INSIGHTS ON THE CES `ARO OPERATOR: SHIFT SEMIGROUPS AND INVARIANT SUBSPACES

. A closed subspace is invariant under the Ces`aro operator C on the classical Hardy space H 2 ( D ) if and only if its orthogonal complement is invariant under the C 0 -semigroup of composition operators induced by the aﬃne maps ϕ t ( z ) = e − t z + 1 − e − t for t ≥ 0 and z ∈ D . The corresponding result also holds in the Hardy spaces H p ( D ) for 1 < p < ∞ . Moreover, in the Hilbert space setting, by linking the invariant subspaces of C to the lattice of the closed invariant subspaces of the standard right-shift semigroup acting on a particular weighted L 2 -space on the line, we exhibit a large class of non-trivial closed invariant subspaces and provide a complete characterization of the ﬁnite codimensional ones, establishing, in particular, the limits of such an approach towards describing the lattice of all invariant subspaces of C . Finally, we present a functional calculus argument which allows us to extend a recent result by Mashreghi, Ptak and Ross regarding the square root of C and discuss its invariant subspaces.


Introduction and preliminaries
Despite the fact that one of the most classical transformations of sequences is the Cesàro operator C, there are still many questions about it unsettled.Recall that C takes a complex sequence a = (a 0 , a 1 , a 2 . . . ) to that with n-th entry: (n ≥ 0).
Upon identifying sequences with Taylor coefficients of power series, C acts formally on f Indeed, if f is a holomorphic function on the unit disc D so is C(f ) and moreover, C is an isomorphism of the Fréchet space H(D) of all holomorphic functions on D endowed with the topology of uniform convergence on compacta.
Nevertheless, this is no longer true when C is restricted to the classical Hardy spaces H p (D), 1 ≤ p < ∞.A classical result of Hardy concerning trigonometric series along with M. Riesz's theorem yields that C is bounded on H p (D) for 1 < p < ∞.Likewise, Siskakis proved that C is bounded on H 1 (D) (providing even an alternative proof of the boundedness on H p (D) for 1 < p < ∞; see [34], [35]).However 0 belongs to the spectrum of C in H p (D) and hence, C is not an isomorphism [34].
Note that (1.1) can be written as There is an extensive literature on the Cesàro operator, and more general, on integral operators, acting on a large variety of spaces of analytic functions regarding its boundedness, compactness or spectral picture (see [1] or [3], for instance).
If we restrict ourselves to the Hilbert space case H 2 (D), Kriete and Trutt proved the striking result that the Cesàro operator is subnormal, namely, C on H 2 (D) has a normal extension.More precisely, if I denotes the identity operator on H 2 (D), they proved that I − C is unitarily equivalent to the operator of multiplication by the identity function acting on the closure of analytic polynomials on the space L 2 (µ, D) for a particular measure µ (see [24]).An alternative proof of the Kriete and Trutt theorem, based on the connection between C and composition operators semigroups, was later established by Cowen [9].
For H p (D), 1 < p < ∞, Miller, Miller and Smith [29] showed that C is subdecomposable, namely, it has a decomposable extension (the H 1 (D) case was proved by Persson [32] ten years later).Decomposable operators were introduced by Foiaş [16] in the sixties as a generalization of spectral operators in the sense of Dunford, and many spectral operators in Hilbert spaces as unitary operators, selfadjoint operators or more generally, normal operators are decomposable (see the monograph [26] for more on the subject).
Normal operators on Hilbert spaces or more generally, decomposable operators on Banach spaces have a rich lattice of non-trivial closed invariant subspaces with a significant description of them.But, very little is known about this description even for concrete examples of subnormal operators as the Cesàro operator, and this will be the main motivation of the present manuscript.
In this context, we discuss invariant subspaces of the Cesàro operator C on the Hardy space H 2 (D).Broadly speaking, we prove a Beurling-Lax Theorem for the Cesàro operator and provide a complete characterization of the finite codimensional invariant subspaces of C. The composition semigroup method has turned out to be a powerful tool to study the Cesàro operator and we will make use of such technique in Section 2 to link the invariant subspaces of C to those of the right-shift semigroup {S τ } τ ≥0 acting on a particular weighted L 2 (R, w(y)dy).In particular, we will establish the limits of our approach towards describing completely the lattice of the invariant subspaces of C.
In Section 3, we discuss Phillips functional calculus (as in Haase's book [22]) which will allow us, in particular, generalize the recent work by Mashreghi, Ptak and Ross [28] regarding the square roots of C. In particular, we will discuss their invariant subspaces.
In order to close this introductory section we collect some preliminaries for the sake of completeness.
1.1.Semigroups of composition operators.The study of semigroups of composition operators on various function spaces of analytic functions has its origins in the work of Berkson and Porta [5], where they characterize their generators on H p (D), proving, indeed, that these semigroups are always strongly continuous.
Recall that a one-parameter family Φ = {ϕ t } t≥0 of analytic self-maps of D is called a holomorphic flow (or holomorphic semiflow by some authors) if it is a continuous family that has a semigroup property with respect to composition, namely 1) ϕ 0 (z) = z, for z ∈ D; 2) ϕ t+s (z) = ϕ t • ϕ s (z), for t, s ≥ 0, and z ∈ D; 3) For any s ≥ 0 and any z ∈ D, lim t→s ϕ t (z) = ϕ s (z).
The holomorphic flow Φ is trivial if ϕ t (z) = z for all t ≥ 0. Otherwise, we say that Φ is nontrivial.We refer to the recent monograph [6] for more on the subject.Associated to the holomorphic flow Φ = {ϕ t } t≥0 is the family of composition operators {C ϕt } t≥0 , defined on the space of analytic functions on D by Clearly, {C ϕt } t≥0 has the semigroup property: Moreover, recall that if an operator semigroup {T t } t≥0 acts on a Banach space X, then it is called strongly continuous or C 0 -semigroup, if it satisfies lim t→0 + T t f = f for any f ∈ X.Given a C 0 -semigroup {T t } t≥0 on a Banach space X, recall that its generator is the closed and densely defined linear operator A defined by

The lattice of the invariant subspaces of the Cesàro operator
The aim of this section is identifying the lattice of the invariant subspaces of the Cesàro operator C acting on the Hardy space H 2 (D).In particular, we will characterize the finite codimensional invariant subspaces of C.
Our first result resembles a Beurling-Lax Theorem for the Cesàro operator.
Theorem 2.1.Let Φ = {ϕ t } t≥0 be the holomorphic flow given by A closed subspace M in H 2 (D) is invariant under the Cesàro operator if and only if its orthogonal complement M ⊥ is invariant under the semigroup of composition operators induced by Φ, namely, {C ϕt } t≥0 .
Before proceeding with the proof, note that each ϕ t in (2.1) is a affine map which is a hyperbolic non-automorphism of the unit disc inducing a bounded composition operator C ϕt on H 2 (D) with norm (2.2) (see, for instance, [10,Theorem 9.4]).
Likewise, the generator of the C 0 -semigroup {C ϕt } t≥0 is given by (see the pioneering work by Berkson and Porta [5], for instance).
Proof.First, let us show that the cogenerator of the C 0 -semigroup {C ϕt } t≥0 given by is a well-defined bounded operator.For such a task, we will prove that 1 ∈ ρ(A), the resolvent of A, or equivalently, We claim that A − I is also surjective.Given g ∈ H 2 (D), in order to find f ∈ H 2 (D) such that Note that the adjoint of the Cesàro operator C * has the following matrix with respect to the canonical orthonormal basis of H 2 (D): Accordingly, T is a well-defined operator in H 2 (D) (as it is -C * ).This in particular implies that the function f in (2.3) belongs to H 2 (D) and hence A − I is surjective.
Accordingly, the cogenerator V of the C 0 -semigroup {C ϕt } t≥0 is a well-defined bounded operator on H 2 (D).Now, having in mind the norm estimate (2.2), we observe that the C 0 -semigroup {e −t C ϕ 2t } t≥0 is contractive on H 2 (D) and its generator is 2A − I. Since the invariant subspaces of the cogenerator are simply the common invariant subspaces of the semigroup (see [17,Chap. 10,Theorem 10.9]) and the statement of the theorem follows.
First, let us remark that a similar argument in the context of C 0 -semigroups of analytic 2isometries was also used in [19].Likewise, recalling that the Hardy space H p (D), 1 ≤ p < ∞, consists of holomorphic functions f on D for which the norm is finite, we note that the previous proof also works in H p (D)-spaces (1 < p < ∞) with the natural identification of the dual space H p (D) * ∼ = H p (D) where p is the conjugate exponent: Therefore, a closed subspace M in H p (D) for 1 < p < ∞ is invariant under the Cesàro operator if and only if its annihilator In this regard, it is worth noting that Remark 2.2.By Dunford and Schwartz [15, Thm.11, p. 622], the resolvent can be expressed in terms of the Laplace transform of the semigroup; that is, Indeed, a consequence of the previous formula is the following: Alternatively, one may use that the adjoint of the generator is the generator of the adjoint semigroup in the context of Hilbert spaces (see [17,Chap. 10], for instance).
Finally, note that the adjoint C * ϕt in H 2 (D) in Corollary 2.3 may be explicitly computed as a weighted composition operator (see [10,Theorem 9.2]).Indeed, expressing ϕ t (z) = e −t z + 1 − e −t in its normal form, namely ϕ t (z) = (a t z + b t )/(0z + a −1 t ) where and h t (z) = a −1 t = e t/2 ; and T gt and T ht denotes the analytic Toeplitz operators acting on H 2 (D) induced by the symbols g t and h t respectively. Accordingly, for f ∈ H 2 (D) and every t ≥ 0.
2.1.Shift semigroups.In order to provide a characterization of the finite codimensional invariant subspaces of C, we will make use of a semigroup of operators acting on the Hardy space of the right half-plane C + .Recall that the Hardy space H 2 (C + ) consists of the functions F analytic on C + with finite norm .
The classical Paley-Wiener Theorem (see [33], for instance) states that H 2 (C + ) is isomorphic under the Laplace transform to L 2 (R + ), the space of measurable functions square-integrable over (0, ∞).More precisely, to each function A first observation already stated in [8, Lemma 4.2] is that each ϕ t (z) = e −t z + 1 − e −t for z ∈ D and t > 0 induces a composition operator in H 2 (D) which is similar under an isomorphism U (indeed unitarily equivalent up to a constant) to e t C φt in H 2 (C + ), where φ t (s) = e t s + (e t − 1), (s ∈ C + ). Namely, (2.5) Since we are interested in studying invariant subspaces, either for the entire semigroup or individual elements, we may disregard factors of the form e λt for a fixed λ ∈ R.
By means of the inverse Laplace transform we are led to consider the semigroup on x g(e −t x), (x > 0, t ≥ 0).Now, proceeding as in [18], we may find a further equivalence with an operator on L 2 (R) using the unitary mapping T : and T −1 g(y) = e y/2 g(e y ), (y ∈ R). Accordingly, (2.7) Denoting by {S t : t ≥ 0} the right-shift semigroup on L 2 (R): and recalling that if w denotes a positive measurable function in R the space L 2 (R, w(y)dy) consists of measurable functions in R square-integrable respect to the measure w(y)dy, a key observation is the following Proposition 2.4.The semigroup {σ t : t ≥ 0} in L 2 (R) given by for h ∈ L 2 (R) is unitarily equivalent to the right-shift semigroup {S t : t ≥ 0} acting on the weighted Lebesgue space L 2 (R, e −2(e y −1) dy).
Proof.Let us denote the weight w(y) = e −2(e y −1) for y ∈ R and consider the unitary mapping for h ∈ L 2 (R).A computation shows that for any function f ∈ L 2 (R, w(y) dy) and t > 0 = W e −(1−e −t )e y f (y − t)e −(e y−t −1) = e e y −1 e −(1−e −t )e y f (y − t)e −(e y−t −1) = f (y − t), for y ∈ R.This yields the statement of the proposition.As a by-product of equations (2.5), (2.6), (2.7), Proposition 2.4 and Theorem 2.1, if we denote by F the unitary isomorphism F = W T −1 L −1 U from H 2 (D) onto L 2 (R, e −2(e y −1) dy), the following result regarding the lattice of invariant subspaces of the Cesàro operator holds: Theorem 2.5.A closed subspace M in H 2 (D) is invariant under the Cesàro operator if and only if FM ⊥ in L 2 (R, e −2(e y −1) dy) is invariant under the right-shift semigroup {S t : t ≥ 0}.
Accordingly, characterizing the lattice of invariant subspaces of the Cesàro operator in the Hardy space reduces to characterize the lattice of the right-shift semigroup in L 2 (R, e −2(e y −1) dy).
Though the lattice of the invariant subspaces of the right-shift semigroup acting on weighted Lebesgue spaces is only characterized for a very restricted subclass of weights (for instance the Beurling-Lax Theorem provides a characterization in L 2 (R + ), where the weight is the characteristic function χ (0,+∞) of (0, +∞)), the question if such a lattice contains non-standard invariant subspaces has been extensively studied (see [11,12,13,14], [20,21], [27] or [31], for instance).
In [11,Equation (8)], Domar proved that if the weight satisfies then the lattice of invariant subspaces of {S t : t ≥ 0} in L 2 (R, w(y) dy) contains non-standard invariant subspaces.
A word about notation: Domar denotes by L 2 (R, w(y) dy) the space of measurable functions f in R such that f w ∈ L 2 (R).Note that this does not affect the previous equation since it is enough to consider the positive function w 1/2 .
In our case, w(y) = e −2(e y −1) for y ∈ R and consequently, {S t : t ≥ 0} has non-standard invariant subspaces in L 2 (R, e −2(e y −1) dy).Indeed, it is possible to exhibit many non-standard invariant subspaces in this case.In order to show them recall that, by means of the unitary equivalence τ ≥ 0, if and only if there exists an inner function Θ ∈ H ∞ (C + ) such that LM = ΘH 2 (C + ) (see [30,Cor. 6.5.5(2), p.149], for instance).Here, recall that an inner function Θ is an analytic function in C + with |Θ(z)| ≤ 1 for z ∈ C + , such that the non-tangential limits exist and are of modulus 1 almost everywhere on the imaginary axis.
for τ ≥ 0, then M ⊕ L 2 ((T, ∞), w(y) dy) is a closed subspace of L 2 (R, w(y) dy) invariant under all right shifts.Now, the Beurling-Lax theorem provides a large class of nonstandard invariant subspaces M : take the "twisted" Laplace transform which gives an isomorphism from L 2 ((−∞, T ), w(y) dy) onto e sT H 2 (C + ).Then any subspace of the form L −1 e sT K Θ is invariant under all truncated right shifts S (−∞,T ), τ where Θ ∈ H ∞ (C + ) is inner and is the associated model space (these calculations are easiest to follow when T = 0, and the general case is a shifted version.) As an explicit example, let K Θ be spanned by the reproducing kernel s → 1/(s + λ) for λ ∈ C + , so that M is the one-dimensional space of L 2 ((−∞, T ), w(y) dy) spanned by e λt ; then M ⊕ L 2 ((T, ∞), w(y) dy) is a non-standard invariant subspace for all right shifts.
While a theorem of Aleman and Koreblum [2] asserts that the analytic Volterra operator is unicellular in H p -spaces, as consequence of the previous considerations we have a new deduction of the following known result: Corollary 2.7.The Cesàro operator C is not a unicellular operator in H 2 (D).
In this regard, the feature that the Cesàro operator C is not a unicellular operator on H 2 (D) can be also deduced from [7], as the referee kindly pointed out to us.Indeed, it follows from the result that the point spectrum of I − C * in H 2 (D) is D along with the fact that any operator on a Hilbert space which has at least two eigenvalues cannot be unicellular.Likewise, in [25,Corollary 6], the authors constructed two nonzero invariant subspaces of C whose intersection is zero space.
On the other hand, it is worth pointing out that the classifying the invariant subspaces turns out to be completely different if one considers other semigroups studied in the context of Cesàro-like operators, as in the following remark: Remark 2.8.In [4], the authors considered the composition operator group on H 2 (C + ) corresponding to the flow on C + given by φ t (s) = e −t s, t ∈ R, in a broader context of studying Cesàro-like operators.
Proceeding similarly as before, the transformed semigroup on L 2 (0, ∞) is given by Ṽt g(x) = e t g(e t x), The subspaces M invariant under the group (τ t ) t∈R = (T −1 Ṽt T ) t∈R were essentially classified by Lax -the factors e t/2 are irrelevant -and can be found, with a slightly different notation in [30, Cor 6.5.4,p. 149].There are two types: (1) 1-invariant subspaces, i.e., τ t M ⊂ M for all t < 0 but not for all t ∈ R.These have the form M = FqH 2 (Π + ), where q is measurable with |q| = 1 almost everywhere and here Π + denotes the upper half-plane; (2) 2-invariant subspaces, i.e., τ t M ⊂ M for all t ∈ R.These have the form M = Fχ E L 2 (R) for some measurable subset E ⊂ R.
Here F denotes the Fourier transform but, alternatively, one can use the bilateral Laplace transform and express the subspaces in terms of L 2 (iR) and the space H 2 (C + ) of the right half-plane.
Likewise, in this case the invariant subspaces of the form L−1 Finally, as an application of Theorem 2.5, we present a characterization of the finite codimensional invariant subspaces of the Cesàro operator C in H 2 (D).Of particular relevance will be a theorem of Domar [12] which states that the lattice of the invariant subspaces of {S τ : τ ≥ 0} consists of just the standard invariant subspaces in L 2 (R + , w(x) dx) whenever: (1) w is a positive continuous function in R + such that log w is concave in [c, ∞) for some c > 0.

Proof.
Suppose first that M is a finite codimensional closed subspace in H 2 (D) invariant under C. Theorem 2.5 yields that N = FM ⊥ is a finite dimensional subspace of L 2 (R, e −2(e y −1) dy) invariant under all right shifts.Thus, P − N , the projection onto L 2 ((−∞, 0), e −2(e y −1) dy) ∼ = L 2 (−∞, 0), is a finite dimensional subspace invariant under all the truncated right shifts.Thus, by the Beurling-Lax Theorem, P − N corresponds to a model space and in particular is spanned by a finite set of functions of the form y k e λy , for y ∈ (−∞, 0) where k = 0, 1, 2, . . ., n λ for λ ∈ Λ ⊂ C + .We now show that N is spanned by what we shall call the "natural extension" to R of such functions as elements of L 2 (R, e −2(e y −1) dy), namely, y k e λy for y ∈
Therefore, there exist h λ,k ∈ N such that P − f λ,k = h λ,k and N is spanned by h λ,k .Let N 1 be spanned by f λ,k with the same k and λ.By (2.10), N 1 is invariant under all right shifts S t .Then span {N, N 1 } is a finite dimensional invariant subspace of all right shifts S t .Upon applying Domar's theorem again, span {N, N 1 } ∩ L 2 ((0, ∞), e −2(e y −1) dy) = {0}.

Since
we conclude that h λ,k = f λ,k .Thus, N is spanned by f λ,k .
For the converse, assume N = FM ⊥ in L 2 (R, e −2(e y −1) dy) is a finite dimensional subspace spanned by a finite subset of the form λ∈Λ {y k e λy : k = 0, 1, 2, . . ., n λ } where λ ∈ Λ ⊂ C + is finite.Equation (2.10) yields that N is invariant under all right shifts S t .Now, Theorem 2.5 and the fact that F is an isomorphism yield that M is a finite codimensional closed subspace invariant under C, which completes the proof.
Remark 2.10.Note that in the case that N = FM ⊥ is infinite-dimensional, the arguments above giving the structure of N in terms of the structure of P − N fail because we can no longer assume that P − N is closed.However, it is of interest to note that the closure P − N in L 2 (−∞, 0) has the same property of invariance under all truncated right shifts, and corresponds to a model space.
For instance, if B is a Blaschke product in C + with the set of zeros Λ and multiplicities n λ + 1 for λ ∈ Λ with n λ ∈ {0, 1, 2, . . .} and we set clearly N B is invariant under all right shifts.Nevertheless, where L is defined in (2.9) with T = 0. Consequently, N B = L 2 (R, e −2(e y −1) dy), and if

2.2.
A final remark regarding the lattice of the invariant subspaces of C. As we have just noted, the approach addressed in the previous theorem fails if P − N is not closed.Indeed, the following example shows that P − N need not be closed even if N is a closed shift-invariant subspace of L 2 (R, e −2(e y −1) dy) showing, somehow, the limits of such an approach.
Let λ > 0 and denote by e λ the function e λ : y ∈ R → e λy .Since 1 < e −2(e y −1) < e 2 for y < 0, we have Now the twisted Laplace transform given in (2.9), with T = 0, provides an isomorphism from L 2 ((−∞, 0), e −2(e y −1) dy) onto H 2 (C + ) transforming e λ to 1/(s + λ).By an argument similar to that used in proving the classical Müntz-Szász theorem it follows that P − N L 2 ((−∞, 0), e −2(e y −1) dy) since there are functions orthogonal to each 1/(s + n 2 ), for example, 1/(s + 2) times a Blaschke product with zeros at {n 2 : n ∈ N}.Now (ker P − ) ∩ N is a closed shift-invariant subspace of L 2 ((0, ∞), e −2(e y −1) dy) and hence, by Domar's theorem, a standard subspace.It must be {0} (again this follows from a Műntz-Szász the Laplace transform of a Borel measure µ on [0, ∞) of bounded variation, by the formula Note that the convention in [22] is that the generator is −A, rather than A, and we have allowed for that in the discussion below.In particular, we have In the case β = 1/2 this agrees with the formula in [28].
Remark 3.2.It is clear from the functional calculus that every subspace for C is also an invariant subspace for C β for Re β > 0. Since invariant subspaces for C 1/n are clearly invariant subspaces for C for n = 1, 2, . .., we may conclude that C and C 1/n have the same lattice of invariant subspaces.

Figure 1 .
Figure 1.The weight e −2(e y −1) as a function of y.

Figure 1
Figure1shows a plot of the weight function described in Proposition 2.4.