First Advanced Course in Operator Theory and Complex Analysis, University of Seville, June 2004

VOLTERRA OPERATORS ON SPACES OF ANALYTIC FUNCTIONS - A SURVEY

ARISTOMENIS G. SISKAKIS

Abstract. We give a short and selective account of results known about operators of the form

V_{g}(f)(z) = \frac{1}{z} ∫_{0}^{z} f(ζ)g′(ζ) dζ,

where g is analytic on the disc and the operator T_{g} = zV_{g} acts on spaces of analytic functions.

1. Introduction

Let 𝔻 denote the unit disc in the complex plane ℂ. For g analytic on the disc consider the linear transformation

(1)  V_{g}(f)(z) = \frac{1}{z} ∫_{0}^{z} f(ζ)g′(ζ) dζ,

acting on the space H(𝔻) of all analytic functions on 𝔻. We also consider the modified transformation

(2)  T_{g}(f)(z) = ∫_{0}^{z} f(ζ)g′(ζ) dζ

which maps every f ∈ H(𝔻) to a function vanishing at 0. In both cases g is the symbol of the transformation, and these transformations have appeared under various names such as Volterra operators, generalized Cesàro operators, Riemann-Stieltjes operators, and integration operators.

2000 Mathematics Subject Classification. Primary 47B38; Secondary 47B35, 30D55, 46E15.

This work was partially supported by Plan Nacional, Ref. BFM2003-00034, Ministerio de Ciencia y Tecnología of Spain.

This class of operators includes the integration operator,

f(z) → \frac{1}{z} ∫_{0}^{z} f(ζ) dζ

obtained with g(z) = z. It also includes the Cesàro operator

C(f)(z) = ∑_{n=0}^{∞} (\frac{1}{n + 1} ∑_{k=0}^{n} a_{k}) z^{n},

which is defined on functions f(z) = ∑_{n=0}^{∞} a_{n}z^{n} ∈ H(𝔻). Indeed a power series calculation shows

C(f)(z) = \frac{1}{z} ∫_{0}^{z} f(ζ)\frac{1}{1 − ζ} dζ
= \frac{1}{z} ∫_{0}^{z} f(ζ) (log (\frac{1}{1 − ζ} ))′ dζ
= \frac{1}{z} ∫_{0}^{z} f(ζ)g′(ζ) dζ,

where g(z) = log(1/(1 − z)). With this g we thus have C = V_{g}.

These operators are closely related to the operation of integration on simply connected domains. Consider a simply connected domain Ω ⊊ ℂ with 0 ∈ Ω. Denote by H(Ω) the space of analytic functions on Ω, and let h : 𝔻 → Ω be a Riemann map with h(0) = 0. Then the operator

C_{h} : H(Ω) → H(𝔻), C_{h}(f)(z) = f(h(z))

is a linear bijection between H(Ω) and H(𝔻). Let

I_{Ω}(f)(z) = ∫_{0}^{z} f(ζ) dζ,

the operator of integration acting on H(Ω). Writing Ĩ_{Ω} = C_{h} ∘ I_{Ω} ∘ C_{h}^{−1} we obtain an operator Ĩ_{Ω} acting on H(𝔻), and we have

Ĩ_{Ω}(f)(z) = C_{h} ∘ I_{Ω} ∘ C_{h}^{−1}(f)(z)
= ∫_{0}^{z} f(ζ)h′(ζ) dζ,

for each f ∈ H(𝔻). Thus Ĩ_{Ω} = T_{h} with h the above Riemann map. Less directly but more interestingly, the averaged integration operator

J_{Ω}(f)(z) = \frac{1}{z} ∫_{0}^{z} f(ζ) dζ

is also related to some V_{g}. Indeed the counterpart of J_{Ω} on H(𝔻) is

J̃_{Ω}(f)(z) = C_{h} ∘ J_{Ω} ∘ C_{h}^{−1}(f)(z)
= \frac{1}{h(z)} ∫_{0}^{z} f(ζ)h′(ζ) dζ.

Next, on H(𝔻) define the transformations

M_{z}(f)(z) = zf(z) (the usual shift),

B_{h}(f)(z) = \frac{1}{zh(z)} ∫_{0}^{z} f(ζ)ζh′(ζ) dζ,

and

R_{h}(f)(z) = \frac{1}{z} ∫_{0}^{z} f(ζ)\frac{ζh′(ζ)}{h(ζ)} dζ.

A calculation shows that the following hold

(3)  (i) M_{z} ∘ B_{h} = J̃_{Ω} ∘ M_{z}, (ii) R_{h} = B_{h} + R_{h} ∘ B_{h}.

From (i) we have J̃_{Ω} ∼ B_{h} and from (ii) that R_{h} ∼ B_{h}, where the symbol ∼ is used in a vague manner to mean that the operators have “similar” properties. This can be made precise when the restrictions of these operators are made to act on appropriate Banach spaces of analytic functions that are subspaces of H(𝔻). For example (ii) says that R_{h} and B_{h} belong to the same operator ideals.

Further since

\frac{zh′(z)}{h(z)} = 1 + z (log \frac{h(z)}{z})′ ,

we find

R_{h}(f)(z) = \frac{1}{z} ∫_{0}^{z} f(ζ) dζ + \frac{1}{z} ∫_{0}^{z} f(ζ)ζ (log \frac{h(ζ)}{ζ})′ dζ
= J_{𝔻}(f)(z) + (V_{g} ∘ M_{z})(f)(z),

where

g(z) = log \frac{h(z)}{z}.

Since the integration J_{𝔻} is a “small” operator, we conclude that with this g, in the vague sense mentioned above we have

J_{Ω} ∼ V_{g}.

In addition to the above connections to integration operators, the Volterra type operators also arise when studying semigroups of composition operators. Indeed they are closely related to the resolvent operators of those semigroups, see [Si3] for details.

2. Some general observations

Before we consider these Volterra type operators to act on specific spaces of analytic functions, let us take a more general point of view. Suppose X and Y are two Banach spaces consisting of analytic functions on 𝔻. We may ask: Question. For what symbols g is

T_{g} : X → Y

a bounded operator? A compact operator? If further X = Y how do the spectral properties of T_{g} : X → X depend on g?

Let us see how far we can go in this generality. Assume for simplicity X = Y and define

V = V_{X} := {g ∈ H(𝔻) : T_{g} : X → X is bounded }

and

V_{0} = V_{0,X} := {g ∈ H(𝔻) : T_{g} : X → X is compact }.

Because

T_{λg} = λT_{g}, T_{g+h} = T_{g} + T_{h},

both V and V_{0} are nontrivial vector spaces (both contain the constants) and V_{0} ⊆ V. We introduce a norm on V

∥g∥ = ∥g∥_{V} = |g(0)| + ∥T_{g}∥_{X→X},

which is also a norm on V_{0} ⊂ V. Then it is easy to show [SiZh] that if the convergence f_{n} → f in X implies uniform convergence on compact subsets of 𝔻 then V and V_{0} are complete under ∥ · ∥_{V} and are therefore Banach spaces. Further assume the multiplication operator M_{z}(f)(z) = zf(z) is bounded on X. Then from

T_{z^{n}}(f)(z) = n ∫_{0}^{z} f(ζ)ζ^{n−1} dζ = n ∫_{0}^{z} M_{z}^{n−1}(f)(ζ) dζ,

we have

T_{z^{n}} = nT_{z} ∘ M_{z}^{n−1},

and it follows that V contains all monomials z^{n} whenever it contains z. Thus if the integration operator T_{z} is bounded on X, then the linear space V contains all polynomials. In the same way it follows that if T_{z} is compact on X, then in fact the smaller space V_{0} contains all polynomials.

Now we consider the Möbius invariance of V and V_{0}. Let

ϕ_{a}(z) = \frac{a − z}{1 − āz} , a ∈ 𝔻.

be the Möbius maps which map the disc conformally onto itself and exchange a with 0. Let C_{a} be the composition operator

C_{a}(f) = f ∘ ϕ_{a},

induced by ϕ_{a} on H(𝔻). For f ∈ H(𝔻) write F (z) = T_{g}(f)(z), then

F′(z) = f(z)g′(z).

Composing with ϕ_{a}(z) and multiplying by ϕ′_{a}(z) we obtain

(F ∘ ϕ_{a})′(z) = (f ∘ ϕ_{a})(z)(g ∘ ϕ_{a})′(z),

thus

(F ∘ ϕ_{a})(z) − (F ∘ ϕ_{a})(0) = ∫_{0}^{z}(f ∘ ϕ_{a})(ζ)(g ∘ ϕ_{a})′(ζ) dζ.

We write this equation in terms of C_{a},

C_{a} ∘ T_{g}(f)(z) − C_{a} ∘ T_{g}(f)(0) = T_{g∘ϕ_{a}} ∘ C_{a}(f)(z),

equivalently,

(4)  I_{0} ∘ C_{a} ∘ T_{g} ∘ C_{a} = T_{g∘ϕ_{a}},

where I_{0} is the operator I_{0}(f)(z) = f(z) − f(0) acting on H(𝔻).

Now consider the restriction of the above to the Banach space X ⊂ H(𝔻). If X contains the constant functions, then I_{0}(f) ∈ X whenever f ∈ X. Using (4) we conclude that if X contains the constants and is preserved by C_{a} (that is f ∘ ϕ_{a} ∈ X for each f ∈ X) then V and V_{0} are also preserved by C_{a}.

Suppose again X contains the constants. Then for the function 1 ∈ X and each g ∈ V we obtain

T_{g}(1)(z) = ∫_{0}^{z} g′(ζ) dζ = g(z) − g(0) ∈ X

so g ∈ X, i.e. V ⊂ X. Further,

∥g∥_{X} = ∥g(0) + g − g(0)∥_{X}
≤ ∥g(0)∥_{X} + ∥g(z) − g(0)∥_{X}
= |g(0)|∥1∥_{X} + ∥T_{g}(1)∥_{X}
≤ |g(0)|∥1∥_{X} + ∥T_{g}∥_{X→X}∥1∥_{X}
= ∥1∥_{X}∥g∥_{V} ,

so ∥g∥_{X} ≤ C∥g∥_{V} with the constant C = ∥1∥_{X} independent of g ∈ V.

We can now iterate the above construction to obtain a sequence of spaces. We start with a Banach space X with the properties:

(i) X contains the constant functions. (ii) If f_{n} → f in X then f_{n} → f uniformly on compact subsets of 𝔻. (iii) If f ∈ X and a ∈ 𝔻 then f → f ∘ ϕ_{a} ∈ X.

Write X_{1} for the space V defined above. Thus X_{1} contains the constants, and is preserved by composition with ϕ_{a}. Further X_{1} ⊂ X and ∥f∥_{X} ≤ C∥f∥_{X_{1}} for each f ∈ X_{1}. It follows that convergence in X_{1} implies convergence in X and by (ii) this implies uniform convergence on compact subsets of 𝔻. We see therefore that X_{1} is a Banach space which also satisfies the properties (i), (ii) and (iii). Proceeding inductively, suppose X_{k−1} has been defined. Define X_{k} by

X_{k} =: {g ∈ H(𝔻) : T_{g} : X_{k−1} → X_{k−1} is bounded },

and equip it with the norm

∥g∥_{X_{k}} = |g(0)| + ∥T_{g}∥_{X_{k−1}→X_{k−1}}.

By induction, X_{k} is a Banach space of analytic functions which satisfies properties (i), (ii) and (iii). We have

X ⊇ X_{1} ⊇ X_{2} ⊇ ⋯ X_{k} ⋯,

and

∥g∥_{X_{k−1}} ≤ ∥g∥_{X_{k}}, k > 1, g ∈ X_{k}.

We may ask various questions about these spaces X_{k}. For example if we start with a specific space for X, what are the subsequent spaces X_{k}? Do they all contain non constant functions? Are they strictly smaller from one step to the next? Does their infinite intersection contain non constant functions?

3. Spaces of functions

Below we will survey some known results about Volterra type operators acting on some classical spaces of analytic functions. We give the definitions and basic properties of the spaces first.

Hardy spaces and BMOA. For 0 < p < ∞ the Hardy space H^{p} contains all analytic functions f : 𝔻 → ℂ for which

∥f∥_{H^{p}} = sup_{0<r<1} (∫_{0}^{2π} |f(re^{iθ})|^{p} \frac{dθ}{2π} )^{1/p} < ∞.

With this norm, H^{p} is a complete metric linear space and a Banach space for p ≥ 1. It is a Hilbert space for p = 2. For p = ∞ the Banach space H^{∞} consists of the bounded analytic functions with norm

∥f∥_{∞} = sup_{z∈𝔻} |f(z)|.

The space BMOA consists of all f ∈ H^{2} such that

∥f∥_{∗} = |f(0)| + sup_{a∈𝔻} ∥f ∘ ϕ_{a} − f(a)∥_{H^{2}} < ∞,

where ϕ_{a}(z) are the Möbius automorphisms of 𝔻. The space V MOA consists of the functions f ∈ BMOA such that

lim_{|a|→1} ∥f ∘ ϕ_{a} − f(a)∥_{H^{2}} = 0.

Equivalently, V MOA is the closure of polynomials in BMOA. Both BMOA and V MOA can be described in terms of Carleson measures.

Recall that a positive measure μ on the disc is a Carleson measure if

(5)  sup_{I⊂∂𝔻} \frac{μ(S(I))}{|I|} < ∞,

where S(I) = {z : 1 − |I| ≤ |z| < 1, z/|z| ∈ I} is the Carleson box based on the arc I ⊂ ∂𝔻 of length |I|. A vanishing Carleson measure μ is one for which μ(S(I)) = o(|I|) as |I| → 0. It is a basic theorem that condition (5) is equivalent to

(6)  ∥f∥_{L^{2}(𝔻,μ)} ≤ C∥f∥_{H^{2}}  f ∈ H^{2},

i.e. the inclusion operator i : H^{2} ↪ L^{2}(𝔻, μ) is bounded. Vanishing Carleson measures are those for which this inclusion is compact.

The Carleson measure characterization of BMOA is as follows. A function f ∈ H^{2} is in BMOA if and only if the measure

dμ(z) = |f′(z)|^{2} log(z^{−1})dm(z)

is a Carleson measure on 𝔻 (here dm(z) is the Lebesgue area measure). And a function f ∈ BMOA is in V MOA if and only if this measure is a vanishing Carleson measure.

Functions f ∈ H^{p} have radial limits almost everywhere on the boundary and a boundary function, denoted by f(e^{iθ}), is well defined. For p ≥ 1 the boundary function is in L^{p}(∂𝔻), and H^{p} embeds in L^{p}(∂𝔻) by this correspondence. H^{p} then can alternatively be described as the subspace of functions in L^{p}(∂𝔻) whose Fourier series have all coefficients of negative index equal to zero. For 1 < p < ∞ the pairing

⟨f, h⟩ = ∫_{0}^{2π} f(e^{iθ})\overline{h(e^{iθ})}\frac{dθ}{2π}

establishes a duality between H^{p} and H^{q} with \frac{1}{p} + \frac{1}{q} = 1. Interpreted in a wider sense, ⟨f, h⟩ = lim_{r→1}⟨f_{r}, h_{r}⟩, where f_{r}(z) = f(rz), this pairing gives also the dualities V MOA^{∗} = H^{1} and (H^{1})^{∗} = BMOA. This pairing can be written by the Littlewood-Paley formula as

⟨f, h⟩ = f(0)\overline{h(0)} + 2 ∫_{𝔻} f′(z)\overline{h′(z)} log (\frac{1}{z}) dm(z).

More information on Hardy spaces and BMOA can be found in [Du], [Bae] and in [Sar].

Bergman and Bloch spaces. Suppose w : [0, 1) → (0, ∞) is a weight function which is integrable on [0, 1). We extend w on 𝔻 by w(z) = w(|z|) and assume that w is normalized so that ∫_{𝔻} w(z)dm(z) = 1.

For 1 ≤ p < ∞ the weighted Bergman space A_{w}^{p} is the space of all analytic functions f : 𝔻 → ℂ such that

∥f∥_{p,w}^{p} = ∫_{𝔻} |f(z)|^{p}w(z)dm(z) < ∞.

These are Banach spaces. We write simply A^{p} when w ≡ 1.

The Bloch space B consists of all f analytic on 𝔻 such that the Bloch norm

∥f∥_{B} := |f(0)| + sup_{z∈𝔻}(1 − |z|^{2})|f′(z)|

is finite, while the little Bloch space B_{0} contains those f ∈ B for which lim_{|z|→1}(1 − |z|^{2})|f′(z)| = 0. Equivalently, B_{0} is the closure of polynomials in the Bloch norm. Both B and B_{0} are Banach spaces under the Bloch norm. Bloch spaces are the area measure versions of BMOA and V MOA. In this connection, a useful characterization is the following. Suppose 1 ≤ p < ∞ and let ϕ_{a} denote the usual Möbius automorphisms of 𝔻. Then we have, see [Axl],

(7)  g ∈ B ⇔ sup_{a∈𝔻} ∥g ∘ ϕ_{a} − g(a)∥_{A^{p}} < ∞,

and

g ∈ B_{0} ⇔ lim_{|a|→1} ∥g ∘ ϕ_{a} − g(a)∥_{A^{p}} = 0.

Besov spaces. For 1 < p < ∞ the Besov space B_{p} consists of all analytic functions f on 𝔻 such that

∥f∥_{B_{p}}^{p} = ∫_{𝔻} |f′(z)|^{p}(1 − |z|^{2})^{p−2} dm(z) < ∞.

The space B_{p} is a Banach space for p > 1. For p = 2, B_{2} is a Hilbert space, also known as Dirichlet space and denoted by D. More information about Bergman, Bloch and Besov spaces can be found in [Zhu].

4. Boundedness and compactness

In this section we will present the main theorems that characterize those g for which T_{g} is bounded or compact. The setting is on Hardy spaces, Bergman spaces and on BMOA.

We consider Hardy spaces first. It was proved by Pommerenke in [Pom, Lemma 1] that T_{g} is bounded on H^{2} if and only if g ∈ BMOA. To see this assume f is a polynomial and use the Littlewood-Paley formula to write

∥T_{g}(f)∥_{H^{2}}^{2} = ⟨T_{g}(f), T_{g}(f)⟩
= 2 ∫_{𝔻} |f(z)|^{2}|g′(z)|^{2} log(z^{−1}) dm(z)
= 2∥f∥_{L^{2}(𝔻, μ_{g}),}^{2}

where

(8)  μ_{g}(z) = |g′(z)|^{2} log(z^{−1})dm(z).

Since polynomials are dense in H^{2}, we see that T_{g} : H^{2} → H^{2} is bounded if and only if the inclusion operator

i : H^{2} ↪ L^{2}(𝔻, μ_{g})

is bounded, and this is equivalent to that μ_{g} is a Carleson measure. In other words, T_{g} is bounded on H^{2} if and only if g ∈ BMOA. The same argument shows that T_{g} is compact on H^{2} if and only if g ∈ V MOA.

The same characterization of boundedness and compactness of T_{g} on H^{p} is valid for any 0 < p < ∞, the proofs however are more difficult. The case p ≥ 1 was proved in [AleSi1] while the extension for 0 < p < 1 was obtained in [AleCi]. Thus we have

Theorem 4.1. Let 0 < p < ∞. Then T_{g} : H^{p} → H^{p} is bounded if and only if g ∈ BMOA.

In [AleCi] the authors in fact consider any pair of indices p, q ∈ (0, ∞) and characterize the symbols g for which T_{g} maps H^{p} into H^{q} (the closed graph theorem then implies that for those g, T_{g} maps H^{p} boundedly into H^{q}). We will not state the details of all cases but we remark that [AleCi, Theorem 1(iii)] is a substantial strengthening of a classical result of Hardy and Littlewood. Indeed they prove: If p < q and \frac{1}{p} − \frac{1}{q} ≤ 1, then T_{g} maps H^{p} into H^{q} if and only if g ∈ Λ_{\frac{1}{p} − \frac{1}{q}}. Here the Lipschitz class Λ_{α}, 0 < α ≤ 1, consists of all analytic functions g on 𝔻 such that

|g′(z)| = O((1 − |z|)^{α−1}), |z| → 1.

Now apply this with \frac{1}{p} − \frac{1}{q} = 1 and g(z) = z ∈ Λ_{1}, to obtain the following theorem of Hardy and Littlewood. If 0 < p < 1, f ∈ H^{p} and F′ = f then F ∈ H^{q} and q = p/(1 − p). Theorem 1(iii) of [AleCi] in its full generality as stated above is a nontrivial strengthened form of the Hardy-Littlewood theorem, see [AleCi] for details.

As a byproduct in the proof of the boundedness of T_{g} one obtains the following corollary for compactness which was proved for p ≥ 1 in [AleSi1] and for all p > 0 in [AleCi].

Corollary 4.2. Let 0 < p < ∞. Then T_{g} : H^{p} → H^{p} is compact if and only if g ∈ V MOA.

Analogous characterization for the symbols g hold for T_{g} to be bounded on Bergman spaces. In the unweighted case, for 1 ≤ p < ∞ and f ∈ A^{p} it is well known that one can write

∥f∥_{A^{p}}^{p} = ∫_{𝔻} |f(z)|^{p} dm(z)
≃ |f(0)|^{p} + ∫_{𝔻} |f′(z)|^{p}(1 − |z|^{2})^{p} dm(z),

where ≃ means the two sides are comparable. Thus we have

∥T_{g}(f)∥_{A^{p}}^{p} ≃ ∫_{𝔻} |f(z)|^{p}|g′(z)|^{p}(1 − |z|^{2})^{p} dm(z).

We see that if g is in the Bloch space B then |g′(z)|(1 − |z|^{2}) ≤ ∥g∥_{B} for all z ∈ 𝔻, and we have

∥T_{g}(f)∥_{A^{p}} ≤ C∥f∥_{A^{p}}∥g∥_{B},

so T_{g} : A^{p} → A^{p} is bounded.

This last inequality together with the fact that integration in A^{p} is a compact operator (and therefore T_{P} is compact on A^{p} for any polynomial P) and the fact that the little Bloch space B_{0} is the closure of polynomials in B, imply that if g ∈ B_{0} then T_{g} : A^{p} → A^{p} is compact.

The converse of this is also true. To prove it we will make use of the characterization of Bloch and little Bloch functions given in (7). Indeed assume 1 < p < ∞ and T_{g} : A^{p} → A^{p} is bounded. We will show that g ∈ B, by showing that sup_{a∈𝔻} ∥g ∘ ϕ_{a} − g(a)∥_{A^{1}} < ∞. We have,

∥g ∘ ϕ_{a} − g(a)∥_{A^{1}} ≃ ∫_{𝔻} |(g ∘ ϕ_{a})′(z)|(1 − |z|^{2}) dm(z)
= ∫_{𝔻} |(g′(ϕ_{a}(z)))||ϕ′_{a}(z)|(1 − |z|^{2}) dm(z)
= ∫_{𝔻} |(g′(ϕ_{a}(z)))|(1 − |ϕ_{a}(z)|^{2}) dm(z)
= ∫_{𝔻} |g′(z)|(1 − |z|^{2})|ϕ′_{a}(z)|^{2} dm(z)
= ∫_{𝔻} |g′(z)|(1 − |z|^{2})\frac{(1 − |a|^{2})^{2}}{|1 − āz|^{4}} dm(z)
= ∫_{𝔻} |g′(z)|(1 − |z|^{2})\frac{(1 − |a|^{2})^{2/p}}{|1 − āz|^{2}} \frac{(1 − |a|^{2})^{2/q}}{|1 − āz|^{2}} dm(z),

where \frac{1}{p} + \frac{1}{q} = 1. Write k_{a}(z) = \frac{(1−|a|^{2})^{2/p}}{(1−āz)^{2}} , h_{a}(z) = \frac{(1−|a|^{2})^{2/q}}{(1−āz)^{2}} and use Hölder’s inequality to obtain

∥g ∘ ϕ_{a} − g(a)∥_{A^{1}} ≤ ||k_{a}||_{L^{q}}∥|g′(z)|(1 − |z|^{2})|h_{a}(z)|∥_{L^{p}}
≃ ∥k_{a}∥_{L^{q}}∥T_{g}(h_{a})∥_{A^{p}}
≤ ∥k_{a}∥_{L^{q}}∥T_{g}∥_{A^{p}→A^{p}}∥h_{a}∥_{A^{p}}
≤ C∥T_{g}∥_{A^{p}→A^{p}}

because ∥k_{a}∥_{L^{q}} ≃ ∥h_{a}∥_{A^{p}} ≃ 1 for each a ∈ 𝔻. By assumption T_{g} is bounded on A^{p}. Thus

sup_{a∈𝔻} ∥g ∘ ϕ_{a} − g(a)∥_{A^{1}} < ∞

so g ∈ B. The case p = 1 can be handled similarly.

An analogous argument, based on weak convergence of test functions gives that if T_{g} : A^{p} → A^{p} is compact then g ∈ B_{0}. We summarize

Theorem 4.3. Let 1 ≤ p < ∞. Then

(i) T_{g} : A^{p} → A^{p} is bounded if and only if g ∈ B. (ii) T_{g} : A^{p} → A^{p} is compact if and only if g ∈ B_{0}.

This theorem can be extended to a class of weighted Bergman spaces for some general weights. The details can be found in [AleSi2]. We will only describe here the weights w for which the above theorem is valid on A_{w}^{p}.

Theorem 4.4. Suppose w : [0, 1) → (0, ∞) is a weight function which is integrable on [0, 1) and satisfies the conditions (P_{1}) There is a constant C such that

w(r) ≥ \frac{C}{1 − r} ∫_{r}^{1} w(u) du, 0 < r < 1.

(P_{2}) There is an s ∈ (0, 1) and a constant D such that

w(sr + 1 − s) ≥ Dw(r), 0 < r < 1.

Then:

(i) T_{g} is bounded on A_{w}^{p} if and only if g ∈ B. (ii) T_{g} is compact on A_{w}^{p} if and only if g ∈ B_{0}.

Condition (P_{1}) in the above theorem is used to prove an inequality of the form

∫_{𝔻} |f(z)|^{p}w(z) dm(z) ≤ C (|f(0)|^{p} + ∫_{𝔻} |f′(z)|^{p}(1 − |z|)^{p}w(z) dm(z)) ,

for f analytic on 𝔻, which clearly gives the sufficiency assertions for T_{g}. Condition (P_{2}) which is a generalized version of w(\frac{1+r}{2} ) ≥ Dw(r) (s = 1/2) is used to prove the necessity. All standard weights w(r) = (1 − r)^{α}, α > −1, satisfy conditions (P_{1}) and (P_{2}). Other weights that satisfy them are,

w(r) = (1 − r)^{α} (log \frac{e}{1 − r} )^{β} , α > −1, β ∈ ℝ,

w(r) = exp (−β (log \frac{e}{1 − r} )^{α}) , 0 < α ≤ 1, β > 0.

However the exponential weight

w(r) = exp (\frac{−β}{(1 − r)^{α}} ) , α, β > 0

does not satisfy condition (P_{2}), see [AleSi2] for details.

We now consider T_{g} acting on BMOA. If T_{g} is bounded on BMOA then T_{g}(1) = g(z) − g(0) ∈ BMOA so g ∈ BMOA. On the other hand, for g(z) = log(\frac{1}{1−z}) ∈ BMOA we have

T_{g}(g)(z) = ∫_{0}^{z} g(ζ)g′(ζ) dζ = \frac{1}{2} log^{2} (\frac{1}{1 − z} )

and this function is not in BMOA. Thus the space of g’s for which T_{g} is bounded is a proper subspace of BMOA. This space was determined in [SiZh] and is a space of functions that are of bounded logarithmically weighted mean oscillation as described in the following

Theorem 4.5. The operator T_{g} maps BMOA boundedly into itself if and only if

(9)  sup_{I⊂∂𝔻}\{\frac{(log \frac{2}{|I|})^{2}}{|I|} ∫_{S(I)}|g′(z)|^{2}(1 − |z|^{2}) dm(z)\} < ∞.

We now show that this condition also characterizes those g for which T_{g} is bounded on V MOA. First observe that under this condition T_{g}(VMOA) ⊂ VMOA. Indeed suppose g satisfies (9) then clearly g ∈ V MOA. Since T_{g}(1) = g − g(0), the constant functions are mapped into V MOA. Let n be a positive integer. An integration by parts gives

T_{g}(z^{n}) = z^{n}g(z) − n ∫_{0}^{z} ζ^{n−1}g(ζ) dζ.

Since multiplication by z and the integration operator are bounded on V MOA we see that T_{g}(z^{n}) ∈ V MOA, and the same is true for T_{g}(p) for any polynomial p. Next let f ∈ V MOA. There is a sequence (p_{n}) of polynomials such that ∥f − p_{n}∥_{∗} → 0 and we have

∥T_{g}(f) − T_{g}(p_{n})∥_{∗} = ∥T_{g}(f − p_{n})∥_{∗} ≤ ∥T_{g}∥∥f − p_{n}∥_{∗}.

This shows that T_{g}(f) can be approximated in the ∥ · ∥_{∗} norm by V MOA functions. Since V MOA is closed in this norm the assertion follows.

Now assume T_{g} : V MOA → V MOA is bounded and recall the dualities between V MOA, H^{1} and BMOA. Let

A_{g} = T_{g}^{∗} : H^{1} → H^{1}

be the adjoint of T_{g} acting on H^{1} = (V MOA)^{∗}, and let

A_{g}^{∗} : BMOA → BMOA

be the adjoint of A_{g}. Because V MOA is weak^{∗} dense in BMOA and since T_{g}(VMOA) ⊂ VMOA, we see that A_{g}^{∗} = T_{g}. It follows that T_{g} is bounded on V MOA if and only if it is bounded on BMOA.

The following theorem from [SiZh] asserts that the little oh condition corresponding to (9) characterizes those g for which T_{g} is a compact operator on BMOA.

Theorem 4.6. T_{g} is compact on BMOA if and only if

(10)  lim_{|I|→0} \{\frac{(log \frac{2}{|I|})^{2}}{|I|} ∫_{S(I)}|g′(z)|^{2}(1 − |z|^{2}) dm(z)\} = 0.

5. Schatten ideals

Let H be a Hilbert space and T : H → H a bounded operator. The singular numbers of T are defined by

λ_{n} = inf{∥T − F ∥ : F : H → H is an operator of rank ≤ n}.

The operator T is compact if and only if λ_{n} → 0, and of finite rank if and only if λ_{n} = 0 for all sufficiently large n. For 0 < p < ∞ the Schatten classes S^{p} contain those bounded linear operators on H for which (λ_{n}) ∈ l^{p}, the space of p-summable sequences. For p ≥ 1 the Schatten norm in S^{p} is defined by |T |_{S^{p}} = ∥(λ_{n})∥_{l^{p}}. Each S^{p} is a two sided self-adjoint ideal in the space of all bounded operators on H and is a Banach space under the Schatten norm. Further T ∈ S^{p} if and only if T^{∗}T ∈ S^{p/2}. The classes S^{2} and S^{1} are the Hilbert-Schmidt and trace class respectively. An operator T is Hilbert-Schmidt if and only if ∑_{n} ∥T (e_{n})∥^{2} < ∞ for some orthonormal basis {e_{n}} of H.

Before we consider the Schatten classes of T_{g} on specific Hilbert spaces let us consider a general case. Suppose the Hilbert space H consists of analytic functions on the disc 𝔻. For each p ≥ 1 define

W_{p} = W_{p,H} = {g ∈ H(𝔻) : T_{g} ∈ S^{p}}.

Since each S^{p} is a Banach space under the Schatten norm, each W_{p} is a vector space always containing the constants. We can give it a norm

∥g∥_{W_{p}} = |g(0)| + ∥T_{g}∥_{S^{p}},

under which W_{p} is a Banach space. This follows from the fact that a Cauchy sequence of operators in the Schatten norm is also Cauchy in the operator norm. For p = 2 the inner product

⟨g, h⟩_{W_{2}} = g(0)\overline{h(0)} + ⟨T_{g}, T_{h}⟩_{S^{2}}

makes W_{2} a Hilbert space.

Further if H contains the constants and the compositions C_{a}(f) = f ∘ ϕ_{a} with Möbius automorphisms of 𝔻 are bounded operators on H, using arguments similar to those in section 2 we find that W_{p} are preserved by composition with ϕ_{a}.

We will now consider the Schatten classes of T_{g} on the Hardy space H^{2}. For the basis {e_{n}(z) = z^{n} : n = 0, 1, 2, … } of H^{2} we have

∑_{n=0}^{∞} ∥T_{g}(e_{n})∥_{H^{2}}^{2} = ∑_{n=0}^{∞} ∥∫_{0}^{z} ζ^{n}g′(ζ) dζ∥_{H^{2}}^{2}

≃ ∑_{n=0}^{∞} ∫_{𝔻} |z|^{2n}|g′(z)|^{2}(1 − |z|) dm(z)

= ∫_{𝔻} (∑_{n=0}^{∞} |z|^{2n}) |g′(z)|^{2}(1 − |z|) dm(z)

= ∫_{𝔻} \frac{1}{1 − |z|^{2}} |g′(z)|^{2}(1 − |z|) dm(z)
≃ ∫_{𝔻} |g′(z)|^{2} dm(z).

This says that T_{g} is Hilbert-Schmidt on H^{2} if and only if g is in the Dirichlet space D. To determine the other Schatten classes observe that for f, h ∈ H^{2} we have by the Littlewood-Paley formula

⟨T_{g}^{∗}T_{g}(f), h⟩ = ⟨T_{g}(f), T_{g}(h)⟩ = 2 ∫_{𝔻} f(z)\overline{h(z)}|g′(z)|^{2} log(z^{−1}) dm(z).

Choosing h(z) = k_{w}(z) = (1 − w̄z)^{−1}, the reproducing kernel of H^{2}, the above equation becomes

T_{g}^{∗}T_{g}(f)(w) = 2 ∫_{𝔻} f(z)\frac{1}{1 − z̄w} dμ(z),

with

dμ(z) = |g′(z)|^{2} log(z^{−1}) dm(z).

Thus the operator T_{g}^{∗}T_{g} is a generalized Toeplitz operator and the Schatten ideals for those operators have been determined in [Lu]. Applying the main theorem of [Lu] we can determine the Schatten class of T_{g}^{∗}T_{g} and therefore also that of T_{g}, see [AleSi1] for details.

Theorem 5.1. Suppose 1 < p < ∞. Then T_{g} ∈ S^{p} on H^{2} if and only if g is in the Besov space B_{p}. Further T_{g} is not in S^{1} unless g is a constant.

We now consider the Schatten classes of T_{g} on the Bergman space A^{2}. Taking \{e_{n}(z) = \sqrt{n + 1}z^{n} : n = 0, 1, ⋯ \} as an orthonormal basis we have

∑_{n=0}^{∞} ∥T_{g}(e_{n})∥_{A^{2}}^{2} ≃ ∑_{n=0}^{∞} ∫_{𝔻}(n + 1)|z|^{2n}|g′(z)|^{2}(1 − |z|)^{2} dm(z)

= ∫_{𝔻} (∑_{n=0}^{∞}(n + 1)|z|^{2n}) |g′(z)|^{2}(1 − |z|)^{2} dm(z)

= ∫_{𝔻} \frac{1}{(1 − |z|^{2})^{2}} |g′(z)|^{2}(1 − |z|)^{2} dm(z)
≃ ∫_{𝔻} |g′(z)|^{2} dm(z),

thus T_{g} is Hilbert-Schmidt on A^{2} if and only if g ∈ D.

More generally consider the weighted Bergman spaces A_{α}^{2} with weights

w(r) = (α + 1)(1 − r^{2})^{α}, α > −1.

The reproducing kernel here is

k_{w}(z) = \frac{1}{(1 − w̄z)^{α+2}} ,

and the inner product can be written

⟨f, h⟩ = (α + 1) ∫_{𝔻} f(z)\overline{h(z)}(1 − |z|^{2})^{α} dm(z)
= f(0)\overline{h(0)} + ∫_{𝔻} f′(z)\overline{h′(z)}v(|z|) dm(z),

where

v(r) = 2 ∫_{r}^{1} \frac{(1 − u^{2})^{α+1}}{u} du.

We can now work as in the Hardy space case to obtain the operator T_{g}^{∗}T_{g} in the form

T_{g}^{∗}T_{g}(f)(w) = ∫_{𝔻} f(z)\frac{1}{(1 − z̄w)^{α+2}} dμ(z)

with dμ(z) = |g′(z)|^{2}v(|z|)dm(z). Applying again the work of [Lu] we obtain

Theorem 5.2. Let α > −1 and 1 < p < ∞. The operator T_{g} is in S^{p} on A_{α}^{2} if and only if g ∈ B_{p}. Further T_{g} is not in S^{1} unless g is a constant.

6. Some final remarks

Volterra type operators and some variations of them have been also studied on various spaces in [Hu], [Xi], [AMN], [You1], [You2], [You3]. We will not recount the results of these papers here. We will make however some remarks about such operators.

1. Cyclicity. Consider T_{g} acting on H^{2} and without loss of generality assume g(0) = 0. Then

T_{g}(1)(z) = ∫_{0}^{z} g′(ζ)dζ = g(z),

T_{g}^{2}(1)(z) = ∫_{0}^{z} g(ζ)g′(ζ) dζ = \frac{1}{2}g(z)^{2}

and inductively

T_{g}^{n}(1)(z) = \frac{1}{n!}g(z)^{n}, n = 1, 2, …

Set also T_{g}^{0}(1)(z) = 1. We see that the linear span of the orbit of 1 under T_{g} is dense in H^{2} if and only if the polynomials in g(z) are dense in H^{2}. Observe that this can happen only if g is univalent. Indeed if there are z, w ∈ 𝔻 such that g(z) = g(w) then for any polynomial P we will have P (g(z)) = P (g(w)) and any limit f of a sequence of such polynomials must satisfy f(z) = f(w), so the set of polynomials in g cannot be dense in H^{2}.

Let Ω = g(𝔻), and let H^{2}(Ω) to be the Hardy space on Ω defined through harmonic majorants i.e. f : Ω → ℂ belongs to H^{2}(Ω) if and only if there is a harmonic function u(z) on Ω such that |f(z)|^{2} ≤ u(z) for each z ∈ Ω. In this situation polynomials in g are dense in H^{2} if and only if the polynomials in z are dense in H^{2}(Ω). We therefore have

Proposition 6.1. The vector 1 is a cyclic vector for T_{g} on H^{2} if and only if polynomials are dense on H^{2}(Ω) with Ω = g(𝔻).

It would be interesting to see if the role of the vector 1 can be removed in the above proposition. In other words to prove that T_{g} is cyclic on H^{2} if and only if polynomials are dense in H^{2}(Ω).

2. A new proof of an old result. It is well known that if g is a function in BMOA then e^{g} is in H^{p} for some p > 0. And if g ∈ V MOA then e^{g} ∈ H^{p} for all p < ∞ [CiSc]. We are going to give a proof of these facts using the operators T_{g}.

Indeed let g ∈ BMOA. Then T_{g} is bounded on H^{2}. Assume without loss of generality g(0) = 0. Applying T_{g} repeatedly to the constant function 1 we have

T_{g}^{n}(1)(z) = \frac{1}{n!}g(z)^{n}, n = 0, 1, 2, …

Let r_{g} denote the spectral radius of T_{g} and let s be a number such that 0 < s < 1/r_{g}. Then the series ∑_{n=0}^{∞} s^{n}T_{g}^{n} converges in the operator norm topology, thus the series

∑_{n=0}^{∞} \frac{(sg(z))^{n}}{n!} = ∑_{n=0}^{∞} s^{n}T_{g}^{n}(1)(z)

converges in H^{2}. The sum of this series coincides with the pointwise sum which is e^{sg(z)}. Thus e^{sg(z)} is in H^{2} and e^{g}(z) is in H^{2s}.

If g ∈ V MOA then T_{g} is compact in H^{2}, and it can be checked by hand that T_{g} has no eigenvalues so its spectrum is {0}, and the spectral radius r_{g} = 0. Thus we can choose the number s in the previous argument to be any number in (0, ∞). It follows that e^{sg(z)} is in H^{2} for all s ∈ (0, ∞) so e^{g(z)} is in every H^{p}, p < ∞.

The same reasoning proves the analogous statements for g in the Bloch or little Bloch space. We have

Theorem 6.2. Let g be analytic on 𝔻, then

(i) If g ∈ BMOA, then e^{g} ∈ H^{p} for some p > 0. (ii) If g ∈ V MOA, then e^{g} ∈ H^{p} for all p > 0. (iii) If g ∈ B, then e^{g} ∈ A^{p} for some p > 0. (iv) If g ∈ B_{0}, then e^{g} ∈ A^{p} for all p > 0.

This theorem can be extended to the spaces X_{k} of section 2. The same reasoning can be applied to prove the following. Suppose X is a Banach space of analytic functions on 𝔻 which contains the constant functions, M_{z} : X → X is bounded, and the point evaluations are bounded on X. Then for the Cesàro operator to be bounded on X it is necessary that X contains the functions 1/(1 − z)^{s} for all sufficiently small s, see [Si2]. This is because the Cesàro operator is obtained as T_{g} with g = log(1/(1 − z)) and e^{sg(z)} = 1/(1 − z)^{s}.

3. Spectral properties. The spectrum of the Cesàro operator has been determined by using its relation to a semigroup of weighted composition operators, see [Si1]. This method can not be applied to the general Volterra operators T_{g}. The spectral properties of these operators remain unknown even though there have been some papers with sporadic results see [You1], [You2], [AMN], [MMN] and [Mil].

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Department of Mathematics, University of Thessaloniki, 54124 Thessaloniki, Greece.

   E-mail address: siskakis@auth.gr