WEIGHTED COMPOSITION OPERATORS BETWEEN DIRICHLET SPACES

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1 Acta Mathematica Scientia 20,3B(2): WEIGHTE COMPOSITION OPERATORS BETWEEN IRICHLET SPACES Wang Maofa ( ) School of Mathematics and Statistics, Wuhan University, Wuhan , China whuwmf@63.com Abstract In this article, we study the boundedness of weighted composition operators between different vector-valued irichlet spaces. Some sufficient and necessary conditions for such operators to be bounded are obtained exactly, which are different completely from the scalar-valued case. As applications, we show that these vector-valued irichlet spaces are different counterparts of the classical scalar-valued irichlet space and characterize the boundedness of multiplication operators between these different spaces. Key words Vector-valued analytic function; irichlet space; weighted composition operator; boundedness 2000 MR Subject Classification 47B33 Introduction Throughout this article, denotes the open unit disk {z C : z < } in the complex plane C. Let X be any complex Banach space and α >, the vector-valued weighted Bergman space A 2 α (X) consists of all analytic functions f : X such that ( /2 f A 2 α (X) = f(z) 2 X α(z)) da <, where da α (z) = (α + )( z 2 ) α da(z) and da is the normalized Lebesgue area measure on. The vector-valued weighted irichlet space α (X) is the collection of all analytic functions f : X for which the derivatives f belong to A 2 α (X). Note that the irichlet space α(x) is a Banach space with the norm /2 f α(x) = ( f(0) 2X + f (z) 2 X α(z)) da. Especially, α (C) is a reproducing kernel Hilbert space with the obvious inner product, that is, for any w in, there is a corresponding reproducing kernel K w in α (C), such that the inner Received November 24, 2007; revised April 6, 200. This work is partially supported by the National Natural Science Foundation of China (09058)

2 642 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B product f, K w α(c) = f(w) for all f in α (C). Also, the analytic function f(z) = a n z n : n C belongs to α (C) if and only if (n + ) α a n 2 <. These kinds of vector-valued n spaces were intensively studied, see e.g., 5 for a recent survey of vector-valued Bergman spaces. A novel feature in the vector-valued function theory is that there often are several natural ways to introduce a vector-valued counterpart of a given scalar-valued function space. The following weak version of vector-valued function spaces were considered by e.g., Blasco 3 and Bonet et al 6: the weak version irichlet space w α (X) consists of those analytic functions f : X for which f wα(x) = sup x f α(c) <, x X here X is the dual space of X. In fact, such kinds of weak version spaces we(x) can be introduced under more general conditions on any Banach spaces E consisting of analytic functions f : C, see e.g., 6, 5, 6. Some strong and weak version spaces are completely different such as Hardy spaces H 2 (X) and wh 2 (X) by constructing some concrete examples in, 6. Others are the same such as Bloch spaces B(X) and wb(x), refer to. It seems reasonable to ask whether the irichlet spaces w α (X) and α (X) are the same. In this article, we will prove indirectly that w α (X) and α (X) are different significantly for any infinite dimensional complex Banach space X by studying the boundedness of weighted composition operators between these strong and weak version vector-valued irichlet spaces. Let u be a scalar-valued analytic function on and ϕ an analytic self-map of. We can define a linear operator uc ϕ, called a weighted composition operator, on the space of analytic functions on by uc ϕ (f) = u(f ϕ). It is seen that this operator can be regarded as a generalization of the multiplication operator M u (that is, ϕ(z) z) and the composition operator C ϕ (that is, u(z) ). Composition operators and multiplication operators can act on various spaces of functions analytic on. In each case, the main goal is to discover the connection between the properties of the inducing symbols ϕ, u, and the operator theoretic properties of C ϕ and M u. References for most of the known results on composition operators can be found in 7, 23. It is well known that uc ϕ is not automatically bounded even on the scalar-valued case α (C), refer to 20, 2, 29. Some necessary and sufficient conditions involving Carleson measure for uc ϕ to be bounded were given in 7, 3, 4, 2. Zorboska 29 also gave the boundedness in terms of an integral average of some determining function for the operator. It is checked that uc ϕ : w α (X) w α (X) is bounded if and only if uc ϕ : α (C) α (C) is bounded if and only if uc ϕ : α (X) α (X) is bounded, see more general case in 6, 6. Now, it is natural to consider whether it is possible to characterize the class of analytic maps ϕ : and analytic functions u on for which the operators uc ϕ to be bounded between these strong and weak version irichlet spaces. This problem is motivated by the fact that this kind of strong and weak Bergman spaces A 2 α (X) and wa 2 α (X) are different significantly for any infinite dimensional Banach space X and α. (Here, we use α = to denote the corresponding Hardy spaces). In fact, A 2 α(x) wa 2 α(x) and the norm A 2 α (X) is not equivalent to wa 2 α (X) on A 2 α (X), see 5 7 for more details and some concrete examples. Thus, we hope the properties of uc ϕ from w α (X) to β (X)

3 No.2 Wang: WEIGHTE COMPOSITION OPERATORS BETWEEN IRICHLET SPACES 643 could reflect some differences between these weak and strong version vector-valued irichlet spaces. Note that w α (C) = α (C) for any α > (for simplification, we often denoted the space α (C) by α ), so it goes back to the classical case. An obvious necessary condition for the boundedness of uc ϕ : w α (X) β (X) is that u, uϕ β. In fact, for any fixed 0 x 0 X, f(z) x 0 w α (X), then, uc ϕ f = u(z)x 0 β (X), that is, u β. Similarly, by setting f(z) = zx 0, then, uϕ β. Under these assumptions, our main results show that, for α > 0, β > and any infinite dimensional Banach space X, the operator uc ϕ : w α (X) β (X) is bounded if and only if ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z) <. Maybe the appearance of the above condition is surprising. As we will see that ϕ and u satisfy the above condition if and only if uc ϕ : α β is a Hilbert-Schmidt operator (that is, uc ϕ e n 2 β < for some orthonormal basis {e n } n= of α ). n= That we focus only on the infinite dimensional complex Banach space X is due to w α (X) = α (X) for α > and any finite dimensional complex Banach space X, which goes back to the classical case again, such as for the case X = C in 7, 3, 4, 20, 29. As a by-product, we answer positively the question mentioned early: these strong and weak version irichlet spaces w α (X) and α (X) are different completely for any infinite dimensional complex Banach space X. Moreover, we obtain the equivalence of some analysis properties of analytic self-map ϕ : and analytic function u : C. Simultaneously, we also obtain immediately the conditions of boundedness of multiplication operators and composition operators between these different vector-valued irichlet spaces respectively, which are also new in the literature. To keep our notation simple, we shall frequently use the letters C or c to denote some absolute positive constant allowing it to change at each occurrence. 2 Main Results and Their Proofs In this section, we first give the following Hilbert-Schmidt characterizations of weighted composition operators between scalar-valued irichlet spaces, up to our knowledge, which seem to have not been made explicit in the literature. Theorem For α, β >, let u be an analytic function on and ϕ an analytic self-map of. Then, the following hold. () For α > 0, uc ϕ : α β is Hilbert-Schmidt if and only if ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z) <. (2) For α = 0, uc ϕ : α β is Hilbert-Schmidt if and only if log ϕ(z) 2 + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) 2 da β (z) <. (3) For α < 0, uc ϕ : α β is Hilbert-Schmidt if and only if u β and u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β(z) <.

4 644 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B Proof () Because every weighted irichlet space α is disk-automorphism invariant, that is, all composition operators whose symbol is an automorphism of are bounded, and the Hilbert-Schmidt class is a two-sided operator ideal of all bounded linear operators on α, we can suppose that ϕ(0) = 0 without loss of generality. We first notice that the above integral condition implies uc ϕ : α β is bounded. In fact, any analytic function f : C satisfies f(z) 2 ( z 2 ) f 2 α+2 A for any z (see e.g. ), so f (z) 2 2 α ( z 2 ) f 2 α+2 α for any z. And for any f α, f(z) = f, K z α f α K z α, K z is the reproducing kernel of α and K z 2 α = n=0 β(n) 2 = z n 2 α = = (α + ) z 2n β(n) 2 ( z 2 ) α, which is due to 2π 0 0 (z n ) 2 (α + )( z 2 ) α da(z) n 2 r 2(n ) ( r 2 ) α rdr dθ π = (α + )n 2 r 2(n ) ( r 2 ) α 2rdr 0 = (α + )n 2 x n ( x) α dx 0 2 Γ(n)Γ(α + ) = (α + )n Γ(n + α + ) n α, here, Γ is the Gamma function and the last estimate comes from the Stirling s formula. The relation A B means c A B c 2 A for some inconsequential constants c, c 2 > 0. In the sequel, we also use A B to denote A cb for some positive constant c. Thus, for any f α, uc ϕ f 2 β = u(0)f(0) 2 + u (f ϕ) + u(f ϕ)ϕ 2 da β (z) u(0)f(0) 2 ( + 2 u (f ϕ) 2 + u(f ϕ)ϕ 2) da β (z) f 2 α u(0) 2 + f 2 α ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z). ( ) That is, uc ϕ : α β is bounded, which implies that u, uϕ β. As { zn β(n) } n=0 is an orthonormal basis of α and ϕ(0) 2n β(n) <, we verify that uc 2 ϕ : α β is Hilbert-Schmidt operator if and only if z n (uϕ n ) (z) 2 > uc ϕ β(n) 2 β n=0 n=0 β(n) 2 da β (z) ( ) = β(n) 2 u ϕ n + unϕ n ϕ 2 + u 2 da β (z) n= ( ) = β(n) 2 u 2 ϕ n 2 + n 2 u 2 ϕ n 2 ϕ 2 + 2Reu ϕ n nuϕ n ϕ + u 2 da β (z) n= ( n +α u 2 ϕ 2n + n +α u 2 ϕ 2 ϕ 2n 2 n= n= n=0

5 No.2 Wang: WEIGHTE COMPOSITION OPERATORS BETWEEN IRICHLET SPACES n α 2Reu ϕ ϕ 2(n ) uϕ )da β (z) ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 + (z)ϕ(z)u(z)ϕ (z) 2Reu ( ϕ(z) 2 ) α+ da β (z), ( ) n= and the last estimate is due to n t ϕ(z) 2n ( ϕ(z) 2 ) t for t > 0 as ϕ(z). Again as u, uϕ β, namely, u ϕ, uϕ L 2 (, da β ), then, u ϕuϕ L (, da β ). Thus, 2Reu (z)ϕ(z)u(z)ϕ (z)da β (z) 2Re u (z)ϕ(z)u(z)ϕ (z) ( ϕ(z) 2 ) α+ da β(z) ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z). So, uc ϕ : α β is Hilbert-Schmidt operator if and only if ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z) <. and The parts (2) and (3) are similar, only need to notice that in the above formula ( ) n t ϕ(z) 2n = log ϕ(z) 2 for t = 0 n= n t ϕ(z) 2n for t < 0. n= Applying the estimates K z 2 α log z for α = 0 and K 2 z 2 α for < α < 0 in the formula ( ) will complete the proof. Now, we estimate the norm of the weighted composition operator uc ϕ : w α (X) β (X) for arbitrary infinite dimensional complex Banach space X and α, β >. We will use the following voretzky s well-known theorem. Lemma 2 8 If X is an infinite dimensional complex Banach space, then for any ε > 0 and n N, there is a linear embedding T n : ln 2 X, such that ( n ) /2 n ( X n ) /2 ( + ε) a j 2 a j T n e j a j 2 j= j= for any scalars a, a 2,,a n and some orthonormal basis {e,, e n } of ln. 2 Theorem 3 Let X be any infinite dimensional complex Banach space, α, β >. u, ϕ are scalar-valued analytic functions on such that ϕ(). Then the following hold. () For α > 0, uc ϕ : w α (X) β (X) is bounded if and only if ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z) <. Moreover, uc ϕ : w α (X) β (X) 2 j= ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z).

6 646 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B (2) For α = 0, uc ϕ : w α (X) β (X) is bounded if and only if Moreover, log ϕ(z) 2 + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) 2 da β (z) <. uc ϕ : w α (X) β (X) 2 log ϕ(z) 2 + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) 2 da β (z). (3) For α < 0, uc ϕ : w α (X) β (X) is bounded if and only if u β and Moreover, uc ϕ : w α (X) β (X) 2 u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β(z) <. + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z). Proof We only prove the case that α > 0. The other cases are similar and we omit those to the reader. As for any f α, and Thus, and f (z) 2 X = f (z) 2 f(z) 2 ( z 2 ) α+2 f 2 α, z, ( z 2 ) α f 2 α, z. sup (x f) (z) 2 x X ( z 2 ) α+2 f 2 w, α(x) f(z) 2 X ( z 2 ) α f 2 w α(x), for any f w α (X). Consequently, uc ϕ f 2 β (X) = u(0)f(ϕ(0)) 2 X + u (f ϕ) + u(f ϕ)ϕ 2 XdA β (z) u(0)f(ϕ(0)) 2 X + ( u (f ϕ) X + u(f ϕ)ϕ X ) 2 da β (z) u(0)f(ϕ(0)) 2 X + ( u (f ϕ) 2 X + u(f ϕ)ϕ 2 ) X daβ (z) f 2 w α(x) u(0) 2 K ϕ(0) 2 α + f 2 w α(x) ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z). So that uc ϕ : w α (X) β (X) 2 ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z).

7 No.2 Wang: WEIGHTE COMPOSITION OPERATORS BETWEEN IRICHLET SPACES 647 For the converse estimate, we first consider the easy case that X = l 2. efine the function f as following f(z) = k α 2 z k+ e k+, z, where {e k } is some fixed orthonormal basis of l 2. Then, we have f (z) 2 l = 2 (k + )k α 2 2 z k e k+ = (k + ) 2 k α z 2k k α+ z 2k In addition, for any x X = l 2, So, Thus, and then, x f(z) = l 2 k α 2 z k+ x e k+. ( z 2 ) α+2. x f 2 α (k + ) α k α 2 x e k+ 2 x e k+ 2 x 2 X. f wα(x) = sup x f α c, x X uc ϕ : w α (X) β (X) 2 c uc ϕ f 2 β (X) c u (f ϕ) + u(f ϕ)ϕ 2 X=l 2dA β(z) = c u 2 f ϕ 2 l + 2 u 2 ϕ 2 f ϕ 2 l + 2 2Reu uϕ f ϕ, f ϕ l 2dA β = c ( u 2 k α ϕ(z) 2(k+) + u 2 ϕ 2 (k + ) 2 k α ϕ(z) 2k +2Reu uϕ c c c ) k α (k + ) ϕ 2k ϕ da β (z) ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 + (z)ϕ(z)u(z)ϕ (z) 2Reu ( ϕ(z) 2 ) α+ da β (z) ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 + 2Reu (z)ϕ(z)u(z)ϕ (z) da β (z) ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z), and the last inequality is due to u, uϕ β whenever uc ϕ : w α (X) β (X) is bounded. For more general complex Banach space X with dimx =, we will use the voretzky s theorem to reduce it to Hilbert case, which is easily managed. For any ǫ > 0 and n N, there exists a linear embedding T n : ln 2 X as in Lemma 2, such that ( n ) /2 n ( X n ) /2 ( + ε) a j 2 a j T n e j a j 2 j= j= j=

8 648 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B for any scalars a, a 2,,a n and some orthonormal basis {e,,e n } of ln 2. efine functions g n by Then, x g n 2 α g n (z) = n n x T n (e k ) 2 = k α 2 z k+ T n e k, z. n Tn x (e k ) 2 Tn x 2 Tn 2 x 2 X, here T n is the adjoint operator of T n. So again g n wα(x) c T n = c T n c and then, uc ϕ : w α (X) β (X) 2 c liminf uc ϕg n 2 n β (X) c liminf u g n (ϕ(z)) + ug n n (ϕ(z))ϕ 2 X da β(z) c n liminf n ( + ǫ) 2 u k α 2 ϕ k+ (z) + uk α 2 (k + )ϕ k (z)ϕ 2 da β (z) liminf n c ( + ǫ) 2 +2Reu uϕϕ c ( + ǫ) 2 +2Reu uϕϕ c ( + ǫ) 2 c ( + ǫ) 2 n ( n n u 2 k α ϕ 2(k+) + u 2 ϕ 2 k α (k + ) 2 ϕ 2k k α (k + ) ϕ 2k )da β ( u 2 k α ϕ 2(k+) + u 2 ϕ 2 k α (k + ) 2 ϕ 2k k α (k + ) ϕ 2k )da β ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 + (z)ϕ(z)u(z)ϕ (z) 2Reu ( ϕ(z) 2 ) α+ da β (z) ( ϕ(z) 2 ) α + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β (z), The sixth line in the above formula is due to Fatou s lemma 0. As ǫ > 0 is arbitrary, we obtain the desired lower bound estimate. From Theorems and 3, it is easy to obtain the following corollary. Corollary 4 Let X be any infinite dimensional complex Banach space, α, β >, u and ϕ be analytic functions on such that ϕ(). Then, uc ϕ : w α (X) β (X) is bounded if and only if uc ϕ : α β is Hilbert-Schmidt. We knew uc ϕ : w α (X) w α (X) is bounded if and only if uc ϕ : α α is bounded, which is strictly weaker than its Hilbert-Schmidt condition. For example, for u(z), ϕ(z) = a z az, a, the operator uc ϕ : α α is bounded, but uc ϕ : w α (X) α (X) is not bounded for any X with dimx =. So, we conclude the following interesting corollary. Corollary 5 For any complex Banach space X with dimx = and α >, the irichlet spaces α (X) and w α (X) are different significantly. By setting u(z) in Theorem 3, we immediately obtain the following corollary.

9 No.2 Wang: WEIGHTE COMPOSITION OPERATORS BETWEEN IRICHLET SPACES 649 Corollary 6 Let X be any infinite dimensional complex Banach space, α, β > and ϕ an analytic self-map of. Then, C ϕ : w α (X) β (X) is bounded if and only if ϕ (z) 2 ( ϕ(z) 2 ) α+2 da β(z) <. By setting ϕ(z) z in Theorem 3, we also obtain the following corollary. Corollary 7 Let X be any infinite dimensional complex Banach space, α, β > and u an analytic function on. Then the following hold. () For α > 0, the multiplication operator M u : w α (X) β (X) is bounded if and only if ( z 2 ) α + u(z) 2 ( z 2 ) α+2 da β (z) <. (2) For α = 0, the multiplication operator M u : w α (X) β (X) is bounded if and only if log z 2 + u(z) 2 ( z 2 ) 2 da β (z) <. (3) For α < 0, the multiplication operator M u : w α (X) β (X) is bounded if and only if u β and u(z) 2 ( z 2 ) α+2 da β(z) <. After characterizing the boundedness of the weighted composition operator uc ϕ : w α (X) β (X), maybe it is a proper question to ask the compactness of uc ϕ. In fact, if uc ϕ : w α (X) β (X) is compact, then for any bounded sequence {x n } in X and f n (z) = x n, {f n } is a bounded sequence in w α (X) since f n wα(x) = x n X, there exists a subsequence {f nk } by the definition of compact operators such that {uc ϕ f nk } is norm convergent in β (X). On the other hand, uc ϕ f n (z) = u(z)x n. So, {u(z)x nk } is norm convergent in β (X). Again f(z) X = sup x f(z) x X, x X K z β f β (X), for any f β (X), {uc ϕ f nk } converges uniformly on any compact subset of, especially it is pointwise convergent. Hence, if u is not the constant 0-function, then for any bounded sequence {x n } in X, there exists a subsequence {x nk }, such that it is norm convergence in X, so X must be finite dimensional Banach space by Bolzano-Weierstrass theorem 9. Namely, for any infinite dimension Banach space X, uc ϕ : w α (X) β (X) is never compact, with the exception of the trivial case that u(z) 0. As for the weak compactness, by a similar analysis, an obvious necessary condition for the weak compactness of uc ϕ : w α (X) β (X) is the reflexivity of X. Under this assumption, the boundedness and the weak compactness of uc ϕ are equivalent, as β (X) is reflexive too. So, nothing needs to be studied any more. 3 Some Comments In the last section, we give some equivalent characterizations of analytic functions u, ϕ on with ϕ(), which are of independent interest. First, it is worth noticing that a similar argument as above in Section 2 can give:

10 650 ACTA MATHEMATICA SCIENTIA Vol.3 Ser.B For α, β >, uc ϕ : wa 2 α (X) A2 β (X) is bounded for any infinite dimensional complex Banach space X if and only if uc ϕ : A 2 α A 2 β is Hilbert-Schmidt if and only if u(z) 2 ( ϕ(z) 2 ) α+2 da β(z) <, where wa 2 α (X) is the space of all analytic functions f : X such that sup x x X f A 2 α (C) <. As for α > the weighted irichlet space α is the weighted Bergman space A 2 α 2 with an equivalent norm (see Lemma 2.3 in 24), so our main results actually give the equivalence of the following analysis properties: For any analytic functions u, ϕ on such that ϕ() and α, β >, then if and only if u(z) 2 ( ϕ(z) 2 ) α+2 da β(z) < ( ϕ(z) 2 ) α+2 + u(z) 2 ϕ (z) 2 ( ϕ(z) 2 ) α+4 da β+2 (z) <. Especially, by setting u(z) and ϕ(z) z, respectively, we have the following facts. For any analytic self-map ϕ : and α, β >, ( ϕ(z) 2 ) α+2 da β(z) < if and only if ϕ (z) 2 ( ϕ(z) 2 ) α+4 da β+2(z) <. For any analytic function u : C and α, β >, u(z) 2 ( z 2 ) α+2 da β(z) < if and only if ( z 2 ) αda β(z) <. The last fact above, which is interesting for β α, can be used to simplify Corollary 7 in the following way, which is a generalization of the main results in 27. Theorem 8 For any infinite dimensional complex Banach space X and α, β >, let u be an analytic function on. Then, the following hold. () For α > 0, M u : w α (X) β (X) is bounded if and only if ( z 2 ) αda β(z) <. (2) For α = 0, M u : w α (X) β (X) is bounded if and only if log z 2dA β(z) <. (3) For α < 0, M u : w α (X) β (X) is bounded if and only if u β.

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