BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE

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1 BLOCH FUNCTIONS ON THE UNIT BALL OF AN INFINITE DIMENSIONAL HILBERT SPACE OSCAR BLASCO, PABLO GALINDO, AND ALEJANDRO MIRALLES Abstract. The Bloch space has been studied on the open unit disk of C and some homogeneous domains of C n. We define Bloch functions on the open unit ball of a Hilbert space E and prove that the corresponding space B(B E) is invariant under composition with the automorphisms of the ball, leading to a norm that- modulo the constant functions - is automorphism invariant as well. All bounded analytic functions on B E are also Bloch functions. Introduction The classical Bloch space B of analytic functions on the open unit disk D of C plays an important role in geometric function theory and it has been studied by many authors. R. M. Timoney ([6] and [7]) etended the notion of Bloch function by considering bounded homogeneous domains in C n, such as the unit ball B n and the polydisk D n. In this article, Bloch functions on the unit ball B E of an infinite-dimensional Hilbert space E are introduced. We prove that a number results about Bloch functions on D and B n can be etended to this infinite dimensional setting. Among them, several characterizations of Bloch functions on B E known to hold in the finite dimensional case. First, we will recall some background about the classical Bloch space and the space of Bloch functions on B n. In Section 2, we will introduce the definition of B(B E ), the space of Bloch functions defined on B E. A function f : B E C is said to be a Bloch function if sup (1 2 ) f() <. B E Section 2 is devoted to the connection between functions in B(B E ) and their restrictions to one-dimensional subspaces seen as functions defined on D or either to their restrictions to finite-dimensional ones, resulting the fact that if for a given n, the restrictions of the function to the n-dimensional subspaces have their Bloch norms uniformly bounded, then the function is a Bloch one and conversely. We also introduce an equivalent norm for B(B E ) obtained by replacing the gradient by the radial derivative. We ehibit in Section 3 another equivalent norm for B(B E ) which is invariant- modulo the constant functions - under the action of the automorphisms of the ball. This is achieved without appealing to the invariant Laplacian and relying only on properties of automorphisms of B E. Further, we are able to show that the space H (B E ) of bounded analytic functions is contractively embedded in B(B E ), as it occurs in the finite dimensional case. Eamples of unbounded Bloch functions are also shown Mathematics Subject Classification. Primary 30D45, 46E50. Secondary 46G20. Key words and phrases. Bloch function, infinite dimensional holomorphy. The first author is supported by Project MTM (MECC. Spain) and the second and third authors are supported by Project MTM (MECC. Spain). 1

2 2 BLASCO, GALINDO, AND MIRALLES 1. Background 1.1. The classical Bloch space B. The classical Bloch space B (see [4]) is the space of analytic functions f : D C satisfying endowed with the norm f B = sup(1 z 2 ) f (z) < z D f Bloch = f(0) + f B < so that (B, Bloch ) becomes a Banach space. It is well-known that the seminorm B is invariant by automorphisms, that is, f ϕ B = f B for any f B and ϕ Aut(D). The following basic result can be proved applying Schwarz s lemma (see for instance [9]). Proposition 1.1. H is properly contained in B and f B f for any f H. For further information and references about the Bloch space B, the reader is referred to [1, 9] The Bloch space on the unit ball of C n. R. M. Timoney etended Bloch functions to bounded homogeneous domains D of C n (see [6] and [7]). K. Zhu studied further these functions on the unit ball B n of C n and many of his results are compiled in [8]. The Bloch space of functions on B n can be defined in several ways. It is natural to consider the space B(B n ) of analytic functions f : B n C satisfying sup (1 z 2 ) f(z) <, z B n where f(z) := ( f z 1 (z),, f z n (z)) and whose norm is the Euclidean one or to consider analytic functions satisfying sup (1 z 2 ) Rf(z) <, z B n where Rf(z) := f(z), z is the so-called radial derivative of f in z, here z = ( z 1,..., z n ) or even to define the space reducing to functions defined on D by the condition sup{ f B : C n, = 1} <, where f (z) = f(z), z D. Adding up f(0) to each of the quantities above to avoid constant functions to have norm 0 we get Banach spaces. It was shown by Timoney that they are equivalent norms in the space B(B n ). However, these definitions using the gradient or the radial derivative, do not allow us to define a semi-norm invariant by automorphisms. It seems that this is why Timoney used the Bergman metric on B n to define the norm Q f (z) of df as a cotangent vector, so f is said to belong to the Bloch functions space on B n, B(B n ), if f B(Bn) := sup z Bn Q f (z) <. This epression satisfies f ϕ B(Bn) = f B(Bn) for any ϕ Aut(B n ) since the Bergman metric is also invariant by automorphisms. Timoney also proved that this is equivalent to the previous formulations and got the bonus of the invariance under automorphisms. Recall that the invariant gradient of a holomorphic function f : B n C at z B n is f(z) := (f ϕ z )(0). Zhu proved that a holomorphic function f : B n C belongs to the Bloch space B(B n ) if and only if sup z Bn f(z) < (see [8]).

3 BLOCH FUNCTIONS. THE INFINITE DIMENSIONAL CASE 3 2. The space of Bloch functions on the unit ball of a Hilbert space E 2.1. Holomorphic functions on E. A function f : B E C is said to be holomorphic if it is Fréchet differentiable at every B E or, equivalently, if f() = n=1 P n() for all B E, where P n is an n homogeneous polynomial, that is, the restriction to the diagonal of a continuous n linear form on the n-fold space E E. The space H (B E ) is given by {f : B E C : f is holomorphic and bounded } and it becomes a uniform Banach algebra when endowed with the sup-norm f = sup{ f() : B E }. It is, obviously, the analogue of the space H for E. Let (e k ) k Γ be an orthonormal basis of E that we fi at once. Then every z E can be written as z = k Γ z ke k and we write z = k Γ z ke k. Given a holomorphic function f : B E C and B E, we will denote, as usual, by f() the gradient of f at, that is, the unique ( element ) in E representing the linear operator f () E. It may be written f() = f k (), and so f ()(z) = k Γ k Γ f k ()z k = z, f(). Bearing in mind the classical Bloch spaces defined on the unit ball of C and C n, and their possible definitions, we set the following possible norms on the space for the unit ball of E. Definition 2.1. We define B(B E ) as the space of holomorphic functions f : B E C such that f B(BE ) := sup B E (1 2 ) f() <. As usual f Bloch(BE ) := f(0) + f B(BE ) is a complete norm on B(B E ). We first observe that the study of Bloch functions defined on the unit ball of E can be reduced to studying functions defined on finite dimensional subspaces. For each z E and each finite subset ν of Γ write z ν = k ν z ke k. Let us use the following notations: Let n N, write z n = (z 1,, z n ) B n and denote SO n = {y = (y 1,, y n ) : y k E, y k, y j = δ k,j }, that is to say the family of orthonormal systems of order n for n 2 and SO 1 the unit sphere of E. Now for each y SO n and f : B E C holomorphic we define (2.1) f y (z n ) = f ( n ) z k y k. We have for each 1 k n This gives (2.2) k=1 f y (z n ) = y k, f ( n ) z j y j. z k j=1 f y (z n ) = f ( n ) z j y j. For each finite subset ν of Γ we denote f ν = f y for y = {e k : k ν}. Proposition 2.2. Let f : B E C be holomorphic. Then the following statements are equivalent: (i) f B(B E ), j=1

4 4 BLASCO, GALINDO, AND MIRALLES (ii) sup{ f ν B(B ν ) : ν Γ finite } <, (iii) sup f y B(Bm) < for all m 2, y SO m (iv) There eists m 2 such that sup f y B(Bm) <. y SO m Moreover, for each m 2, f B(BE ) = sup f n B(Bn) = sup f y B(Bm). n N y SO m Proof. (i) = (ii) Let ν Γ finite, n = ν and z n B n. According to (2.2), (2.3) f ν (z n ) = f ( ) z j e j. Since j ν z jy j = z n we obtain f ν B(Bn) f B(BE ). In particular j ν sup{ f ν B(B ν ) : ν Γ finite } f B(BE ). (ii) = (i) Let = k Γ z ke k B E, actually a series whose partial sums we denote s n. Since f is holomorphic one has that f() = lim f(s n ) sup{ f ν (z ν ) : ν Γ finite}. n Hence, since s n = z ν with n = ν, we obtain (1 2 ) f() sup { (1 z ν 2 ) f n (z ν ) : ν Γ finite }. (i) = (iii) follows analogously to (i) = (ii). (iii) = (iv) is obvious. (iv) = (i) Assume now that sup f y B(Bm) < for some m 2. Fi B E \ {0} y SO m and choose y E a unit vector such that f() = y, f(). Now write y = m j=1 α jy j for some (α 1,, α m ) C m where y SO m such that y 1 =. Using now (2.2) for z m = (, 0,, 0) we obtain z m = and f y (z m ) = f(). This gives f B(BE ) sup y SO m f y B(Bm). We now show that descriptions in terms of the radial derivative and the one dimensional case can be obtained as well. Definition 2.3. For a holomorphic function f : B E C we set f R := sup B E (1 2 ) Rf(), where R(f)() = f(),, B E and f weak := sup f y B, y =1 where f y (z) = f(zy), z < 1, for each y E with y = 1. Notice that zf y(z) = Rf(zy). We denote B R (B E ) and B weak (B E ) the space of holomorphic functions on B E for which f R < and f weak < respectively. As usual, f(0) + f R and f(0) + f weak are complete norms in these spaces. Our aim is to show that the three spaces, B(B E ), B R (B E ) and B weak (B E ) coincide and that their norms are equivalent actually. Comparing B R (B E ) and B weak (B E ) is rather elementary.

5 BLOCH FUNCTIONS. THE INFINITE DIMENSIONAL CASE 5 Proposition 2.4. The spaces B R (B E ) and B weak (B E ) coincide. Moreover there eists C > 0 such that for every f in the spaces Proof. Let f B weak (B E ). Since f R f weak C f R. Rf() = f, B E \ {0} we have that f R f weak for any holomorphic f defined on B E, so f B R (B E ). Assume now that f B R (B E ). We check that f B weak (B E ). Let y E with y = 1. Since f is holomorphic at 0, its derivative f : B E E is also holomorphic and thus, bounded on some ball B(0, r), 0 < r < 1, hence there is M > 0 such that f () M for any B(0, r). Thus sup (1 z 2 ) f y(z) M. z r While for z > r and taking into account that zf y(z) = Rf(zy) and that the function 1 t t is decreasing, we have Hence (1 z 2 ) f y(z) = (1 z 2 )(1 z ) f y(z) + (1 z 2 ) z f y(z) (1 z 2 ) z 1 r f r y(z) + (1 zy 2 ) Rf(zy) ( (1 zy 2 ) 1 r ) + (1 zy 2 ) Rf(zy) r sup (1 z 2 ) f y(z) 1 z >r r sup (1 2 ) Rf(). <1 Therefore f B weak (B E ) as wanted, and so the spaces coincide. Now the open mapping theorem yields the equivalence of both complete norms. To compare B(B E ) and B weak (B E ) we follow the arguments used by Timoney (see Theorem 4.10 in [6]) applying the following lemma for functions in two variables. Lemma 2.5 (Lemma 4.11 in [6]). Let F : B 2 C be an analytic function. If there eists M 0 such that for any (z 1, z 2 ) B 2, the function F (z1,z 2 )(z) := F (zz 1, zz 2 ), z < 1, satisfies F (z1,z 2 ) B M, then F (z 1, 0) (1 z 1 2 ) 3M log 2 for all z 1 C, z 1 < 1. Although the coming proof follows the same pattern as Theorem 4.10 in [6], we include it for the sake of completeness. Theorem 2.6. The spaces B(B E ), B R (B E ) and B weak (B E ) coincide. Moreover f R f B(BE ) (3 log 2) f weak. Proof. Of course Rf() f() and therefore f R f B(BE ). Let us show that f B(BE ) (3 log 2) f weak. The result follows using Proposition 2.4. Let f B weak (B E ) with sup y =1 f y B = 1. Fi B E. Choose y E a unit vector such that f() = y, f(). Now consider two orthogonal unit vectors 1, 2 B E such that = α 1 and y = α α 2 2 for some α, α 1, α 2 C. Define F : B 2 C by F (z 1, z 2 ) = f(z z 2 2 ). If L : C 2 E is the linear mapping L(z 1, z 2 ) := z z 2 2, F = f L, and the chain rule yields F (z 1, z 2 ) = ( f(z 1 1 +z 2 2 ), 1, f(z 1 1 +z 2 2, 2 ). Then F satisfies the assumptions of Lemma 2.5, so (α 1, α 2 ), F (α, 0) (1 α 2 ) 3 log 2 and we have the aimed inequality f() (1 2 ) = y, f() (1 2 ) 3 log 2.

6 6 BLASCO, GALINDO, AND MIRALLES Using Theorem 2.6 and classical results on Bloch functions on the unit disk, one can obtain the following result. Proposition 2.7. There eists C > 0 such that for the Taylor series f() = n=0 P n() of any f B(B E ) one has (2.4) P n C f B(BE ) for every n = 1, 2,... Proof. It suffices to notice that for any z D, f y (z) = P n (zy) = P n (y)z n n=0 n=0 and to use that for any ϕ B one has (see [1, Lemma 2.1]) a n e ϕ B, n 1. The result follows now considering a n = P n (y) and applying Theorem 2.6. Proposition 2.8. Let (P k ) be a sequence of 2 k homogeneous polynomials on E with Then f() = k=0 P k() B(B E ). M = sup P k (y) <. k N,y B E Proof. To see that f is holomorphic in the unit ball, simply observe that if < 1, then P k () M k k 2k C 1. From that we get the uniform convergence on compact sets and therefore f is holomorphic. To finish the proof use again that f y (z) = P k (y)z 2k and now the fact (see [1, Lemma 2.1]) n=0 ϕ B C sup a k k whenever ϕ(z) = k a kz 2k together with Theorem 2.6. Notice that f H (B E ) is equivalent to sup y =1 f y = f. The use of Proposition 1.1 and the fact that B(B E ) = B weak (B E ) imply H (B E ) B(B E ). Let us finish this section with a basic eample of an unbounded Bloch function on B l2. Eample 2.9. Let f(z) = Then f B(B l2 ) \ H (B l2 ). 1 n n + 1 ( zk 2 )n, z = n=0 k=1 z k e k B l2. k=1

7 BLOCH FUNCTIONS. THE INFINITE DIMENSIONAL CASE 7 Proof. Let 0 < r < 1 and z r. The series 1 n=0 n+1 ( n k=1 z2 k )n converges uniformly in z r because 1 n + 1 sup n ( zk 2 )n 1 1 z r n + 1 r2n log( 1 r 2 ). n=0 k=1 n=0 Hence f is holomorphic on the unit ball. On the other hand, for j N, f 2n n (z) = z j n + 1 ( zk 2 )n 1 z j. Hence and f z j (z) n=1 n=j k=1 2n n n + 1 ( z k 2 ) n 1 z j k=1 f(z) 2 1 z 2. n=1 2n n + 1 z 2n 2 z j 1 Finally we observe that selecting = (z, 0,, ) we have f() = log( ) and therefore 1 z 2 f / H (B l2 ). 3. A Möbius invariant norm for the Bloch space on the unit ball B E Now our aim is to prove that the Bloch space B(B E ) is invariant under the action of the automorphisms of the ball. We begin by collecting the necessary information about such automorphisms Automorphisms and the pseudohyperbolic distance on B E. The analogues of Möbius transformations on E are the mappings ϕ a : B E B E, a B E, defined according to (3.1) ϕ a () = (s a Q a + P a )(m a ()) where s a = 1 a 2, m a : B E B E is the analytic map (3.2) m a () = a 1, a, P a : E E is the orthogonal projection along the one-dimensional subspace spanned by a, that is,, a P a () = a, a a and Q a : E E, is the orthogonal complement, Q a = Id P a. Recall that P a and Q a are self-adjoint operators since they are projections, so P a (), y =, P a (y) and Q a (), y =, Q a (y) for any, y E. The automorphisms of the unit ball B E turn to be compositions of such analogous Möbius transformations with unitary transformations U of E, that is, self-maps of E satisfying U(), U(y) =, y for all, y E. We will also need the following facts about the pseudohyperbolic distance in B E. It is given by (3.3) and it satisfies that (3.4) ρ E (, y) = ϕ y () for any, y B E ρ(f(), f(y)) ρ E (, y) for f H (B E ) with f 1,

8 8 BLASCO, GALINDO, AND MIRALLES where ρ(z, w) = (3.5) z w 1 zw for any z, w D is the pseudohyperbolic distance in D. Actually, ρ E (, y) 2 = 1 (1 2 )(1 y 2 ) 1, y 2 For all these facts and further information on the automorphisms of B E and the pseudohyperbolic distance, see [3] The invariance under automorphisms. As it happens in the finite dimensional case n 2, it is not true that f ϕ B = f B for all f B(B E ). This is also false for the norms R and weak. Thus, in Timoney spirit, we are led to find a semi-norm on B(B E ) that is invariant under the automorphisms of the ball. Our goal is to give a direct proof using only properties of automorphisms of B E and not the invariant Laplacian. Definition 3.1. Let f : B E C be a holomorphic function. The invariant gradient f is defined by f() = (f ϕ )(0) for any B E. Let us first relate the invariant gradient with the standard one and the radial derivative. We need first following easy fact. Lemma 3.2. Let B E. Then, ϕ (0) = s 2 P s Q. Proof. Recall that ϕ (y) = (P + s Q )(m (y)), where m (y) = The derivative of m is given by m (y)(t) = so m (0) = Id + 2 P and, by the chain rule, y 1,y. (1, y )t + t, t, y (1, y ) 2, ϕ (0) = (P + s Q ) (m (0)) m (0) = (P + s Q ) ( Id + 2 P ) = P s Q + 2 P = s 2 P s Q. Lemma 3.3 is a generalization of Lemma 2.13 in [8] for the infinite dimensional case. Lemma 3.3. Let f : B E C be a holomorphic function. Then, f() 2 + (1 2 ) Rf() 2 = (1 2 ) f() 2. Proof. By Lemma 3.2, ϕ (0) = s 2 P s Q, so by the chain rule and bearing in mind that P, Q are self-adjoint, we get ( f() 2 = = sup ( f() ϕ (0))(y) y B E ( sup ϕ (0)(y), f() y B E = ϕ (0)( f()) 2 ) 2 ) 2 = s 4 P ( f()) 2 + s 2 Q ( f()) 2 = s 4 P ( f()) 2 + s 2 f() 2 s 2 P ( f()) 2 = s 2 (s 2 1) P ( f()) 2 + s 2 f() 2 = s 2 f() 2 s 2 2 f(), 2 2 = s 2 ( f() 2 Rf() 2 )

9 BLOCH FUNCTIONS. THE INFINITE DIMENSIONAL CASE 9 Definition 3.4. Denote B inv (B E ) the set of holomorphic functions f : B E C such that f inv := sup B E f() <. It is clear that f ϕ inv = f inv for any f B(B E ) and any automorphism ϕ of B E. Note that Rf() f() and then Hence, using Lemma 3.3, we have f() 2 Rf() 2 (1 2 ) f() 2. (3.6) (1 2 ) f() f() 1 2 f(). In particular, B inv (B E ) B(B E ) and f B(BE ) f inv. We shall show that B(B E ) = B inv (B E ). We will neither use the facts related to the Bergman metric or Q f (z) of the finite dimensional case used by Timoney nor the properties of the invariant gradient and its connection to the invariant Laplacian used by Zhu. We shall use the following lemma, which generalizes Theorem 3.1 in [8] for the infinite dimensional case. It gives a different eplicit formula for f() which is closely related to the epression of Q f (z) for the finite dimensional case. This calculation will help us in proving Theorem 3.8. Lemma 3.5. Let f : B E C be a holomorphic function. Then, Proof. Notice that f() f(), w (1 2 ) = sup w 0 (1 2 ) w 2 + w,. 2 f() 2 = ϕ (0)( f()) 2 ϕ = sup (0)( f()), w w 0 w and bearing in mind that ϕ (0) is a linear invertible and self-adjoint operator, we have that f() 2 ϕ = sup (0)( f()), ϕ (0) 1 (w) f(), w w 0 ϕ (0) 1 = sup (w) w 0 ϕ (0) 1 (w). Since ϕ (0) 1 = 1 P s 2 1 s Q, we have that ϕ (0) 1 (w) 2 = 1 s 4 P (w) s 2 Q (w) 2 = 1 s 4 P (w) s 2 ( w 2 P (w) 2 ) so we conclude that = 1 s2 s 4 w, s2 s 4 w 2 = (1 2 ) w 2 + w, 2 (1 2 ) 2, f() f(), w (1 2 ) = sup w 0 (1 2 ) w 2 + w,. 2 We shall use a result about holomorphic functions in two comple variables due to Timoney.

10 10 BLASCO, GALINDO, AND MIRALLES Lemma 3.6. (see Lemma 4.8 in [6]) Let F : B 2 C be a holomorphic function satisfying Then for each z D, F B(B2 ) := sup (1 z 1 2 z 2 2 ) F (z 1, z 2 ). (z 1,z 2 ) B 2 F (z, 0) 31 w (1 z 2 ) 1/2 F 2 B(B2 ). The previous lemma was also generalized in [6] to the unit ball of C n. We give the proof for B E for the sake of completeness. Lemma 3.7. Let f : B E C be a holomorphic function satisfying f B(BE ) <. Let 0 B E and y E, y = 1, such that 0, y = 0. Then 31 f( 0 ), y (1 0 2 ) 1/2 2 f B(B E ). Proof. Set 0 = 0/ 0. We consider the linear mapping L : C 2 E given by L(z 1, z 2 ) = 0 z 1 + yz 2 and define the holomorphic function F : B 2 C by F (z 1, z 2 ) = (f L)(z 1, z 2 ). Applying the Chain Rule, we have F (z 1, z 2 ) = ( f(z z 2 y), 0, f(z z 2 y), y ). Using that 0 and y are orthonormal vectors in E we conclude that F B(B 2 ) f B(BE ). Therefore, we can apply Lemma 3.6 to obtain F ( 0, 0) 31 z 2 (1 0 2 ) 1/2 2 f B(B E ). Since we finish the proof. F z 2 ( 0, 0) = f( 0 ), y Theorem 3.8. Let f : B E C be a holomorphic function. Then f B(B E ) if and only if f B inv (B E ). The underlying semi-norms satisfy for some constant C (1 + f B(BE ) f inv C f B(BE ) 31 2 ). Proof. We already mentioned that f B(BE ) f inv. Let us show that f inv C f B(BE ). By Lemma 3.5, we have that f() f(), w (1 2 ) = sup w 0 (1 2 ) w 2 + w,. 2 Fi B E. Every v E can be decomposed as v = λ + y, where y is orthogonal to. Then, f(), v (1 2 ) (1 2 ) v 2 + v, = f(), v (1 2 ) 2 λ (1 2 ) y 2 = f(), λ (1 2 ) λ (1 2 ) y 2 + f(), y (1 2 ) λ (1 2 ) y 2 f(), λ (1 2 ) λ f(), y (1 2 ) (1 2 ) y 2 = f(), (1 y 2 ) + f(), y 1 2.

11 BLOCH FUNCTIONS. THE INFINITE DIMENSIONAL CASE 11 Hence, applying Lemma 3.7, ( f inv = sup B E sup v 0 f(), v (1 2 ) (1 2 ) v 2 + v, 2 ) ( ) f B(B E ). 4. The embeddeding of H (B E ) into B(B E ) As we remarked before, H (B E ) B(B E ). However, the use of the equivalence of the norms f B(BE ) and f R does not give the best constant for the embedding. To improve the result we give firstly an elementary lemma. Lemma 4.1. Let, y B E. Then, Proof. We have ρ E (, y) y 1, y. ρ E (, y) 2 = 1 (1 2 )(1 y 2 ) 1, y 2 = 2R, y +, y y 2 2 y 2 1, y 2. Using now, y 2 2 y 2 0 we have ρ E (, y) 2 2R, y y 2 1, y 2 = y 2 1, y 2. Theorem 4.2. Let f H (B E ) such that f 1. For any B E, we have that (1 2 ) f() 1 f() 2. Proof. Notice that f () is the functional on E given by f ()(y) = lim t 0 f( + ty) f() t for any y E. There is y E, y = 1 such that f () = f ()(y). Put δ := 1 and consider only t such that t < δ. For those t, + ty + δ < 1, so + ty B E and we have, applying Lemma 4.1, that t ρ E ( + ty, ) if t < δ. 1, + ty On the other hand f( + ty) f() t f( + ty) f() 1 f()f( + ty) = 1 f()f( + ty) t = ρ ( f( + ty), f() ) 1 f()f( + ty) t 1 f()f( + ty) ρ E ( + ty, ) t 1 f()f( + ty), 1, + ty where we have used that the pseudohyperbolic distance is contractive for f.

12 12 BLASCO, GALINDO, AND MIRALLES Hence Therefore, f ()(y) 1 f()f( + ty) lim sup = 1 f() 2 t 0 1, + ty 1 2. (1 2 ) f () 1 f() 2. So we get the following etension of Proposition 1.1. Corollary 4.3. The inclusion i : H (B E ) B(B E ) is a linear operator satisfying f B(BE ) f. Corollary 4.4. Let f H (B E ) with f = 1 and ϕ B. Then g = ϕ f B(B E ) and g B(BE ) ϕ B. In particular, f() = log(1, e 1 ) B(B E ) \ H (B E ). Proof. Using the chain rule for g() = ϕ(f()) we have g() = ϕ (f()) f(). Therefore, using Theorem 4.2 we conclude g() = ϕ (f()) f() ϕ B f() 1 f() 2 ϕ B. References [1] J. M. Anderson, J. G. Clunie, Ch. Pommerenke, On Bloch functions and normal functions, J. reine angew. Math. 270 (1974), [2] T. K. Carne, B. Cole and T. W. Gamelin, A uniform algebra of analytic functions on a Banach space, Trans. Amer. Math. Soc. 314 (1989), [3] K. Goebel and S. Reich, Uniform conveity, hyperbolic geometry, and nonepansive mappings, Marcel Dekker, Inc., New York and Basel (1984). [4] Ch. Pommerenke, On Bloch functions, J. London Math. Soc. 2(2) (1970), [5] W. Rudin, Function theory in the unit ball of C n, Reprint of the 1980 edition, Classics in Mathematics, Springer-Verlag, Berlin (2008). [6] R. M. Timoney, Bloch functions in several comple variables I, Bull. London Math. Soc. 12 (1980), [7] R. M. Timoney, Bloch functions in several comple variables II, J. Reine Angew. Math. 319 (1980), [8] K. Zhu, Spaces of holomorphic functions in the unit ball, Graduate Tets in Mathematics 226, Springer- Verlag, New York (2005). [9] K. Zhu, Operator theory in function spaces, Mathematical Surveys and Monographs 138, American Mathematical Society, Providence, RI (2007). Oscar Blasco. Departamento de Análisis Matemático, Universidad de Valencia, Valencia, Spain. e.mail: oscar.blasco@uv.es Pablo Galindo. Departamento de Análisis Matemático, Universidad de Valencia, Valencia, Spain. e.mail: pablo.galindo@uv.es Alejandro Miralles. Instituto Universitario de Matemáticas y Aplicaciones de Castellón (IMAC), Universitat Jaume I de Castelló (UJI), Castelló, Spain. e.mail: mirallea@uji.es