CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE

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1 CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS Abstract. In this article we study the Cesàro operator C(f)() = d, and its companion operator T on Hardy spaces of the upper half plane. We identify C and T as resolvents for appropriate semigroups of composition operators and we find the norm and the spectrum in each case. The relation of C and T with the corresponding Cesàro operators on Lebesgue spaces L p (R) of the boundary line is also discussed.. Introduction Let U = { C : Im() > } denote the upper half of the complex plane. For < p <, the Hardy space H p (U) is the space of analytic functions f : U C for which f H p (U) = sup y> ( ) f(x + iy) p p dx <. For p = we denote by H (U) the space of all bounded analytic functions on U with the supremum norm. The spaces H p (U), p, are Banach spaces and for p = 2, H 2 (U) is a Hilbert space. For the rest of the paper we use the notation f p for f H p (U). Let p < and suppose f H p (U). Then f satisfies the growth condition (.) f() p C f p p Im(), U, where C is a constant. Further the limit lim y f(x + iy) exists for almost every x in R and we may define the boundary function on R, denoted by f, as f (x) = lim y f(x + iy). Date: June 8, 2. 2 Mathematics Subject Classification. Primary 47B38; Secondary 3D55, 47D3. Key words and phrases. Cesàro operators, Hardy spaces, Semigroups, Composition operators. The first author was supported with a graduate fellowship from Alexander S. Onassis Foundation.

2 2 ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS This function is p-integrable and (.2) f p p = f (x) p dx. Thus H p (U) can be viewed as a subspace of L p (R). For more details on Hardy spaces H p (U) see [Du], [Ga], [Ho]. For f L p (R) the well known Cesàro transformation is defined by (.3) C(f)(x) = x x f(u) du for x R, with an appropriate convention if x =. This defines a bounded operator on L p (R) for p > [BHS]. In particular if f L p (R) is the boundary function of f H p (U) the question arises whether the transformed function C(f ) is also the boundary function of an f H p (U). In this article we consider the half-plane version of the Cesàro operator, which is formally defined C(f)() = d, f H p (U). It turns out that this formula defines an analytic function on U for each f H p (U) and the resulting operator is bounded on H p (U), p >. In addition the companion operator T (f)() = d, f Hp (U), is also shown to be bounded on H p (U) for p. We find the norm and the spectrum of C, T and we show that, for the boundary functions, we have C(f) = C(f ). The whole discussion is based on the observation that C and T can be obtained as resolvent operators for appropriate strongly continuous semigroups of simple composition operators on H p (U). 2. Related semigroups, strong continuity. For each t R consider the analytic self maps of U φ t () = e t, U, and for p <, the corresponding weighted composition operators on H p (U), T t (f)() = e t p f(φ t ()), f H p (U). For f H p (U) we have ( T t (f) p p = e t + ) sup f(e t x + ie t y) p dx y> ( + ) = sup f(u + iv) p du v> = f p p,

3 CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE 3 thus each T t is an isometry on H p (U). Further it is easy to see that the family {T t } satisfies T t T s = T t+s for each t, s R and T = I the identity operator, so {T t } is a group of isometries. We will use this group or the positive and negative semigroups {T t, t } and {T t, t } in our study of the operators C and T. Proposition 2.. For p < the group {T t } is strongly continuous on H p (U). The infinitesimal generator Γ of {T t } is given by Γ(f)() = f () p f(), and its domain is D(Γ) = {f H p (U) : f () H p (U)}. Proof. For the strong continuity we need to show (2.) lim t T t (f) f p = lim t e t p f φ t f p = for every f H p (U). Given such an f consider the functions f s () = f( + is), s >. It is easy to see that f s H p (U) for every s and we have T t (f) f p T t (f) T t (f s ) p + T t (f s ) f s p + f s f p ( T t + ) f s f p + T t (f s ) f s p = 2 f s f p + e t p f s φ t f s p. Now it is clear that the boundary function of f s () is f(x + is) and the boundary function of e t p f s (e t ) is e t p f(e t x + is) for all x R. Since f s f p p = f(x + is) f (x) p dx, as s (see for example [Du, Theorem.4]), it suffices to show lim t e t p f s φ t f s p = for each s. For a fixed such s and a > (to be specified later) we have e t p f s φ t f s p p = = 2a + + 2a 2a 2a e t p f(e t x + is) f(x + is) p dx e t p f(e t x + is) f(x + is) p dx e t p f(e t x + is) f(x + is) p dx e t p f(e t x + is) f(x + is) p dx = I (t) + I 2 (t) + I 3 (t).

4 4 ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS To estimate the integral I (t) we use the standard inequality e t p f(e t x + is) f(x + is) p 2 p (e t f(e t x + is) p + f(x + is) p ) and the change of variable u = e t x to obtain 2ae t 2a I (t) 2 p f(u + is) p du + 2 p f(x + is) p dx a 2 p+ f(x + is) p dx, valid for < t < log 2. An analogous estimate for I 3 (t) gives I 3 (t) 2 p+ f(x + is) p dx, a for < t < log 2. Let ɛ > be given. Since f s H p (U) we can choose a > such that for < t < log 2, I (t) + I 3 (t) 2 p+ f(x + is) p dx < ɛ. R\[ a, a] We now deal with the second integral I 2 (t). Using the growth estimate (.) we have e t p f(e t x + is) f(x + is) p 2 p e t p f(e t x + is) p + 2 p f(x + is) p t 2 p C e p f(e t x + is) p p Im(e t + 2 p C f(x + is) p p x + is) Im(x + is) 2 p C f p p s = 2 p+ C f p p s. + 2 p C f p p s This says in particular that e t p f(e t x + is) f(x + is) p is a uniformly bounded function of x on the interval [ 2a, 2a] for each t >. Since this function has pointwise limit as t, an application of Lebesgue s dominated convergence theorem gives lim t 2a 2a e t p f(e t x + is) f(x + is) p dx =. Putting together all previous estimates we conclude lim t T t (f) f p = and the strong continuity follows. By definition the domain D(Γ) of Γ, consists of all f H p (U) for which the limit lim t e tp f φ t f t exists in H p (U) and e t p f φ t f Γ(f) = lim, f D(Γ). t t

5 CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE 5 The growth estimate (.) shows that convergence in the norm of H p (U) implies in particular pointwise convergence, therefore for each f D(Γ), Γ(f)() = lim e t t = t p f(e t ) f() t ( e t p f(e t ) = f () p f(). ) t= This shows that D(Γ) {f H p (U) : f () H p (U)}. Conversely let f H p (U) such that f () H p (U). Then we can write for U, t ( ) T t (f)() f() = e s p f(φ s ()) ds s = = t t e s p φ s ()f (φ s ()) p e s p f(φ s ()) ds T s (F )() ds, where F () = f () p f() is a function in Hp (U). Thus T t (f) f lim t t = lim t t t T s (F ) ds. From the general theory of strongly continuous (semi)groups, for F H p (U) the latter limit exists in H p (U) and is equal to F. Thus T t (f) f lim t t = F. This says that D(Γ) {f H p (U) : f () H p (U)} completing the proof. Remark. An immediate corollary of the previous proposition is that the two semigroups {T t, t } and {T t, t } are strongly continuous on H p (U) and their infinitesimal generators are Γ and Γ respectively. Lemma 2.2. Let < p < and λ C, then (i) e λ () = λ / H p (U). (ii) h λ () = (i + ) λ H p (U) if and only if Re(λ) < p. Proof. This can be proved by direct calculation of the norms. We give an alternative argument involving the following well known characteriation of membership in the Hardy space of the half plane. For a function f analytic on U, f H p (U) if and only if ψ () /p f(ψ()) H p (D),

6 6 ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS where ψ() = i +, a conformal map from the unit disc D = { < } onto U, and H p (D) is the usual Hardy space of the disc. We find (2.2) ψ () /p c h λ (ψ()) =, ( ) 2 p +λ and (2.3) ψ () /p e λ (ψ()) = c 2 ( + ) λ ( ) 2 p +λ, where c, c 2 are nonero constants. Next recall that if ν is a complex number, ( ) ν H p (D) if and only if Re(ν) > /p (this follows for example from [Du, page 3, Ex. ]). Applying this to (2.2) we obtain h λ H p (U) if and only if Re( 2 p λ) > /p and this gives the desired conclusion. In the case of e λ, the right hand side of (2.3) belongs to H p (D) if and only if both terms ( +) λ and ( ) 2 p λ belong to H p (D), because each of the two terms is analytic and nonero at the point where the other term has a singularity. Thus for e λ to belong to H p (U), both conditions Re(λ) > /p and Re( 2 p λ) > /p must be satisfied, which is impossible. For an operator A denote by σ π (A) the set of eigenvalues of A, by σ(a) the spectrum of A and by ρ(a) the resolvent set of A on H p (U). Proposition 2.3. Let p < and consider {T t } acting on H p (U). Then: (i) σ π (Γ) is empty. (ii) σ(γ) = ir. In particular Γ is an unbounded operator. Proof. (i) Let γ be an eigenvalue of Γ and let f be a corresponding eigenvector. The eigenvalue equation Γ(f) = γf is equivalent to the differential equation f () + (γ + )f() =. p The nonero analytic solutions of this equation on U have the form f() = c (γ+ p ) with c, which by Lemma 2.2 are not in H p (U). It follows that σ π (Γ) =. (ii) Because each T t is an invertible isometry its spectrum satisfies σ(t t ) D. The spectral mapping theorem for strongly continuous groups ([Pa, Theorem 2.3]) says e tσ(γ) σ(t t ). Thus if w σ(γ), then e tw D, so that σ(γ) ir. We will show that in fact σ(γ) = ir.

7 CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE 7 Let µ ir and assume that µ ρ(γ). Let λ = µ + p and consider the function f λ () = iλ(i + ) λ. Since Re( λ ) = /p < /p, this function is in H p (U). Since µ ρ(γ) the operator R µ = (µ Γ) : H p (U) H p (U) is bounded. The image function g = R µ (f λ ) satisfies the equation (µ Γ)(g) = f λ or equivalently, (µ + p )g() + g () = f λ (), U. Thus g solves the differential equation λg() + g () = iλ(i + ) λ, U. It is easy to check that the general solution of this equation in U is G() = c λ + (i + ) λ, Using the notation of Lemma 2.2 we find c a constant. (ψ ()) /p G(ψ()) = c + cc 2 ( + ) λ ( ) 2 p λ, where c, c 2 are nonero constants. This last expression represents an analytic function on the unit disc which however, by an argument similar to that in the proof of Lemma 2.2, is not in the Hardy space H p (D) of the unit disc for any value of c because Re(λ) = /p. Thus G() is not in H p (U) for any c and this is a contradiction. It follows that σ(γ) = ir, and this completes the proof. 3. The Cesàro operators It follows from the above that when < p < the point λ p = /p is in the resolvent set of the generator Γ. The resolvent operator R(λ p, Γ) is therefore bounded. Let f H p (U) and let g = R(λ p, Γ)(f). It follows that (λ p Γ)(g) = f or equivalently ( /p)g() + g () + (/p)g() = f(). Thus g satisfies the differential equation g() + g () = f(), U. Fix a point w on the imaginary axis. Then g() = w d + c, U, with c a constant. Now let = iy along the imaginary axis. Since g H p (U) with p >, the estimate (.) implies that g() therefore c = w d

8 8 ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS (the existence of this integral is also a consequence of (.)). It follows that the integral of f on the segment [, ] exists and we have g() = R(λ p, Γ)(f)() = d. Theorem 3.. Let < p < and let C be the operator defined by C(f)() = d, Then C : H p (U) H p (U) is bounded. Further and C = σ(c) = {w C : w Proof. As found above, for < p <, C = R(λ p, Γ), p p, f H p (U). p 2(p ) = p 2(p ) }. λ p = /p ρ(γ). The spectral mapping theorem [DS, Lemma VII.9.2] gives σ(c) = { : σ(γ)} {} λ p = { : r R} {} /p ir p = {w C : w 2(p ) = p 2(p ) }, giving the spectrum of C on H p (U). Since the spectral radius of C is equal to p/(p ) it follows that C p p. On the other hand we can apply the Hille-Yosida-Phillips theorem to the group {T t } of isometries [DS, Corollary VIII..4] to obtain and the proof is complete. C = R(λ p, Γ) p = p p, Remark. It follows from Lemma (2.2) that h() = (i + ) 2 belongs to H (U). The transformed function i(i + ) = h() d is analytic on U but does not belong to H (U). Thus C does not take H (U) to H (U). We now consider the negative part {T t : t } of the group {T t } and we rename it {S t }. That is, for f H p (U), S t (f)() = e t p f(φ t ()) = e t p f(e t ), t.

9 CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE 9 It is clear that {S t } is strongly continuous on H p (U) and that its generator is = Γ. It follows from Proposition 2.3 that σ( ) = ir, σ π ( ) =. Let µ p = p ρ( ), and consider the bounded resolvent operator R(µ p, ). Let f H p (U) and let g = R(µ p, )(f). Then µ p g (g) = f or equivalently g () = f(), U. Fix a point a U, then we have g() = a d + c with c a constant. Now let within a half-plane Im() δ >. Then g() [Du, Corollary, page 9], thus the following limit exists and gives the value of c, w c = lim d. w Im(w) δ a We therefore find a g() = = lim w Im(w) δ w d + lim w Im(w) δ d. w a We now show that the above limit exists unrestrictedly when w within the half-plane U. Indeed for δ > we have, w d = w+iδ d + w w+iδ d d = J w + I w. With the change of variable = w + isδ, s, we have I w = iδ δ δ w = w f(w + isδ) w + isδ f(w + isδ) ds w + isδ y+δ y f(w + isδ) ds ds f(x + iu) du, where w = x+iy and u = y +sδ. For p = the Fejér-Ries inequality for the half-plane [Du, Exercise 6, page 98] implies, through the last inequality, I w w f(x + iu) du 2 w f, as w,

10 ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS while for p >, the growth estimate (.) gives ( y+δ ) I w C p u p du f p w y C p (y + δ) /q f p, (/q = /p + ), w C p ( w + δ) /q f p, as w. w Thus in all cases I w and we have w lim w d = lim w J w + lim w I w = g() i.e. the unrestricted limit exists. For this reason we can write g() = d. Theorem 3.2. Let p < and let T be the operator defined by T (f)() = d, Then T : H p (U) H p (U) is bounded. Further f H p (U). and T = p, σ(t ) = {w C : w p 2 = p 2 }. Proof. As found above we have T = R(µ p, ), µ p = /p ρ( ). From the spectral mapping theorem we find σ(t ) = { : σ( )} {} µ p = { : r R} {} /p ir = {w C : w p 2 = p 2 }, giving the spectrum of T on H p (U). Since the spectral radius of T is equal to p it follows that T p. On the other hand we can apply the Hille-Yosida-Phillips theorem to the semigroup {S t } of isometries to obtain T = R(µ p, ) /p = p, and the proof is complete.

11 CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE Suppose now < p < and let q be the conjugate index, /p + /q =. Recall the duality (H p (U)) = H q (U) which is realied through the pairing (3.) f, g = f (x)g (x) dx. For the semigroups T t (f)() = e t p f(e t ) and S t (g)() = e t q g(e t ) acting on H p (U) and H q (U) respectively we find (3.2) T t (f), g = = = f, S t (g). e t p f (e t x)g (x) dx f (x)e t q g (e t x) dx Thus {T t } and {S t } are adjoints of each other. From the general theory of operator semigroups this relation of being adjoint, on reflexive spaces, is inherited by the infinitesimal generators and subsequently by the resolvent operators [Pa, Corollaries.2 and.6]. It follows that C and T are adjoints of each other on the reflexive Hardy spaces of the half-plane. 4. Boundary correspondence We now examine the boundary correspondence between C and its real line version C, as well as between T and the corresponding real line operator T defined on L p (R) by T(f)(x) = x f(u) u du, x >, x f(u) u du, x <, while T(f)() can be chosen arbitrarily. It is well known (and easy to prove) that T is bounded on L p (R) for p. Theorem 4.. Consider the operators C, T on the Hardy spaces H p (U) and the operators C, T on the spaces L p (R). Then (i) For < p < and f H p (U), (ii) For p < and f H p (U), C(f) = C(f ). T (f) = T(f ). Proof. (i) For f H p (U) and δ > consider the function f δ () = f( + iδ), Im() > δ.

12 2 ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS It is clear that f δ H p (U) and f δ (x) = f δ(x), and we have C(f) C(f ) L p (R) C(f) C(f δ ) L p (R) + C(f δ ) C(f δ ) L p (R) + C(f δ ) C(f ) L p (R) ( C + C ) f δ f H p (U) + C(f δ ) C(f δ ) L p (R). We can make f δ f H p (U) as small as we wish by choosing δ close enough to. Thus in order to prove that C(f) (x) = C(f )(x) a.e. on R, it suffices to show C(f δ ) (x) = C(fδ )(x), i.e. C(f δ ) (x) = x x fδ (u) du for almost all x. Now for = x + iy U, since f δ () is analytic on {Im() > δ}, its integral on the segment [, ] can be obtained by integrating over the path [, x] [x, ], so we have C(f δ )() = f δ () d = f δ () d + [, x] [x, ] f δ () d. If x then clearly the limit of the first integral as y is lim f δ () d = x f δ (u) du. y [, x] x The limit of the second integral vanishes. Indeed since f H p (U), f is bounded over every half-plane { : Im() δ} and we find therefore sup f δ () sup f() = M <, [x, ] Im() δ [x, ] as y. It follows that f δ () d x ( sup f δ () )y M [x, ] x y C(f δ ) (x) = lim y C(f δ )() = x x f δ (u) du = x x fδ (u) du and the proof of (i) is complete. (ii) We argue as in part (i) and use the same notation. If f H p (U), p <, let f δ be defined as in part (i). Using the triangle inequality as in part (i) we see that it suffices to show that for almost all x, T (f δ ) (x) = T(f δ )(x). Let = x + iy U with x > (the case x < is similar), choose s >, and consider the path [, x] [x, x + s] [x + s, + s( + i)] as an alternative

13 CESARO OPERATORS ON THE HARDY SPACES OF THE HALF-PLANE 3 path of integration to obtain T (f δ )() = lim s +s(+i) = x f δ () d + lim s f δ () d ( x+s x f δ () For the first integral inside the limit it is clear that Write I(s) = +s(+i) x+s x+s lim s x f δ () I(s) = f δ () d = + x ) +s(+i) f δ () d + d. x+s f δ (u) u du. d, the second integral inside the limit, then y+s y+s x + s f δ (x + s + it) i dt x + s + it f δ (x + s + it) dt x + s + it y+s f δ (x + s + it) dt. If p = then the Fejér -Ries inequality for the upper half-plane gives y+s f δ (x + s + it) dt f δ (x + s + it) dt 2 f δ 2 f, thus I(s) 2(x+s) f as s. If p > then using (.) we obtain y+s f δ (x + s + it) dt C p f δ p y+s dt t/p C p f p (y + s) p +, which implies I(s) C p f p (y+s) p + T (f δ )() = x+s x f δ () as s. We have shown d + + x f δ (u) u for each = x + iy U. Taking the limit of the above as y we find T (f δ ) (x) = and the proof is complete. + x f δ (u) u du du = T(f δ )(x) As a consequence we have the following Corollary 4.2. For p < let H p be the closed subspace of L p (R) consisting of all boundary functions f of f H p (U). Then C(H p ) H p for < p <, and T(H p ) H p for p <.

14 4 ATHANASIOS G. ARVANITIDIS AND ARISTOMENIS G. SISKAKIS References [BHS] A. Brown, P. R. Halmos and A. L. Shields, Cesàro operators, Acta Sci. Math. (Seged) 26 (965) [Du] P. L. Duren, Theory of H p Spaces, Pure and Applied Mathematics, Vol. 38 Academic Press, New York-London 97. [DS] N. Dunford and J. T. Schwart, Linear Operators, Part I, Pure and Applied Mathematics, Vol. 7 Interscience Publishers, Inc., New York, 958. [Ga] J. B. Garnett, Bounded Analytic Functions, Pure and Applied Mathematics, Vol. 96, Academic Press, Inc. New York-London, 98. [HLP] G. Hardy, J. Littlewood and G. Polya, Inequalities, 2d ed., Cambridge University Press, 952. [Ho] K. Hoffman, Banach Spaces of Analytic Functions, Reprint, Dover Publications, Inc., New York, 988. [Ma] V. Matache, Composition operators on Hardy spaces of a half-plane, Proc. Amer. Math. Soc. 27 (999), [Ma2] V. Matache, Weighted composition operators on H 2 and applications, Complex Anal. Oper. Theory 2 (28), [Pa] A. Pay, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, Vol. 44, Springer-Verlag, New York, 983. [Si] A. G. Siskakis, Composition semigroups and the Cesàro operator on H p, J. London Math. Soc. (2) 36 (987), [Zy] A. Zygmund, Trigonometric Series I & II, 3rd edition, Cambridge Mathematical Library, Cambridge University Press, 22. Department of Mathematics, University of Thessaloniki, 5424 Thessaloniki, Greece address: arvanit@math.auth.gr Department of Mathematics, University of Thessaloniki, 5424 Thessaloniki, Greece address: siskakis@math.auth.gr