weighted Bergman spaces. These are given in terms of Cessàro-like operators defined formally on H(U) by Cν f (z) :=

1 zν

Cν f (z) := −



1 zν

z

f (ω)ων−1 dω, if (ν) > γ , and

0



∞

f (ω)ων−1 dω, if (ν) < γ .

(2) (3)

z

These operators are analogous to those considered in [4] in the setting of the unit disc. We employ semigroup methods similar to those used by Arvanitidis and Siskakis p to obtain norms and spectral properties for Cν on the spaces L a (U, μα ) and H p (U), 1 ≤ p < ∞, −1 < α; see Theorem 3.4 below. The group of isometries considered by Arvanitidis and Siskakis corresponds to a certain group of automorphisms of U. But, by Theorem 2.2 below, there are in fact only three distinct continuous one-parameter groups (ϕt )t∈R ⊂ Aut(U). Thus it is natural to extend our analysis to the other two cases, completing the program. If X is an arbitrary Banach space, and if T is a linear operator with domain D(T ) ⊂ X , denote the spectrum and point spectrum of T by σ (T, X ) and σ p (T, X ), respectively. The resolvent set of T is ρ(T, X ) = C\σ (T, X ). Let L(X ) denote the algebra of bounded operators on X , and if T ∈ L(X ), denote the spectral radius of T by r (T ).

2 Continuous One-Parameter Groups in Aut(U) and Associated Isometries The function ψ(z) :=

i(1 + z) 1−z

maps the unit disc D conformally onto U with inverse ψ −1 (ω) =

ω−i . ω+i

S. Ballamoole et al.

It is well-known that f ∈ H p (U), 0 < p < ∞, if and only if Sψ f := (ψ  )γ f ◦ ψ ∈ H p (D), [3, page 4]; in fact, for 1 ≤ p < ∞, Sψ is an isometry from H p (U) onto H p (D) with inverse Sψ −1 g = ((ψ −1 ) )γ g ◦ ψ −1 . Moreover, a calculation shows that for all z ∈ D, |ψ  |(1 − |z|2 ) = 2(ψ(z)), p

and therefore, for every f ∈ L a (U, μα ), change of variables yields 2−α/ p Sψ f L p (m α ) = f L p (μα ) . Similarly, for every g ∈ L a (D, m α ), we obtain that 2α/ p Sψ −1 g L p (μα ) = g L p (m α ) . In particular, Sψ−1 = Sψ −1 in the setting of Bergman spaces as well as Hardy spaces. More generally, let {V1 , V2 } = {D, U}, and let let L F(Vi , V j ) denote the collection of conformal mappings from Vi onto V j . Then L F(Vi , Vi ) = Aut(Vi ), and if h ∈ L F(Vi , V j ), then g ∈ Aut(V j ) → h −1 ◦ g ◦ h ∈ Aut(Vi ) is an isomorphism from Aut(Vi ) onto Aut(V j ). For each h ∈ L F(Vi , V j ), define a weighted composition operator Sh : H(V j ) → H(Vi ), by (4) Sh f (z) = (h  (z))γ f (h(z)) (z ∈ Vi ). p

If g ∈ L F(Vi , V j ) and h ∈ L F(V j , Vi ), then Sh Sg = Sgh and Sh−1 = Sh −1 by the chain rule. Proposition 2.1 1. If g ∈ Aut(U), then for every p, 1 ≤ p < ∞ and α > −1, the p operator Sg is a surjective isometry on L a (U, μα ). 2. If g ∈ Aut(U), then for every p, 1 ≤ p < ∞, the operator Sg is a surjective isometry on H p (U). 3. If g ∈ L F(U, D), then Sg is a surjective isometry from H p (D) onto H p (U). p 4. If g ∈ L F(U, D), then 2α/ p Sg is a surjective isometry from L a (D, m α ) onto p L a (U, mu α ) with inverse 2−α/ p Sg−1 . Proof For (1), recall that every automorphism g of U is a linear map of the   a fractional b with det(A) = 1. form g(z) = az+b corresponding to a real 2 × 2 matrix A = cd cz+d In fact, this representation is unique up to multiples of ±1. If g and A are as above, then a computation yields that |g  (z)|(z) = (g(z)), and therefore, for every f ∈ p L a (U, μα ),  Sg f p = | f (g(z))| p (|g  (z)|(z))α |g  (z)|2 d A(z) U = | f (ω)| p ((ω))α d A(ω) = f p . U p

Thus Sg is an isometry on L a (U, μα ), and Sg Sg−1 = I by the remarks preceding the proposition.

Cesàro-Like Operators on the Hardy and Bergman…

g ∈ Aut(U) if and only if h = ψ −1 ◦ g ◦ ψ ∈ Aut(D), and in this case, Sh is a surjective isometry on H p (D). Since Sψ is an isometry from H p (U) onto H p (D), (2) follows. If g ∈ L F(U, D), then h = ψ ◦ g ∈ Aut(U) and so Sh = Sg ◦ Sψ is a surjective isometry on H p (U). Thus Sg = Sh Sψ −1 is a surjective isometry from H p (D) onto H p (U). Finally, if g is a conformal map from U onto D, then h = ψ ◦ g ∈ Aut(U) and Sg = Sh Sψ −1 . Sh is a surjective isometry by (1), and from the remarks preceding p p the proposition, 2−α/ p Sψ −1 is an isometry from L a (D, m α ) onto L a (U, μα ). Thus 2−α/ p Sg is also a surjective isometry.   The following is an immediate consequence of [1, Theorems 1.4.22 and 1.4.23]. Theorem 2.2 Let ϕ : R → Aut(U) be a nontrivial continuous group homomorphism. Then exactly one of the following cases holds: 1. There exists k > 0, k = 1, and g ∈ Aut(U) so that ϕt (z) = g −1 (k t g(z)) for all z ∈ U and t ∈ R. 2. There exists k ∈ R, k = 0, and g ∈ Aut(U) so that ϕt (z) = g −1 (g(z) + kt) for all z ∈ U and t ∈ R. 3. There exists k ∈ R, k = 0, and a conformal mapping g of U onto D such that ϕt (z) = g −1 (eikt g(z)) for all z ∈ U and t ∈ R. Equivalently, there exist θ ∈ R\{0} and h ∈ Aut(U) so that   h(z) cos(θ t) − sin(θ t) ϕt (z) = h −1 . h(z) sin(θ t) + cos(θ t) Proof Let ϕ : R → Aut(U) be a nontrivial continuous group homomorphism. Then, by Theorems 1.4.22 and 1.4.23 of [1], exactly one of the following cases holds: (a) there is a conformal map h of U onto a horizontal strip V = {z : c < (z) < d} such that ϕt (z) = h −1 (h(z) + t) for all z ∈ U, t ∈ R; moreover, h is unique up to an additive constant. (b) There is a conformal map h of U onto a horizontal half plane W = {z : c < (z)} or W = {z : (z) < c} such that ϕt (z) = h −1 (h(z) + t)for all z ∈ U, t ∈ R; moreover, h is unique up to an additive constant. (c) There is a unique real number k = 0 and a conformal map h of U onto a disc D = {z : |z| < R} such that ϕt (z) = h −1 (eikt h(z)) for all z ∈ U, t ∈ R. The mapping h is unique up to an multiplicative constant. If k > 0, k = 1, then the automorphism group ϕt (z) = k t z has the form (a) corresponding to h(z) = log(z)/ log(k). Conversely, for ϕ and h as in (a), we may, by adding a constant to the conformal mapping h, assume that V = {z : 0 < (z) < c} for some c > 0. Let u(z) = πc log(z). Then g = u −1 ◦ h ∈ Aut(U), and g ◦ ϕt ◦ g −1 (z) = eπ/c t z. Thus ϕt (z) = g −1 (k t g(z)) for k = eπ/c . For ϕ and h as in (b), we may again add a constant to h to obtain ran(h) = U, in which case we are done, or ran(h) = −U. In the latter case, taking g = −h yields g ◦ ϕt ◦ g −1 (z) = z − t, or ϕt (z) = g −1 (g(z) + bt) with b = −1. The first statement in (3) follows from case (c) above by re-scaling. For the second part of (3),

S. Ballamoole et al.

suppose that ϕt (z) = g −1 (eikt g(z)) for some k ∈ R\{0} and g ∈ L F(U, D). Then h := ψ ◦ g ∈ Aut(U) and a calculation shows that ψ(e

ikt

ψ

−1

    ω cos − k2 t − sin − k2 t (ω)) =    . ω sin − k2 t + cos − k2 t  

The second statement of (3) follows.

Since every continuous one-parameter semigroup ϕ : R+ → Aut(U) extends uniquely to a continuous one-parameter group via ϕ(t) = ϕ −1 (−t) for t < 0, continuous one-parameter semigroups of automorphisms are also of three types. Combining Theorem 2.2 and the Proposition 2.1, we obtain three similarity classes of isometries associated with continuous one-parameter groups of automorphisms. Theorem 2.3 Let ϕ : R → Aut(U) be a nontrivial continuous group homomorphism. and let Tt = Sϕt be the associated isometry. Then (Tt )t∈R is strongly continuous in L(X ). p

Proof First consider the case that X = L a (U, μα ), 1 ≤ p < ∞ and α > −1. Let p f ∈ L a (U, μα ) and suppose that tn → 0 in R. Let f n = Ttn f . Then f n (z) → f (z) for each z ∈ U, and f n p = f p for each n. If gn := 2 p−1 (| f | p + | f n | p ) − | f − f n | p , then gn ≥ 0 and gn (z) → 2 p | f (z)| p on U. It now follows from Fatou’s lemma that  lim sup n→∞

U

| f − f n | p dμα = 0,

and therefore Ttn f − f L ap (μα ) → 0 as n → ∞. For the Hardy space case, since each Tt is an isometry, it suffices to show that Tt f − f H p (U) → 0 as t → 0 on the dense subset { f ∈ H p : f is continuous on U}. In each case of Theorem 2.2, ϕt (x) → 1 and ϕt (x) → x for all x ∈ R\{g −1 (∞), g(∞)}, and therefore, for f ∈ H p such that f is continuous on U, Tt f (x) → x for almost all x ∈ R as t → 0. We now apply Fatou’s lemma as above to conclude that 

∞

−∞

| f (x) − Tt f (x)| p d x → 0 as t → 0.  

We may now apply the theory of strongly continuous semigroups of Banach space operators. Specifically, the positive semigroup (Sϕt )t≥0 associated with a continuous homomorphism ϕ : R → Aut(U) has as an infinitesimal generator a closed, densely S f−f defined linear operator . The domain D( ) consists of all f ∈ X for which ϕt t + is norm convergent as t → 0 , and

f := lim

t→0+

Sϕt f − f t

Cesàro-Like Operators on the Hardy and Bergman…

for each f ∈ D( ). The negative semigroup (Sϕ−t )t≥0 is also strongly continuous with generator − . We refer to as the generator of the group (Sϕt )t∈R . See, for example, [7, Chapter VIII], [9] or [11] for the basic theory of one-parameter semigroups of Banach space operators. If X and Y are arbitrary Banach spaces and U ∈ L(X, Y ) is an invertible operator, then clearly (At )t∈R ⊂ L(X ) is a strongly continuous group if and only if Bt := U At U −1 , t ∈ R, is a strongly continuous group in L(Y ). In this case, if (At )t∈R has generator , then (Bt )t∈R has generator  = U U −1 with domain D() = U D( ). Moreover, σ p (, Y ) = σ p ( , X ), and σ (, Y ) = σ ( , X ) since for λ in the resolvent set ρ( , X ) = C\σ ( , X ), we have that R(λ, ) = U R(λ, )U −1 ; see, for example [9, Chapter II].

3 Cesàro-Like Operators The spectral theory of isometries associated with the automorphisms ϕt (z) = k t z differ essentially only according to whether 0 < k < 1 or k > 1. Therefore, we consider only ϕt (z) = e−t z, and let Tt = Sϕt , t ∈ R. This is the group considered by Arvanitidis and Siskakis on H p (U), 1 ≤ p < ∞. Proposition 3.1 The infinitesimal generator of (Tt )t∈R ⊂ L(X ) is given by  

( f )(z) = −z f (z) − γ f (z) with domain D( ) = { f ∈ X : z f (z) ∈ X }. Proof The Hardy space case is [3, Proposition 2.1], and the Bergman space case p follows similarly. Indeed, if f ∈ D( ) in L a (U, μα ),then the growth condition Eq. (1) implies that for all z ∈ U,   ∂  −γ t e−γ t f (e−t z) − f (z) −t = e f (e z) 

f (z) = lim t ∂t t→0+ t=0 

= −z f (z) − γ f (z). Therefore, D( ) ⊂ { f ∈ X : z f  (z) ∈ X }. Conversely let f ∈ L a (U, μα ) be such that z f  (z) ∈ X . Then for z ∈ U we have p



 ∂  −γ s e f (ϕs (z)) ds 0 ∂s  t  −γ s  = −e ϕs (z) f  (ϕs (z)) − γ e−γ s f (ϕs (z)) ds 0  t = Ts F(z)ds, where F(z) = −z f  (z) − γ f (z). t

Tt ( f )(z) − f (z) =

0

Thus, Tt ( f ) − f 1 = lim lim t t→0+ t→0+ t



t

Ts F ds, 0

S. Ballamoole et al.

and strong continuity of (Ts )s≥0 implies that 1t D( ) ⊇ { f ∈ X : z f  (z) ∈ X }.

t 0

Ts F − F ds → 0 as t → 0+ .Thus   p

Lemma 3.2 Let X denote one of the spaces H p (U) or L a (U, μα ), 1 ≤ p < ∞ and α > −1 (α = −1 if X = H p (U)), and let γ = (α + 2)/ p. If c ∈ R and λ, ν ∈ C, then 1. f (ω) = (ω−c)λ (ω+i)ν ∈ X if and only if (λ+ν) < −γ < (λ). In particular, (ω − c)λ ∈ X for any λ ∈ C, and (ω + i)ν ∈ X if and only if ν < −γ . 2. f (ω) = eiω /ωc ∈ X if and only if 1/ p < c < γ . In particular, eiω /ωc ∈ H p (U) for any c ∈ R. Proof Let f (ω) = (ω − c)λ (ω + i)ν . Then f ∈ X if and only if Sψ f ∈ X (D), the corresponding space of analytic functions on D. There is a constant K = 0 so that g(z) = K

c−i λ c+i ) . 2γ z) +λ+ν

(z − (1 −

It is well known that for every real θ , (z − eiθ )η ∈ L p (∂D) if and only if η > −1/ p, and it follows from [12, Lemma 3.10] that (z − eiθ )η ∈ L p (D, m α ) if and only if

η > −γ . The first statement now follows and the other two assertions in (1) are special cases of the first. (2) follows from (1) in the case that X = H p (U) by identifying f and its boundary values on R. The function f (ω) = eiω /ωc is analytic on U for all c ∈ R, and letting ω = x + i y, 

 U

| f (ω)| p dμα (ω) =

0

∞

e−yp y α−cp+1 dy



∞

−∞

(u 2

1 du. + 1)cp/2

The first integral converges if and only if c < γ , and the second converges if and only if cp > 1.   p

Proposition 3.3 Let X denote one of the spaces H p (U) or L a (U, μα ), 1 ≤ p < ∞. Then σ p ( , X ) = ∅ and σ ( , X ) = iR. In particular is an unbounded operator. Proof Once again, the case X = H p (U) is Proposition 2.4 of [3]. Suppose that p f ∈ L a (U, μα ) is in the ker( − λ) for some λ ∈ C. Then the equation (λ − ) f = 0 is equivalent to the differential equation z f  (z) + (λ + γ ) f (z) = 0, whose general solution is f (z) = cz −(λ+γ ) . It follows from Lemma 3.2, that f = 0 p and therefore σ p ( , L a (U, μα )) = ∅. Since each Tt is an invertible isometry, its spectrum satisfies σ (Tt ) ⊆ ∂D, and the spectral mapping theorem for strongly continuous groups [11, Theorem 2.3] implies p that etσ ( ) ⊆ σ (Tt ). It follows that σ ( , L a (U, μα )) ⊆ iR.

Cesàro-Like Operators on the Hardy and Bergman… p

To show that iR ⊆ σ ( , L a (U, μα )), we argue as in [3, Proposition 2.4]. Fix p λ ∈ iR and consider the function h(ω) = (ω + i)−(λ+γ +1) . Then h ∈ L a (U, μα ) by Lemma 3.2, and the general solution of the equation (λ − ) f = h has form f (ω) =

1 (cω−(λ+γ ) + (i + ω)−(λ+γ ) ) i(λ + γ )

for some constant c. It follows that Sψ f has form Sψ f (z) = (c1 + c2 c(1 + z)−λ )(1 − z)λ−γ , p

where c1 , c2 = 0. Since (λ − γ ) = −γ , Sψ f ∈ L a (D, m α ) for any value of c, and p p therefore f ∈ L a (U, μα ) for any c. Thus h ∈ ran(λ− ), and σ ( , L a (U, μα )) = iR.   If (ν) > γ , then ν − γ ∈ ρ( , X ), and the resolvent operator is given by the Laplace transform, [7, VIII.1.12]: for every h ∈ X ,  ∞ e−(ν−γ )t Tt h dt, R(ν − γ , )h = 0

with convergence in norm. Alternatively, if f = R(ν − γ , )h, then f solves the differential equation ν f (z) + z f  (z) = ((ν − γ ) − ) f (z) = h(z) (z ∈ U), which implies that (z ν f ) (z) = z ν−1 h(z). Integrating along the path ω = e−t z, 0 ≤ t < ∞, we obtain the concrete representation as a Cesàro-like operator:  z  ∞ e−(ν−γ )t Tt h(z) dt = z1ν ων−1 h(ω) dω = Cν h(z), (5) R(ν − γ , )h(z) = 0

0

for all h ∈ X and z ∈ U. Similarly, in the case that (ν) < γ , then −(ν − γ ) ∈ ρ(− , X ), where −

generates the strongly continuous semigroup T−t h(z) = eγ t h(et z), t ≥ 0. Again, the resolvent operator is given by the Laplace transform,  ∞ e(ν−γ )t T−t h dt, R(ν − γ , )h = −R(−(ν − γ ), − )h = − 0

with convergence in norm. For each z ∈ U, we integrate from z to ∞ along the path ω = et z, 0 ≤ t < ∞, to obtain  ∞  ∞ e(ν−γ )t (T−t h)(z) dt = − z1ν ων−1 h(ω) dw R(ν − γ , )h(z) = − 0

= Cν h(z),

z

(6)

for all h ∈ X and z ∈ U. The following theorem is now an immediate consequence of the spectral mapping theorem and the Hille–Yosida Theorem, [7, VII.9.2 and VIII.1.13].

S. Ballamoole et al. p

Theorem 3.4 Let X denote one of the spaces H p (U) or L a (U, μα ), with α > −1 and 1 ≤ p < ∞, (α = −1 in the Hardy space case), and let γ = (α + 2)/ p. Then  z ων−1 f (ω) dω if (ν) > γ , and Cν f (z) := z1ν 0  ∞ 1 ων−1 f (ω) dω if (ν) < γ . Cν f (z) := − z ν z

are bounded operators on X. Moreover, 1. σ (Cν , X ) = {w ∈ C : |w − 2. Cν =

1 | (ν−γ )| .

1 2 (ν−γ ) |

=

1 2| (ν−γ )| },

and

Proof (1) follows from the spectral mapping theorem and Proposition 3.3. Since Cν 1 1 has spectral radius r (Cν ) = | (ν−γ )| , it follows from Hille–Yosida that | (ν−γ )| ≤ Cν ≤

1 | (ν−γ )| ,

 

and so (2) holds as well.

We end this section with a representation of the adjoints of Cesàro-like operators Cν on the weighted Bergman and Hardy spaces. Let 1 < p < ∞ and let q be conjugate to p, 1p + q1 = 1. We will write γ p = (α + 2)/ p, α ≥ −1 to make the dependence p on p apparent. If X = L a (U), then the boundedness of the Bergman kernel [5, q Theorem 1.34], implies that X ∗ ≈ L a (U, μα ), under the sesquilinear pairing  p q f (ω)g(ω) dμα ( f ∈ L a (μα ), g ∈ L a (μα )).  f, g = U

It is also well known that (H p (U))∗ ≈ H q (U) via   f, g = f (x)g(x) d x ( f ∈ H p (U), g ∈ H q (U); R

see for example [10, Theorem VI.4.2]. Notice that under these pairings, the adjoint operator from L(X ) to L(X ∗ ) is conjugate linear. If Tt f (z) = e−γ p t f (e−t z) for f ∈ X , then a computation yields Tt∗ g(z) = T−t g(z) = eγq t g(et z) for g ∈ X ∗ . Since X is reflexive, it follows from [11, Corollaries 10.2, 10.6] that ∗p = − q and that R(λ, p )∗ = R(λ¯ , − q ). p

Theorem 3.5 Let X be one of the spaces H p (U) or L a (U, μα ), 1 < p < ∞, α > −1, and let q be conjugate to p. If (ν) = γ p , then Cν∗ = −Cα+2−¯ν . Proof By definition, if (ν) = γ p , then Cν∗ = R(ν − γ p , p )∗ = R(γq − (α + 2 − ν¯ ), − q ) = −R((α + 2 − ν¯ ) − γq , q ) = −Cα+2−¯ν .  

Cesàro-Like Operators on the Hardy and Bergman…

4 The Translation Group The spectral theory of isometries associated with the automorphisms ϕt (z) = z + kt differ significantly only according to the sign of k ∈ R\{0}. Therefore, we consider only ϕt (z) = z + t. Again let Tt = Sϕt for all t ∈ R, and denote the generator of (Tt )t∈R by . p

Lemma 4.1 For each of the spaces X = H p (U) or X = L a (U, μα ), there is a constant K so that for every f ∈ X , | f  (ω)| ≤

K f . (ω)γ +1

1+γ

Proof The generalized Bloch space B∞ (D) is the space of functions g ∈ H(D) for which the seminorm g B 1+γ (D),1 = supz∈D (1 − |z|2 )γ |g(z)| < ∞. An equiv∞ alent seminorm is g B 1+γ (D),2 = supz∈D (1 − |z|2 )γ +1 |g  (z)|, [13, Proposition 7]. ∞

1+γ

Moreover, the spaces X (D) are continuously embedded in B∞ (D). If f ∈ X , then Sψ f ∈ X (D) and (Sψ f ) (z) =

2γ (ψ  (z))γ f (ψ(z)) + (ψ  (z))γ +1 f  (ψ(z)). 1−z

Thus Sψ f B 1+γ (D),2 + 4γ Sψ f B 1+γ (D),1 ∞ ∞   1 − |z|2 2 γ ≥ Sψ f B 1+γ (D),2 + 2γ sup (1 − |z| ) |Sψ f (z)| ∞ |1 − z| z∈D ≥ sup(1 − |z|2 )γ +1 | f  (ψ(z))|. z∈D

Since |ψ  (ψ −1 (ω))|(1 − |ψ −1 (ω)|2 ) = 2(ω), it follows that for some K depending on X , K f ≥ sup (ω)γ +1 | f  (ω)|. ω∈U

  A similar bound holds for the corresponding spaces on the disc. Specifically, for each X (D) there is a constant K such that |g  (z)| ≤

K g (1 − |z|2 )γ +1

for every g ∈ X (D) and z ∈ D. Theorem 4.2 The generator of the group (Tt )t∈R is f (z) = f  (z) with domain D( ) = { f ∈ X : f  ∈ X }. Moreover, σ p ( , X ) = ∅ and σ ( , X ) = {is : s ≥ 0}.

S. Ballamoole et al.

Proof If f ∈ D( ), then, just as in the proof of Proposition 3.1,   ∂ f (z + t) = f  (z).

f (z) = ∂t t=0 Thus D( ) ⊆ { f ∈ X : f  ∈ X }. Conversely, if f ∈ X is such that f  ∈ X , then Tt f − f 1 = t t



t

∂s Ts f ds,

0

and, for each z ∈ U, ∂s Ts f (z) = f  (z + s) = Ts f  (z). Thus  Tt f − f 1 t  ≤ − f Ts f  − f  ds → 0 t t 0

as t → 0

by strong continuity. Thus { f ∈ X : f  ∈ X } ⊆ D( ). Let λ ∈ C and f, h ∈ H(U). Then (λ − ) f = h if and only if (e−λz f (z)) = −e−λz h(z) (z ∈ U).

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In particular, (λ − ) f = 0 if and only if f (z) = ceλz for some constant c. But ceλz ∈ X only if c = 0. Thus σ p ( , X ) = ∅. Since each Tt is an invertible isometry, σ ( , X ) ⊆ iR as in Proposition 3.3. Let λ = is for some s ≥ 0. We consider the Bergman and Hardy spaces separately. p Let c satisfy 1/ p < c < min{1 + 1/ p, γ }, c = 1, so that h(z) = eλz /z c ∈ L a (μα ) by Lemma 3.2. Then by Eq. (7), (λ − ) f = h implies that f (z) = K eλz − (1 − p c)−1 eλz /z c−1 for some constant K . But K eλz − (1 − c)−1 eλz /z c−1 ∈ L a (μα ) for any p K , and therefore h ∈ ran(λ − ). Thus {is : s ≥ 0} ⊆ σ ( , L a (μα )). In the Hardy space case, let h(z) = eλz (z + i)−(1+1/ p) . Then h ∈ H p (U) and λz every solution of Eq. (7) has form f (z) = K eλz − p ze1/ p for some constant K . λz

Again, K eλz − p ze1/ p ∈ H p (U) for any K , and therefore (λ − ) is not surjective; {is : s ≥ 0} ⊆ σ ( , H p (U)). Now fix λ with (λ) < 0 and let h ∈ X . Let f ∈ H(U) be given by f (z) = eλz



∞

e−λω h(ω) dω,

z

integrating along the path w = z + f (z) =

λ¯ |λ|



λ¯ |λ| t, ∞

0

0 ≤ t < ∞. Then

e−|λ|t h z +

λ¯ |λ| t

 ) dt,

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Cesàro-Like Operators on the Hardy and Bergman…

and the integral is absolutely convergent by Eq. (1). Fix ω ∈ U. Then for all z = ω with (z) ≥ 21 (ω), Lemma 4.1 implies that    γ +1  f (z) − f (ω)  2 ≤  K h ,   z−ω (ω) and so a standard dominated convergence argument implies that f ∈ H(U). The function f also satisfies satisfies Eq. (7). Indeed, let g(z) = eλz = eλz



i

z ∞



e−λω h(ω) dω, +



∞

e−λω h(ω) dω



i

e−λω(t) h(ω(t))ω (t)dt

0 ¯

λ where w(t) = (1 − t)z + it if 0 ≤ t ≤ 1and w(t) = i + |λ| (t − 1) for t ≥ 1. Then g satisfies Eq. (7), and Cauchy’s theorem gives that   u  ∞

   λz −λω(t)  −|λ|t λ¯ λ¯ e  e h(ω(t))ω (t) dt − e h z + t  |λ| |λ| ) dt  0 0      z+λ¯ /|λ|u  λz  −λξ = e e h(ξ ) dξ  → 0   ω(u)

as u → ∞ by Eq. (1). For every t > 0, the composition operator h(z) → hz + λ¯ /|λ|t is clearly a conp traction on X . In the case that X = L a (U, μα ), then, by the integral version of Minkowski’s inequality [10, page 14],  f L p (μα ) ≤ ≤

∞

0 ∞ 0

e−|λ|t h(z + λ¯ /|λ|t) L p (μα (z)) dt e−|λ|t h L p (μα ) dt =

1 h L p (μα ) . |λ|

A similar computation shows that if h ∈ H p (U), then f ∈ H p (U) with f H p (U ) ≤

1 h H p (U) . |λ|

1 Thus in either case R(λ, ) ∈ L(X ) whenever (λ) < 0, and R(λ, ) ≤ |λ| . In particular, {is : s ≥ 0} = σ ( , X ).      1  1 Proposition 4.3 If (λ) = 0, denote the circle z − 2 (λ) by Cλ , and if  = |2 (λ)| λ = ib for some b < 0, take Cλ to be the imaginary axis.

1. If (λ) = 0 and λ ≥ 0, then σ (R(λ, )) is the arc of the circle Cλ from that contains the upper half of Cλ .

1 λ

to 0

S. Ballamoole et al.

2. If (λ) < 0, then σ (R(λ, )) is the arc of the circle Cλ from 1/λ to 0 contained in the upper half of Cλ . 1 in either case. 3. R(λ, ) = r (R(λ, )) = | (λ)| Proof The spectral

theorem and Theorem 4.2 imply that for all λ ∈ ρ( ), 1 mapping : s ≥ 0 . Thus, if λ ∈ ρ( ) with (λ) ≥ 0, then σ (R(λ, )) is the σ (R(λ, )) = λ-is arc of the circle Cλ from λ1 to 0 that contains the upper half of Cλ , and r (R(λ, )) = 1 1 | (λ)| . The Hille–Yosida theorem yields r (R(λ, )) ≤ R(λ, ) ≤ | (λ)| . In case λ ∈ ρ( ) has (λ) < 0, then σ (R(λ, )) is the arc of the circle Cλ 1 from 1/λ to 0 contained in the upper half of Cλ , and r (R(λ, )) = |λ| . The norm estimate for R(λ, ) in the proof of Theorem 4.2 along with Hille–Yosida yields 1 in this case as well.   r (R(λ, )) ≤ R(λ, ) ≤ |Re(λ)|

5 The Rotation Group Finally, we consider isometries on the spaces X (D) of the form Tt f (z) = eict f (eikt z) with c, k ∈ R, k = 0. We restrict our attention to the group Tt f (z) = eict f (eit z), t ∈ R since other values of the constant k = 0 can be handled in the same way as this specific case. Arguing as in Theorem 2.3, we see that (Tt )t∈R is a strongly continuous group of isometries on X (D). We again denote its generator by . Recall that the operator Mz f (z) := z f (z) is bounded and bounded below on each p of the spaces X (D) = H p (D), 1 ≤ p < ∞ or X (D) = L a (D, m α ), 1 ≤ p < ∞ and α > −1, with ran(Mz ) = { f ∈ X (D) : f (0) = 0}. The left inverse of Mz is the operator Q f (z) := f (z)−z f (0) . For every m ∈ N, X (D) = ran(Mzm ) ⊕ span{z n : n ∈ Z+ , n < m}, and Pm = Mzm Q m is the projection of X (D) onto ran(Mzm ) with kernel span{z n : n ∈ Z+ , n < m}.   Theorem 5.1 1. f (z) = i c f (z) + z f  (z) with D( ) = { f ∈ X (D) : f  ∈ X (D)}. 2. σ ( , X ) = σ p ( , X ) = {i(n + c) : n ∈ Z+ }, and for each n ≥ 0, ker(i(n + c) −

) = span(z n ). 3. If λ ∈ ρ( ), then ran(Mzm ) is R(λ, )-invariant for every m ∈ Z+ such that m + c > (λ). In fact, if h ∈ ran(Mzm ), then R(λ, )h(z) = z −(c+λi)



z 0

 ωc+λi−1 h(ω) dω = z m

1

t m+c+λi−1 Q m h(t z) dt.

0

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Proof As in the previous cases, 

f (z) = ∂t (eict f (eit z))t=0 = i(c f (z) + z f  (z)) with domain D( ) ⊂ { f ∈ X (D) : z f  ∈ X (D)}. But z f  ∈ X (D) implies that z f  ∈ ran(Mz ) and therefore f  ∈ X (D). Thus { f ∈ X (D) : z f  ∈ X (D)} = { f ∈ X (D) : f  ∈ X (D)}.

Cesàro-Like Operators on the Hardy and Bergman…

Conversely if f ∈ X (D) is such that z f  ∈ X , then F(z) = i(c f (z) + z f  (z)) ∈ X (D), and for all t > 0  1 t Tt f (z) − f (z) = ∂s (Ts f (z)) ds t t 0  t  1 1 t ics is is  is = e (ic f (e z) + i(e z) f (e z)) ds = Ts F(z) ds. t 0 t 0  t Again, strong continuity of (Ts )s≥0 implies that 1t 0 Ts F ds − F → 0 as t → 0+ .

Thus, D( ) = { f ∈ X (D) : f  ∈ X (D)}. As before, that Tt is a surjective isometry implies that σ ( , X (D)) ⊆ iR. Let λ ∈ C and h ∈ H(D). Then the equation (λ − ) f (z) = h(z) is equivalent to f  (z) + or

i (c + iλ) f (z) = h(z), z z

(z c+iλ f (z)) = i z c+iλ−1 h(z).

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In particular, (λ − ) f = 0 implies that f (z) = K z −(c+iλ) for some constant K . Since z −(c+iλ) ∈ H(D) if and only if −(c + iλ) ∈ Z+ , it follows that σ p ( , X (D)) = σ p ( , H(D)) = {i(c + n) : n ∈ Z+ }, with ker(i(c + n) − ) = span(z n ). Notice that for λ ∈ C\σ p ( ) and f ∈ H(D), the functions f and (λ − ) f have the same order of zero at 0. Thus ran(Mzm , H(D)) is invariant under λ − . Fix λ ∈ C\σ p ( ) and let m ∈ Z+ be such that (λ) < c + m. Then, for h(z) = z m g(z) ∈ ran(Mzm , H(D)), the integral  i

z

 ωc+λi−1 h(ω) dω = i z m+c+λi

0

1

t m+c+λi−1 g(t z) dt

0

is absolutely convergent. Thus Eq. (10) has unique solution  f (z) = z

1

m

t m+c+λi−1 g(t z) dt.

0

For every g ∈ X (D) and t, 0 ≤ t < 1, we have g(t z) ≤ g since, for every  2π p 1 iθ p g ∈ H(D) and p > 0, the function M p (r, g) := 2π 0 |g(r e )| dθ is an increasing function of r , [8, Theorem 1.5]. Minkowski’s inequality again implies that for h ∈ ran(Mzm , X (D)),  f ≤ 0

1

t m+c−(λ)−1 Q m h dt ≤

Q m h , m + c − (λ)

S. Ballamoole et al.

and so (λ − )|ran(Mz ,X (D)) has bounded resolvent 

1

Rm (λ, )h(z) = z m

t m+c−(λ)−1 Q m h(t z) dt  z −(c+λi) =z ωc+λi−1 h(ω) dω. 0

0 n For every n ∈ Z+ , the equation (λ − ) f =  z has uniquen solution f (z) = i n n∈Z+ , n
R(λ, )h(z) =

 n∈Z+ , n
an

i zn + Rm (λ, )Pm h(z) n + c + λi

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is bounded on X (D). Thus σ ( ) = σ p ( ).

 

Theorem 5.2 For every λ ∈ ρ( , X (D)), the resolvent R(λ, ) is compact. Proof Fix λ ∈ ρ( ) and let m ∈ Z+ be such that (λ) < c + m. Since ran(Mzm ) has finite codimension in X (D), it suffices to show that Rm (λ, ) = R(λ, )|ran(Mzm ) is compact. If r > 0, let A(r D) be the disc algebra A(r D) = C(r D) ∩ H(r D), equipped with the supremum norm, and for each t, 0 ≤ t < 1, and f ∈ H(D), let Ht f (z) = f t (z) = f (t z). Then, as in the proof of Theorem 5.1, for every t, 0 ≤ t < 1, Ht is a contraction on X (D). 1 By Eq. (9), Rm (λ, ) = i Mzm 0 t c+m+λi−1 Ht Q m dt with convergence in norm. r For each r , 0 < r < 1, let Cr = i Mzm 0 t c+m+λi−1 Ht Q m dt on ran(Mz , X (D)). Then  Rm − Cr ≤

1

t c+m−(λ)−1 Q m dt =

r

Q m (1 − r c+m−(λ) ) → 0 c + m − (λ)

as r → 1− . Choose s so that 1 < s < r −1 . Then Cr : ran(Mz , X (D)) → ran(Mz , X (D)) factors through A(sD). If B denotes the closed unit ball of ran(Mz , X (D)), then the growth conditions Eq. (1) and Lemma 4.1, imply that for all f ∈ B and z ∈ sD, K r c+m−(λ) and (1 − r s)γ (c + m − (λ)) K r c+m−(λ) |(Cr f ) (z)| ≤ . (1 − r s)γ +1 (c + m − (λ)) |Cr f (z)| ≤

Thus Cr B is pre-compact in A(sD) by Arzela–Ascoli. Since A(sD) is continuously embedded in X (D), it follows that Cr B is pre-compact in X (D) as well. Thus each Cr is compact in L(ran(Mz , X (D))). It follows that Rm (λ, ) = (norm) limr →1− Tr is compact as well.  

Cesàro-Like Operators on the Hardy and Bergman…

References 1. Abate, M.: Iteration theory of holomorphic maps on taut manifolds. Mediterranean Press, Cosenza. http://www.dm.unipi.it/~abate/libri/libriric/libriric.html (1989). Accessed Oct 2014 2. Albrecht, E., Miller, T.L., Neumann, M.M.: Spectral properties of generalized Cesàro operators on Hardy and weighted Bergman spaces. Arch. Math. (Basel) 85(5), 446–459 (2005) 3. Arvanitidis, A.G., Siskakis, A.G.: Cesàro Operators on the Hardy spaces of the half-plane. Can. Math. Bull. 153, 1–12 (2011) 4. Ballamoole, S., Miller, T.L., Miller, V.G.: Spectral properties of Cesàro-like operators on weighted Bergman spaces. J. Math. Anal. Appl. 394, 656–669 (2012) 5. Békollé, D., Bonimi, A., Garrigós, G., Nana, C., Peloso, M., Ricci, F.: Lecture notes on Bergman projections in tube domais over cones: an analytic and geometric viewpoint. In: IMHOTEP J. Afr. Math. Pures Appl., vol. 5. http://webs.um.es/gustavo.garrigos/papers/workshop5 (2004). Accessed Oct 2014 6. Brown, A., Halmos, P., Shields, A.: Cesàro operators. Acta Sci. Math. (Szeged) 26, 125–137 (1965) 7. Dunford, N., Schwartz, J.T.: Linear Operators Part I. Interscience Publishers, New York (1958) 8. Duren, P.: Theory of H p Spaces. Academic Press, New York (1970) 9. Engel, K.-J., Nagel, R.: A short course on operator semigroups. In: Universitext. Springer, New York (2006) 10. Garnett, J.: Bounded analytic functions. In: Graduate Texts in Mathematics, Revised First Edition, Springer, Berlin (2010) 11. Pazy, A.: Semigroups of linear operators and applications to partial differential equations. In: Applied Mathematical Sciences, vol. 44. Springer, New York (1983) 12. Zhu, K.: Operator theory in function spaces. In: Mathematical Surveys and Monographs, vol. 138. Amer. Math. Soc, Providence (2007) 13. Zhu, K.: Bloch type spaces of analytic functions. Rocky Mt. J. Math 23, 1143–1177 (1993)