TOEPLITZ OPERATORS ON BLOCH-TYPE SPACES

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1 PROCEEINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 34, Number, ecember 006, Pages S ( Article electronically published on May 3, 006 TOEPLITZ OPERATORS ON BLOCH-TYPE SPACES ZHIJIAN WU, RUHAN ZHAO, AN NINA ZORBOSKA (Communicated by Joseph A. Ball Abstract. We characterize complex measures µ on the unit disk for which the Toeplitz operator T α µ,α>0, is bounded or compact on the Bloch type spaces B α.. Introduction and preliminaries Let be the unit disk on the complex plane. Let da(z = π dxdy be the normalized Lebesgue measure on. For a complex measure µ, α>0, and b L, define a Toeplitz operator as follows: Tµ α ( w α b(w (b(z =α ( wz α+ dµ(w. We also define the general Bergman projection of the measure µ and for α> as follows: ( w α P α (µ(z =(α + dµ(w. ( wz α+ The general Bergman projection of the function b is P α (b(z =(α + ( w α b(w ( wz α+ da(w. Thus Tµ α (b(z = P α (µ b (z, where dµ b (z = b(zdµ(z. Note that these choices of the indexes provide the standard notation for the general Bergman projections and for the standard case α = as follows: When α = and the measure µ is such that dµ(z =f(zda(z, with f L, we have that Tµ is the standard Toeplitz operator defined by Tµ(b =T f (b =P (fb, b L. Here P = P 0 denotes the standard Bergman projection. Recall that if f is in L, then T f is bounded on the Bergman spaces L p a, p>. Toeplitz operators have been studied extensively on the Bergman spaces by many authors. For references, see for example [9]. In this paper we study the boundedness and compactness of general Toeplitz operators on the α-bloch spaces. Received by the editors October 9, 004 and, in revised form, June 5, Mathematics Subject Classification. Primary 47B35; Secondary 3A8. Key words and phrases. Toeplitz operators, α-bloch spaces. The research of the first author was supported in part by NSF grant MS The research of the third author was supported in part by an NSERC grant. 353 c 006 American Mathematical Society Reverts to public domain 8 years from publication

2 353 ZHIJIAN WU, RUHAN ZHAO, AN NINA ZORBOSKA For α>0, the α-bloch spaces B α are the spaces of analytic functions f on such that M α (f =sup f (z ( z α <. z Each B α is a Banach space with norm of f equal to f B α = M α (f+ f(0. An analytic function on belongs to the little α-bloch space B0 α, α>0, whenever lim f (z ( z α =0. z The spaces B0 α are the subspaces of B α that are the closure of the polynomials with respect to the B α norm. The α-bloch spaces and operators on them have been studied in many different contexts. For more references and details on the next few facts stated below see [9] and [0]. For α =,B = B is the classical Bloch space. There is a natural connection between the Bloch space and the Toeplitz operators via the fact that P (L =B. A necessary condition for boundedness of T f on B is that Pf B, andsothe condition is satisfied whenever f L.Aswewillseelater,thisisnotasufficient condition for boundedness of T f on B. In general, the growth condition of a function f in B α with f(0 = 0 is determined by f(z f B α z dt ( z t.thus,forα =,wegetthat α 0 f(z f B log z, while for α>wehavethat f(z α f B α ( z. For 0 <α<, the α spaces B α are Lipschitz spaces Lip α and B α H. For 0 <α<β,wehavethatb α B β. The Bloch space B is included in all Bergman spaces L p a, p, but for large α, such as α, B α gets so large that, for example, it includes the Bergman space L a. In presenting our results we will also refer to the logarithmic Bloch space. The logarithmic Bloch space LB is the space of analytic functions f on such that sup f (z ( z log z z <. Correspondingly, the little logarithmic Bloch space LB 0 is the space of analytic functions f on such that lim f (z ( z log z z =0. Throughout the paper, in order to have the operator Tµ α,α>0 well defined, we will assume that the complex measure µ is such that, for all g in B α g(w ( w α dµ(w <. Note that in the case when dµ(z =f(zda(z, a sufficient condition on µ, sothat the integral above is finite, is that f L,whenα>, f L (log w da(w, when α =andf L (( w α da(w, when 0 <α<.

3 TOEPLITZ OPERATORS ON BLOCH-TYPE SPACES Bounded Toeplitz operators In this section we will present our main characterization of bounded Toeplitz operators on general α-bloch spaces. The result includes a restriction on the inducing measure that is a generalization of several known cases. For a complex measure µ and α>0, we will say that µ satisfies the condition R α if R α (µ(w =α( w ( z α (z w( w z α+ dµ(z L. We get another form of R α using the identity ( w = w z + w( z w: R α (µ(w =α ( z α (z w( w z α dµ(z+wp α (µ(w. The special case when the measure µ is such that dµ(z = f(zda(z, with f H, is the case when the Toeplitz operator is a multiplication operator. For f B α,aswewillseebelow,µ satisfies the condition R α if and only if f H. Arazy s ([] and Zhu s ([8] and [0] results provide a complete characterization of bounded multiplication operators on α-bloch spaces. We will state them in terms of multipliers. Theorem A ([0]. Let M(X denote the space of multipliers of the Banach space of functions X, i.e., M(X ={f X : fg X, g X }. (i If 0 <α<, thenm(b α =B α and M(B0 α=bα 0. (ii If α =,thenm(b α =M(B0 α =H ( LB. (iii If α>, thenm(b α =M(B0 α =H (. As a consequence of the above theorem we can see that, for example, not every bounded function f induces a bounded Toeplitz operator T f on the Bloch space B. As we will see later, for f L, T f is bounded on B if and only if Pf LB, which is a generalization of the analytic case of the above theorem. We will also see that there exist unbounded L functions that induce bounded Toeplitz operators on the Bloch spaces. In the next proposition we give more details about the condition R α in some special cases. Proposition.. Let µ be a measure such that dµ(z =f(zda(z, withf L. Then: (i If f L, µ satisfies the condition R α, α>0. (ii If f is analytic and in B α,thenr α (µ(w =wf(w. (iii If f is conjugate analytic, f is in B α and f(0 = 0, then R α (µ(w = w f(w. Proof. Let ψ w (z = w z wz,forz, w in. (i For α>0andf L, R α (µ(w = α( w α f ( w ( z α f(z da(z (z w( wz α+ ( z α da(z z w wz α+ cα f ( w ( w = cα f.

4 3534 ZHIJIAN WU, RUHAN ZHAO, AN NINA ZORBOSKA The second inequality follows from the fact that ( z β z w wz da(z isequivalent γ to ( w β γ+, whenever β γ +< 0, which can be easily derived from the well-known Forelli-Rudin estimates (see [6], page 7, or [8], page 53. The half-plane version of the proof of this equivalency can be found in [7]. Thus R α (µ(w L, and (i is proved. (ii Let f be in B α. With a change of variable z = ψ w (u, we get that R α (µ(w = α = α ( ψ w (u α ψ w(u ū( w ψ f ψ w (u α+ w (uda(u+wp α (f(w ( u α ū( wu α+ f ψ w(uda(u+wf(w =I + wf(w. f ψ For a fixed w we have that w (u ( wu belongs to B α whenever f B α,andso α+ ( u α f ψ w (u ( wu must be in L. Using Taylor series expansion it is not hard to α+ see that then I = 0. Hence R α (µ(w =wf(w. (iii Let f be conjugate analytic and let f(0 = 0. Then P α (f(w = 0, and as in the proof of part (ii, we have that ( u α R α (µ(w = α u( ūw α+ f ψ w(uda(u ( u α f ψ w (u f(w = α ( ūw α+ da(u, u ( u α since u( ūw da(u = 0. α+ integrable function, we get that Using the fact that f ψ w(u f(w u is an analytic R α (µ(w = f ψ w(w f(w w = f(0 f(w w = w f(w. Remark. If f from the previous proposition is harmonic, then R α (µ isalsoharmonic. If also f is a harmonic extension of an L (T function, then R α (µ(e iθ = e iθ f(e iθ. Thus, for f harmonic and in L,wehavethatR α (µ isalsoharmonic and in L,with R α (µ = f. We will use the following lemma in the proof of our next theorem. The result is known in a more general form and can be found, for example, in []. We include a proof for completeness. Lemma.. Let h L a and g B α, α>0. Then we have that Hg ᾱ h(z =(I P α (ḡh(z = g (w h(w ( w α+ (z w( wz α+ da(w.

5 TOEPLITZ OPERATORS ON BLOCH-TYPE SPACES 3535 Proof. We will use the following three formulas which can be found in [3] and [0]. For g B α and h L a : (a Hg ᾱ (ḡ(z ḡ(u ( u α h(z =(α + ( ūz α+ h(uda(u, (b g(z =g(0 + ( u α+ g (u ū( ūz α+ da(u, ( = Hg ψ ᾱ z (( (c H ᾱ g h ψ z (ψ z α+ (h ψ z (ψ z α+ Without loss of generality, let us assume that g(0 = 0. Then Hg ᾱ h(0 = (α + ( ḡ(u( u α h(uda(u ( ( w = (α + α+ g (w w( wu α+ ( w α+ ( g = (α + (w w = ( w α+ g (w h(wda(w. w Since ψ z (v = ( v ψ z(v and ψ z(v = H ᾱ g h(z = = = = = ( z α+ ( z α+ ( z α+ (( H ᾱ g h ψ z (0 (ψ z (0 α+ (H ᾱ g ψz ((h ψ z (ψ z α+ (0. da(w ( u α h(uda(u ( u α h(u ( ūw α+ da(u da(w z ( zv = ψ z (ψ z(v,wehave: ( w α+ (g ψ z (w h ψ z (w(ψ w z(w α+ da(w g (v h(v ( v α+ ( z ( zv( vz (z v( vz α+ ( z zv da(v g (v h(v ( v α+ (z v( vz α+ da(v. We have the following theorem: Theorem.. Suppose µ satisfies the condition R α, i.e., ( R α (µ(w =α( w ( z α (z w( w z α+ dµ(z L. Then we have: (i If 0 <α<, thentµ α is bounded on B α if and only if P α (µ B α. (ii If α =,thentµ α is bounded on B α if and only if P α (µ LB. (iii If α>, thentµ α is bounded on B α if and only if P α (µ B. Proof. Let g B α and h L a. It is known that (L a = B α under the integral pairing h, g α = α h(zg(z( z α da(z. See [0, Theorem 4] for more details.

6 3536 ZHIJIAN WU, RUHAN ZHAO, AN NINA ZORBOSKA Thus, by Fubini s Theorem, h, Tµ α (g α = α h(ztµ α (g(z( z α da(z = α h(zg(z( z α d µ(z = α P α (hḡ(z( z α d µ(z + α [(I P α (hḡ](z( z α d µ(z =I + I. From Lemma. we have that [(I P α (hḡ](z = g (wh(w( w α+ (z w( wz α+ da(w. Hence, by (, I = α g (wh(w( w α+ (z w( wz α+ da(w( z α d µ(z = g (wh(w( w α R α (µ(w da(w g B α h R α (µ C<. On the other hand, by Fubini s Theorem, I = h(wg(wq α (µ(w( w α da(w, where Q α (µ(w =α(α + Hence Tµ α (g Bα if and only if ( z α ( zw α+ dµ(z. ( ( w α g(wq α (µ(w L. The relation between Q α (µ andp α (µ is (3 Q α (µ(w =(α +P α (µ(w+wp α (µ(w. It is easy to see that P α (µ isinb α, LB and B as 0 <α<, α =andα>, respectively, implies that Q α (µ satisfies, respectively: Q α (µ(w( w α L, if 0 <α<; Q α (µ(w( w log w L, if α =; Q α (µ(w( w L, if α>. Notice that B α H as 0 <α< and using the fact that, as α>, g B α if and only if ( w α g(w L, we easily see that ( is true for all these cases. Thus, Tµ α is bounded on B α. Conversely, if Tµ α is bounded on B α,thentµ α ( = P α (µ B α. If R α (µ L, as above, we get that there is a constant c>0, independent of g, such that ( w α g(wq α (µ(w c g B α. For 0 <α<, we have already shown that P α (µ B α.

7 TOEPLITZ OPERATORS ON BLOCH-TYPE SPACES 3537 For α, using test functions g a (w =log āw and g a(w =( āw α,as α = andα >, respectively, we get Q α (µ(w( w (4 log w L if α =; (5 Q α (µ(w( w L if α>. If α =,P α (µ B implies that P α (µ(w( w log w L. Thus, using (3 and (4, we easily see that P α (µ (w( w log w L. If α>, then P α (µ B α implies that (6 P α (µ(w( w α L. For α, this implies that P α (µ(w( w L, and using (3 and (5, we get that P α (µ B. For α >, combining (3, (5 and (6 we get P α (µ (w( w α L, which is equivalent to (7 P α (µ(w( w α L. If α >, combining (3, (5 and (7 we get that P α (µ (w( w α is in L, and we continue the process from above until we reach a positive integer n such that α n andp α (µ (w( w α n L. Then, P α (µ (w( w L, and so P α (µ B. This completes the proof. Note that in the case dµ(z =f(zda(z, with f B α, it follows from part (ii of Proposition. that R α (µ(w =wf(w. Thus µ satisfies the condition R α if and only if f H. Corollary.. Let f L and let Tf α(g =P α (fg for g B α. Then we have: (i If 0 <α<, thentf α is bounded on Bα if and only if P α (f B α. (ii If α =,thentf α is bounded on Bα if and only if P α (f LB. (iii If α>, thentf α is bounded on Bα if and only if P α (f B. Proof. Follows from Proposition. and Theorem.. Since the Bloch space B is the dual of L a and T µ = T µ under this duality, we get the following corollary. Similar results including a few other restrictions can be found in [4] and [8]. Corollary.. Let µ be a complex measure satisfying condition R α. Then T µ is bounded on L a if and only if P (µ LB. Remark. The proof of Theorem. implies that the condition R α in the theorem can be replaced by a more general condition involving Carleson measures. Recall that for a positive Borel measure ν on we say that ν is a Carleson measure on the Bergman space if there exists c>0 such that, for all h L a, h(z dν(z c h(z da(z. Let ν α (µ be the positive measure defined by dν α (µ(z = R α (µ(z da(z. If µ satisfies condition R α, then it is easy to see that ν α (µ is a Carleson measure. By inspecting the proof of Theorem. we can see that, whenever R α (µ isinl,

8 3538 ZHIJIAN WU, RUHAN ZHAO, AN NINA ZORBOSKA the statements (i, (ii and (iii of Theorem. are still true if the condition R α is replaced by a weaker one, namely, by requiring only that the measure ν α (µ isa Carleson measure for the Bergman space. Note that the measure ν α (µ is a Carleson measure for the Bergman space if and only if there exists r>0, such that sup z (z, r (z,r R α (µ(w da(w <, where (z, r denotes the hyperbolic disk with center z and radius r. See [9, Theorem 6..] for details. Next we give an example of an unbounded L function that induces a bounded Toeplitz operator on B α. We present part of the calculations as a separate lemma that covers a wider class of functions. Lemma.. Let f(z = n=0 a n( z n be a function in R α (µ(w =d α w ᾱ w where w = {z : w < z < }. n=0 a n n+α L (( z α da(z, for α > 0, and such that is finite. Let d α = α a n n=0 n+α and let F α(f = f(z( z α da(z. Then, for dµ(z =f(zda(z we have ( F α (f ( w α f(z( z α da(z, w Proof. As in the proof of part (ii of Proposition. we have that, for w 0, ( u α R α (µ(w =wp α (f(w α ū( wu α+ f ψ w(uda(u. Using the given series expansion of f and the identity ψ w (u = ( w ( u we get that P α (f(w =α a n n=0 n+α = d α, and that the above integral equals a n ( w n ( u n+α ( ūw n ( wu n+α+ da(u. ū n=0 Since ( ūw n ū (n + α is a conjugate analytic, integrable function, we have that ( ūw n ( u n+α ū ( wu n+α+ da(u =( w n. ū ( u n+α wu, Since also ū( wu da(u = 0, we have that, furthermore, n+α+ a n R α (µ(w = d α w α n + α ( w n ( w n w n= ( = d α w ᾱ a n w n + α a n n + α ( w n n=0 n=0 = d α w ᾱ ( F α (f w ( w α f(z( z α da(z. w

9 TOEPLITZ OPERATORS ON BLOCH-TYPE SPACES 3539 Example. For α>0, let c α = α f α (z = c α ( n= n= n(n+α and for z 0,let n ( z n =+c α log( z. The function f α is a radial, unbounded, L function. Since P α (f α isaradial analytic function, it must be a constant. It is easy to check that P α (f α (0 = 0 and so P α (f α =0. In order to show that T fα is bounded on B α we only have to prove that µ α,with dµ α (z =f α (zda(z, satisfies the condition R α. Using Lemma. and the fact that d α =0,wegetthat ( R α (µ α (w = w +αc α ( w α ( t α log tdt. w It is easy to see that lim w R α (µ(w =. Since 0 ( tα log tdt = αc α,we also have that lim w 0 R α (µ(w = α. Thus, R α (µ α L,andbyTheorem., T fα is bounded on B α. Note that we can also use Lemma. to show that R α (µ L is not a necessary condition for the boundedness of the Toeplitz operator T µ. For example, let dµ(z = f(zda(z withf(z =. Note that for this radial, positive function F z 3/ α (f is equal to Γ(/4Γ(α Γ(α+/4, and d α is equivalent to α n= n /4 (n+α. UsingLemma.we can get that R α (µ(w c and so R w α (µ isnotinl. Since the measure ν α (µ is a Carleson measure, T µ is still bounded on B α. 3. Compact Toeplitz operators For the next result, we need the following lemma from [5]. Lemma 3.. Let 0 <α< and let T be a bounded linear operator from B α into a normed linear space Y.ThenT is compact if and only if Tg n Y 0, whenever (g n is a bounded sequence in B α that converges to 0 uniformly on. Theorem 3.. Suppose (8 lim w R α(µ(w =0. Then we have: (i If 0 <α<, thentµ α is compact on B α if and only if P α (µ B α. (ii If α =,thentµ α is compact on Bα if and only if P α (µ LB 0. (iii If α>, thentµ α is compact on B α if and only if P α (µ B 0. Proof. For the case α = or α >, let {g n } be a sequence in B α such that g n B α andg n (z 0 uniformly on compact subsets of. Let h be in the

10 3540 ZHIJIAN WU, RUHAN ZHAO, AN NINA ZORBOSKA unit ball of L a. Similarly, as in the proof of Theorem., we have h, Tµ α (g n α = α P α (hḡ n (z( z α d µ(z + α [(I P α (hḡ n ](z( z α d µ(z = I + I. For 0 <r<, and r = {z : z r}, I = α g n(wh(w( w α R α (µ(w da(w ( = α g n(wh(w( w α R α (µ(w da(w =K + K. r + \ r For a fixed ε>0, using condition (8, let r be sufficiently close to so that R α (µ(w <εas w \ r.then K ε g n B α h ε. Since g n (z 0 as n, we can choose n big enough so that g n (z ( w α <ε. Therefore, K ε R α (µ h ε R α (µ. Hence I <Cε,whereC does not depend on h, and so lim n sup h I =0. ThusTµ α is compact on B α if and only if sup h I 0asn. Similarly, as in the proof of Theorem., we have I = h(wg n (wq α (µ(w( w α da(w ( = h(wg n (wq α (µ(w( w α da(w =M + M. r + \ r It is easy to see that if P α (µisinlb 0 and B 0 as α =andα>, respectively, then, as w, Q α satisfies, respectively: Q α (µ(w( w log 0, if α =; w Q α (µ(w( w 0, if α>. Again, notice that, as α>, g B α if and only if ( z α g(z L, and we may choose r sufficiently close to such that whenever w \ r, Q α (µ(w ( w log w <εas α =;and Q α (µ(w ( w α <εas α>, respectively. We see that M εα g n B α h Cε, where C does not depend on h. Since g n (z 0 uniformly on compact subsets of, wecanchoosen big enough so that g n (w <ε. Hence we easily see that, as n is big enough, M <Cε,where C does not depend on h. Therefore, sup h I 0asn,andsoTµ α is compact on B α as α =orα>. Now let 0 <α<. Let {g n } be a sequence in B α such that g n B α and g n (z 0 uniformly on. Leth be in the unit ball of L a. By a similar discussion as above, Tµ α is compact on B α if and only if sup h I 0asn. However,

11 TOEPLITZ OPERATORS ON BLOCH-TYPE SPACES 354 since g n B α andg n (z 0 uniformly on, wecanchoosen big enough such that g n (w <εuniformly for w. Thus, as n is big enough, I εα P α (µ B α h. Therefore, sup h I 0asn,andsoTµ α is compact on B α. Conversely, if Tµ α is compact on B α. Suppose first that α =orα>. Then sup h I 0asn for any sequence g n in B α such that g n B α and g n (z 0 uniformly on compact subsets of. Let h z (w = ( z α ( zw +α.then h z C, and I = ( z α g n (wq α (µ(w( w α ( zw +α da(w = (α + ( z α g n (zq α (µ(z. Thus, sup h I 0asn implies (9 lim sup n z ( z α g n (z Q α (µ(z =0. As α =orα>, using testing functions g a (z = ( ( log a log and g a (z =( a ( āz α respectively in (9, as in the proof of Theorem., we easily see that P α (µ isinlb 0 and B 0, respectively. Now let 0 <α<andlettµ α be compact on B α.thentµ α is bounded on B α. By Theorem., P α (µ B α. The proof is completed. Remark. The requirement that lim w R α (µ(w = 0, in the proof of Theorem 3., can be replaced by a more general requirement that the measure ν α (µ, defined in Section, is a compact Carleson measure. Recall that for a positive Borel measure ν on we say that ν is a compact Carleson measure whenever lim z dν(w =0, (z, r (z,r āz where (z, r denotes the hyperbolic disk with center z and radius r. See [9, Theorem 6..5] for more details. If the measure µ is such that lim w R α (µ(w = 0, then it is easy to see that ν α (µ isacompactcarlesonmeasure. Note that in the case dµ(z =f(zda(z, with f B α, it follows from part (ii of Proposition. that R α (µ(w =wf(w. Thus the condition (8 in Theorem 3. is satisfied only when f =0. References [] J. Arazy, Multipliers of Bloch Functions, University of Haifa Mathem. Public. Series 54 (98. [] J. Arazy, S. Fisher, S. Janson, J. Peetre, An Identity for Reproducing Kernels in a Planar omain and Hilbert-Schmidt Hankel Operators, J. Reine Angew. Math. 406 (990, [3] J. Arazy, S. Fisher, J. Peetre, Hankel Operators on Weighted Bergman Spaces, American J. Math. 0 (988, MR09709 (90a:47067 [4] K.R.M.Attele,Toeplitz and Hankel Operators on Bergman One Space, Hokkaido Math. J. (99, MR69795 (93d:4705 [5] S. Ohno, K. Stroethoff, R. Zhao, Weighted Composition Operators between Bloch-Type spaces, Rocky Mountain J. Math. 33 (003, 9-5. MR (004d:47058

12 354 ZHIJIAN WU, RUHAN ZHAO, AN NINA ZORBOSKA [6] W. Rudin, Function Theory in the Unit Ball in C n, Springer-Verlag, New York, 980. MR (8i:300 [7] J. L. Wang, Z. Wu, Minimum solutions of δ k+ and Middle Hankel Operators, J. Funct. Anal. 8 (993, MR4560 (94j:4704 [8] K. Zhu, Multipliers of BMO in the Bergman Metric with Applications to Toeplitz Operators, J. Funct. Anal. 87 (989, MR0588 (90m:404 [9] K. Zhu, Operator Theory on Function Spaces, Marcel ekker, New York, 990. MR (9c:4703 [0] K. Zhu, Bloch-Type Spaces of Analytic Functions, Rocky Mountain J. Math. 3 (993, MR4547 (95d:4600 epartment of Mathematics, University of Alabama, Tuscaloosa, Alabama address: zwu@gp.as.ua.edu epartment of Mathematics, University of Toledo, Toledo, Ohio address: Ruhan.Zhao@utoledo.edu Current address: epartment of Mathematics, SUNY Brockport, Brockport, New York 440 address: rzhao@brockport.edu epartment of Mathematics, University of Manitoba, Winnipeg, Manitoba, Canada R3T N address: zorbosk@cc.umanitoba.ca