On a new integral-type operator from the Bloch space to Bloch-type spaces on the unit ball

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Journal of Mathematical Analysis and Applications

www.elsevier.com/locate/jmaa

On a new integral-type operator from the Bloch space to Bloch-type

spaces on the unit ball

Stevo Stevi ´c

Mathematical Institute of the Serbian Academy of Science, Knez Mihailova 36/III, 11000 Beograd, Serbia

a r t i c l e i n f o a b s t r a c t

Article history: Received 22 July 2008 Available online 8 January 2009 Submitted by M.M. Peloso Keywords: Integral-type operator Bloch space Bloch-type space Boundedness Compactness Essential norm Unit ball

We introduce the following integral-type operator on the spaceH(B) of all holomorphic functions on the unit ball_B⊂Cn

Pϕg(f)(z)= 1 0 f

ϕ

(tz)g(tz)dt t, z∈B,

where g∈H(B),g(0)=0 and

ϕ

is a holomorphic self-map of B. The boundedness and compactness of the operator from the Bloch spaceB or the little Bloch spaceB0 to the Bloch-type spaceBμ or the little Bloch-type space Bμ,0, are characterized. In the main results we calculate the essential norm of the operatorsPϕg:B(orB0)→Bμ(orBμ,0)in an elegant way.

©2009 Elsevier Inc. All rights reserved.

1. Introduction and preliminaries

Let

B

be the open unit ball in

C

n_,

D

_{the unit disk in}

C

,

_H

(

B

)

_{the class of all holomorphic functions on the unit ball and}

H∞

=

H∞

(

B

)

the space of all bounded holomorphic functions on

B

with the norm

f

_∞

=

sup

z∈B

f

(

z

).

Letz

=

(

z1

, . . . ,

zn)andw

=

(

w1

, . . . ,

wn)be points in

C

n,

z

,

w

=

n

k=1zkw

¯

kand

|

z

| =

√

z

,

z

. For f

∈

H

(

B

)

with the Taylor expansion f

(

z

)

=

_|β|0aβzβ, let

f

(

z

)

=

|β|0

|

β

|

aβzβ

be the radial derivative of f

,

where

β

=

(β

1

, β

2

, . . . , βn)

is a multi-index,

|

β

| =

β

1

+ · · · +

βn

andzβ

=

zβ11

· · ·

z

βn

n (see [29]). A positive continuous function

μ

on

[

0

,

1

)

is called normal [30] if there is

δ

∈ [

0

,

1

)

andaandb, 0

<

a

<

bsuch that

μ

(

r

)

(

1

−

r

)

a is decreasing on

[

δ,

1

)

and _rlim_→1

μ

(

r

)

(

1

−

r

)

a

=

0

,

μ

(

r

)

(

1

−

r

)

b is increasing on

[

δ,

1

)

and _rlim_→1

μ

(

r

)

(

1

−

r

)

b

= ∞

.

If we say that a function

μ

:

B

→ [

0

,

∞

)

is normal we will also assume that it is radial, that is,

μ

(

z

)

=

μ

(

|

z

|

)

,z

∈

B

.

E-mail address:[email protected].

0022-247X/$ – see front matter ©2009 Elsevier Inc. All rights reserved. doi:10.1016/j.jmaa.2008.12.059

The weighted space H∞_μ

=

H∞_μ

(

B

)

consists of all f

∈

H

(

B

)

such that sup

z∈B

μ

(

z

)

f

(

z

)

<

∞

,

where

μ

is normal. For

μ

(

z

)

=

(

1

− |

z

|

2

)

β,

β >

0 we obtain the (classical) weighted spaceH∞_β

=

H∞_β

(

B

)

. The little weighted spaceH∞_μ,0

=

H∞_μ,0

(

B

)

is a subspace of H_μ∞consisting of all f

∈

H

(

B

)

such that

lim

|z|→1

μ

(

z

)

f

(

z

)

=

0

.

The Bloch-type space, denoted by

B

μ

=

B

μ

(

B

)

, consists of all f

∈

H

(

B

)

such that

Bμ

(

f

)

=

sup z∈B

μ

(

z

)

f

(

z

)

<

∞

,

where

μ

is normal. With the norm

f

_Bμ

=

f

(

0

)

+

Bμ

(

f

)

the Bloch-type space becomes a Banach space.

The

α

-Bloch space

B

α is obtained for

μ

(

z

)

=

(

1

− |

z

|

2

)

α,

α

∈

(

0

,

∞

)

(see, e.g., [26,35,38] and references therein). The little Bloch-type space

B

μ,0is a subspace of

B

μconsisting of those f such that

lim

|z|→1

μ

(

z

)

f

(

z

)

=

0

.

Bearing in mind the following asymptotic relation from [36] (see also [8] for the case of the

α

-Bloch space)

bμ

(

f

)

:=

sup z∈B

μ

(

z

)

∇

f

(

z

)

sup z∈B

μ

(

z

)

f

(

z

)

(1)

we see that

B

μ can be defined as the class of all f

∈

H

(

B

)

such that bμ

(

f

)

is finite. Also the little Bloch-type space is equivalent with the subspace of

B

μconsisting of all f

∈

H

(

B

)

such that

lim

|z|→1

μ

(

z

)

∇

f

(

z

)

=

0

.

From this observation and for some technical benefits, for the norm of the

α

-Bloch space we choose the second definition, that is, f

∈

B

α if and only if

f

_Bα

=

f

(

0

)

+

sup z∈B

1

− |

z

|

2

α

∇

f

(

z

)

<

∞

.

If

μ

(

z

)

=

(

1

− |

z

|

2

)

, then the quantitybμ

(

f

)

in (1), will be denoted byb

(

f

).

Let

ϕ

be a holomorphic self-map of

B

.

For f

∈

H

(

B

)

the composition operator is defined byCϕf

(

z

)

=

f

(

ϕ

(

z

))

(see, e.g., the monograph [9] or recent papers [10,18,27,37]).

Let g

∈

H

(

D

)

and

ϕ

be a holomorphic self-map of

D

.

Products of integral and composition operators on H

(

D

)

were introduced by S. Li and S. Stevi ´c in a private communication (see, e.g., [23,24] and [25], as well as related paper [20]) as follows CϕJgf

(

z

)

=

ϕ(z)

0 f

(ζ )

g

(ζ )

d

ζ

and JgCϕf

(

z

)

=

z

0 f

ϕ

(ζ )

g

(ζ )

d

ζ.

(2)

Operators in (2) are extensions of the following integral operator

Tg(f

)(

z

)

=

z

0

f

(ζ )

g

(ζ )

d

ζ

which was introduced in [28]. Some other results on the operator Tg can be found, e.g., in [1–3,31]. For some results on

n-dimensional extensions of the operator, see [4–7,11–17,19,21,22,32–34,36] and references therein.

One of the interesting questions is to extend operators in (2) in the unit ball settings and to study their function theoretic properties between spaces of holomorphic functions on the unit ball in terms of inducing functions.

Assume thatg

∈

H

(

B

)

, g

(

0

)

=

0 and

ϕ

is a holomorphic self-map of

B

, then we introduce the following operator on the unit ball Pϕg

(

f

)(

z

)

=

1

0 f

ϕ

(

tz

)

g

(

tz

)

dt t

,

f

∈

H

(

B

),

z

∈

B

.

(3)

Ifn

=

1, then g

∈

H

(

D

)

andg

(

0

)

=

0, so thatg

(

z

)

=

zg0

(

z

),

for someg0

∈

H

(

D

).

By the change of variable

ζ

=

tz, it follows that Pϕgf

(

z

)

=

1

0 f

ϕ

(

tz

)

tzg0

(

tz

)

dt t

=

z

0 f

ϕ

(ζ )

g0

(ζ )

d

ζ.

Thus operator (3) is a natural extension of the second operator in (2).

Here we study the boundedness and compactness of operator P_ϕg from the Bloch space

B

or the little Bloch space

B

0 to the Bloch-type space

B

μ or the little Bloch-type space

B

μ,0

.

In Sections 4 and 5 we calculate the essential norm of the operators P_ϕg

:

B

(

or

B

0

)

→

B

μ

(

or

B

μ,0

)

.

Throughout the paperC will denote a positive constant not necessarily the same at each occurrence. The notationA

B

means that there is a positive constantC such that A

/

C

B

C A

.

The following lemmas are used in the proofs of the main results.

Lemma 1.Suppose g

∈

H

(

B

)

, g

(

0

)

=

0,

μ

is normal and

ϕ

is a holomorphic self-map of

B

. Then the operator Pϕg

:

B

(

or

B

0

)

→

B

μ

is compact if and only if Pϕg

:

B

(

or

B

0

)

→

B

μis bounded and for any bounded sequence

(

fk)k∈Nin

B

(

or

B

0

)

converging to zero

uniformly on compacts of

B

, we have

Pϕgfk

_Bμ

→

0as k

→ ∞

.

The proof of Lemma 1 follows by standard arguments (see, for example, the proofs of Proposition 3.11 in [9] and Lemma 3 in [34]). Hence, we omit its proof.

Lemma 2.Suppose f

,

g

∈

H

(

B

)

and g

(

0

)

=

0. Then

Pϕg

(

f

)(

z

)

=

f

ϕ

(

z

)

g

(

z

).

Proof. Assume that the holomorphic function f

(

ϕ

(

z

))

g

(

z

)

has the expansion

_βaβzβ

.

_{Since g}

(

₀

)

=

_{0, note thata}

0

=

0. Then

Pϕg

(

f

)

(

z

)

=

1

0

β=0 aβ

(

tz

)

β dt t

=

β=0 aβ

|

β

|

z β

=

β=0 aβzβ

,

which is what we wanted to prove.

2

Lemma 3.Let f

∈

B(

B

).

Then the following inequality holds

f

(

z

)

f

_Bmax

1

,

1 2ln 1

+ |

z

|

1

− |

z

|

.

(4)

Proof. The proof of the lemma follows from the following inequality

f

(

z

)

−

f

(

0

)

=

1

0

∇

f

(

tz

),

z

¯

dt

b

(

f

)

1

0

|

z

|

dt 1

− |

z

|

2_t2

=

b

(

f

)

1 2ln 1

+ |

z

|

1

− |

z

|

,

whereb

(

f

)

=

sup_z∈B

(

1

− |

z

|

2

)

|∇

_f

(

_z

)

|

.

2

2. The norm of the operatorPϕg

:

B

(or

B

0

)

→

B

μ

In this section we calculate the norms

P_ϕg

_B→Bμ and

P_ϕg

_B0→Bμ.

Theorem 1.Assume g

∈

H

(

B

)

, g

(

0

)

=

0,

μ

is normal,

ϕ

is a holomorphic self-map of

B

and Pϕg

:

B

(

or

B

0

)

→

B

μis bounded. Then

Pϕg

_B→Bμ

=

Pϕg

_B0→Bμ

=

sup z∈B

μ

(

z

)

g

(

z

)

max

1

,

1 2ln 1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(5)

Proof. If f

∈

B

,

then by Lemma 2 and (4) we obtain

Pϕgf

_Bμ

=

sup z∈B

μ

(

z

)

g

(

z

)

f

ϕ

(

z

)

f

_Bsup z∈B

μ

(

z

)

g

(

z

)

max

1

,

1 2ln 1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

,

(6)

from which it follows that

Pϕg

_B→Bμ

sup z∈B

μ

(

z

)

g

(

z

)

max

1

,

1 2ln 1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(7)

The same inequality holds for P_ϕg

:

B

0

→

B

μ

.

Now we prove the reverse inequality. By taking the function given by f0

(

z

)

≡

1

∈

B

0 and using the boundedness of

Pϕg

:

B

→

B

μ, we obtain

P_ϕg

_B→B μ

=

f0

B

P g ϕ

_B→Bμ

Pϕgf0

_B μ

=

sup_z∈B

μ

(

z

)

g

(

z

)

f0

ϕ

(

z

)

=

sup z∈B

μ

(

z

)

g

(

z

)

.

(8) The same inequality holds for Pϕg

:

B

0

→

B

μ

.

Forw

∈

B

, set fw

(

z

)

=

1 2ln 1

+

z

,

w

1

−

z

,

w

,

(9)

with ln 1

=

0. Since fw

(

0

)

=

0 and

1

− |

z

|

2

∇

fw

(

z

)

=

(

1

− |

z

|

2

)

|

_w

|

|

1

−

z

,

w

2

|

1

− |

z

|

2 1

− |

w

|

2

|

_z

|

2

min

1

,

1

− |

z

|

2 1

− |

w

|

2

,

it follows that supw∈B

fw

_B

1, and fw

∈

B

0 for each fixedw

∈

B

.

From this and the boundedness of Pϕg

:

B

(

or

B

0

)

→

B

μ we have that when

ϕ

(

w

)

=

0 and for every t

∈

(

0

,

1

)

the following inequality holds

Pϕg

_B0→Bμ

Pϕgftϕ(w)/|ϕ(w)|

_Bμ

=

sup z∈B

μ

(

z

)

g

(

z

)

1 2

ln1

+

t

ϕ

(

z

),

ϕ

(

w

)/

|

ϕ

(

w

)

|

1

−

t

ϕ

(

z

),

ϕ

(

w

)/

|

ϕ

(

w

)

|

1 2

μ

(

w

)

g

(

w

)

ln 1

+

t

|

ϕ

(

w

)

|

1

−

t

|

ϕ

(

w

)

|

.

(10)

Note that (10) obviously holds if

ϕ

(

w

)

=

0

.

Lettingt

→

1 in (10), we obtain that for eachw

∈

B

,

P_ϕg

_B 0→Bμ

1 2

μ

(

w

)

g

(

w

)

ln 1

+ |

ϕ

(

w

)

|

1

− |

ϕ

(

w

)

|

.

From this and since wis an arbitrary element of

B

, it follows that

P_ϕg

_B 0→Bμ

1 2supz∈B

μ

(

z

)

u

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(11)

Note also that

Pϕg

_B→Bμ

Pϕg

_B0→Bμ

.

(12)

From (8), (11) and (12) we obtain that

P_ϕg

_B→B μ

P g ϕ

_B0→Bμ

sup z∈B

μ

(

z

)

g

(

z

)

max

1

,

1 2ln 1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(13)

From (7) and (13), equalities in (5) follow.

2

Corollary 1.Assume g

∈

H

(

B

)

, g

(

0

)

=

0,

μ

is normal and

ϕ

is a holomorphic self-map of

B

. Then P_ϕg

:

B

(

or

B

0

)

→

B

μis bounded

if and only if sup z∈B

μ

(

z

)

g

(

z

)

max

1

,

1 2ln 1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

<

∞

.

(14)

Proof. If Pϕg

:

B

(

or

B

0

)

→

B

μ is bounded, then (14) follows from Theorem 1. If (14) holds, then the boundedness of

3. The boundedness of the operatorPϕg

:

B

0

→

B

μ,0

Here we characterize the boundedness of the operator P_ϕg

:

B

0

→

B

μ,0.

Theorem 2.Assume g

∈

H

(

B

)

, g

(

0

)

=

0,

μ

is normal and

ϕ

is a holomorphic self-map of

B

.

Then P_ϕg

:

B

0

→

B

μ,0is bounded if and

only if P_ϕg

:

B

0

→

B

μis bounded and g

∈

H_μ∞,0

.

Proof. Assume that Pϕg

:

B

0

→

B

μ,0 is bounded. Then clearly Pϕg

:

B

0

→

B

μ is bounded. Taking the test function

f0

(

z

)

=

1

∈

B

0 we obtaing

∈

H∞_μ,0

.

Conversely, assume P_ϕg

:

B

0

→

B

μis bounded and g

∈

H∞_μ,0

.

Then, for every polynomialp, we have

μ

(

z

)

P_ϕgp

(

z

)

=

μ

(

z

)

g

(

z

)

p

ϕ

(

z

)

μ

(

z

)

g

(

z

)

p

_∞

→

0

,

as

|

z

| →

1

.

Since the set of all polynomials is dense in

B

0

,

for each f

∈

B

0there is a sequence of polynomials

(

pk)k∈N such that lim

k→∞

f

−

pk

B

=

0

.

(15)

From (15) and since the operator Pϕg

:

B

0

→

B

μis bounded, it follows that

Pϕgf

−

Pϕgpk

_Bμ

P

g

ϕ

_B0→Bμ

f

−

pk

B0

→

0

,

ask

→ ∞

. HencePϕg

(

B

0

)

⊂

B

μ,0

.

Since

B

μ,0 is a closed subset of

B

μthe boundedness ofPϕg

:

B

0

→

B

μ,0 follows.

2

4. Essential norm ofP_ϕg

:

B

(or

B

0

)

→

B

μ

LetXandY be Banach spaces, andL

:

X

→

Ybe a bounded linear operator. The essential norm of the operatorL

:

X

→

Y, denoted by

L

e,X→Y, is defined as follows

L

e,X→Y

=

inf

L

+

K

X→Y: K is compact fromX toY

,

where

·

X→Y denote the operator norm.

From this definition and since the set of all compact operators is a closed subset of the set of bounded operators it follows that operator Lis compact if and only if

L

e,X→Y

=

0

.

Here we prove the main result in the paper, namely, we calculate the essential norm of the operator Pϕg:

B

(

or

B

0

)

→

B

μ.

Theorem 3.Assume g

∈

H

(

B

)

, g

(

0

)

=

0

,

μ

is normal,

ϕ

is a holomorphic self-map of

B

and P_ϕg

:

B

(

or

B

0

)

→

B

μis bounded. If

ϕ

∞

=

1, then

P_ϕg

_e,B→B μ

=

P g ϕ

e,B0→Bμ

=

1 2_|lim supϕ(z)|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

,

(16) while if

ϕ

∞

<

1, then

P_ϕg

_e,B→B μ

=

P g ϕ

e,B0→Bμ

=

0

.

(17)

Proof. First assume that

ϕ

∞

=

1. Set the following family of test functions

fwε

(

z

)

=

ln

(

1

+ |

w

|

)

2 1

−

z

,

w

ε+1

ln1

+ |

w

|

1

− |

w

|

−ε

,

w

∈

B

\ {

0

}

.

It is easy to see that

f_wε

(

0

)

ln

1

+ |

w

|

2

ε+1

ln1

+ |

w

|

1

− |

w

|

−ε

2ε+1ln 2 and lim |w|→1

f_wε

(

0

)

=

0

.

(18)

Further we have

1

− |

z

|

2

∇

f_wε

(

z

)

=

(

ε

+

1

)

(

1

− |

z

|

2

)

|

_w

|

|

1

−

z

,

w

|

ln

(

1

+ |

w

|

)

2 1

−

z

,

w

ε

ln1

+ |

w

|

1

− |

w

|

−ε

(

ε

+

1

)

(

1

− |

z

|

2

)

|

_w

|

1

− |

z

||

w

|

ln

(

1

+ |

w

|

)

2 1

− |

z

||

w

|

+

2

π

ε

ln1

+ |

w

|

1

− |

w

|

−ε (19)

(

ε

+

1

)

1

+ |

z

|

|

w

|

ln

(

1

+ |

w

|

)

2 1

− |

w

|

+

2

π

ε

ln1

+ |

w

|

1

− |

w

|

−ε

.

(20)

From (20) it follows that lim sup

|w|→1

b

fwε

2

(

ε

+

1

)

(21)

and from (19) that, for each fixed w

∈

B

\ {

0

}

, f_wε

∈

B

0

.

Hence (18) and (21) imply

lim

|w|→1

f_wε

_B

2

(

ε

+

1

).

(22)

Now, assume that

(

ϕ

(

zk))k_∈Nis a sequence in

B

such that

|

ϕ

(

zk)

| →

1 ask

→ ∞

. Note that from (22) it follows that the sequence Fk(z

)

=

f_ϕε_(z

k)

(

z

)

,k

∈

N

is such that lim

k→∞

Fk

B

2

(

ε

+

1

),

(23)

and thatFkconverges to zero uniformly on compacts of

B

ask

→ ∞

. By Theorem 3.16 in [38] it follows thatFk

→

0 weakly in

B

0ask

→ ∞

. Hence for every compact operatorK

:

B

0

→

B

μwe have that

lim

k→∞

K Fk

Bμ

=

0

.

(24)

Assume that K

:

B

0

→

B

μis an arbitrary compact operator. Then from the boundedness of Pϕg

:

B

0

→

B

μwe have that for eachk

∈

N

Fk

B

Pϕg

+

K

_B0→Bμ

Pϕg

+

K

(

Fk)

_B μ

P g ϕFk

_B μ

−

K Fk

Bμ

.

(25)

Lettingk

→ ∞

in (25) and using (24) we obtain lim k→∞

Fk

B

P g ϕ

+

K

_B0→Bμ

lim sup k→∞

PϕgFk

_Bμ

−

K Fk

Bμ

=

lim sup k→∞

PϕgFk

_B μ

=

lim sup k→∞ supz∈B

μ

(

z

)

g

(

z

)

Fk

ϕ

(

z

)

lim sup k→∞

μ

(

z k)g

(

zk)Fk

ϕ

(

zk)

=

lim sup k→∞

μ

(

z k)g

(

zk)ln1

+ |

ϕ

(

zk)

|

1

− |

ϕ

(

zk)

|

.

From this and (23) we have

2

(

ε

+

1

)

Pϕg

+

K

_B0→Bμ

lim sup k→∞

μ

(

zk)

g

(

zk)

ln1

+ |

ϕ

(

zk)

|

1

− |

ϕ

(

zk)

|

.

(26)

Taking the infimum in (26) over the set of all compact operators K

:

B

0

→

B

μ and since

ε

is an arbitrary positive number, we obtain

Pϕg

_e,B0→Bμ

lim sup k→∞ 1 2

μ

(

zk

)

g

(

zk

)

ln 1

+ |

ϕ

(

zk)

|

1

− |

ϕ

(

zk)

|

,

which implies the inequality

Pϕg

e,B0→Bμ

lim sup |ϕ(z)|→1 1 2

μ

(

z

)

g

(

z

)

ln 1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(27)

Now we prove the reverse inequality. Assume that

(

rl)l_∈N is a sequence which increasingly converges to 1. Consider the operators defined by Prglϕ

(

f

)(

z

)

=

1

0 g

(

tz

)

f

rl

ϕ

(

tz

)

dt t

,

l

∈

N

.

(28)

Assume that

(

hk)k_∈N is a bounded sequence in

B

(

or

B

0

)

converging to zero uniformly on compacts of

B

. Since g

∈

H∞μ

,

we have

μ

(

z

)

Prglϕ

(

hk)(z

)

=

μ

(

z

)

g

(

z

)

hk

rl

ϕ

(

z

)

g

H∞_μ sup |w|rl

hk(w

)

→

0

,

ask

→ ∞

.

Hence by Lemma 1, for eachl

∈

N

, Prglϕ

:

B

(

or

B

0

)

→

B

μis compact.

Let

ρ

∈

(

0

,

1

)

be fixed for a moment. Employing Lemma 2, the fact that g

∈

H_μ∞, and the following formula (see, [38, Theorem 3.14]) sup f∈B,fB1

f

(

z

)

−

f

(

w

)

=

1 2ln 1

+ |

ϕ

w

(

z

)

|

1

− |

ϕ

w

(

z

)

|

,

z

,

w

∈

B

(where

ϕ

w is the involutive automorphism of

B

that interchanges 0 andw), we have

Pϕg

−

Prglϕ

B→B_μ

=

sup fB1 sup z∈B

μ

(

z

)

g

(

z

)

f

ϕ

(

z

)

−

f

rl

ϕ

(

z

)

sup fB1 sup |ϕ(z)|ρ

μ

(

z

)

g

(

z

)

f

ϕ

(

z

)

−

f

rl

ϕ

(

z

)

+

sup fB1 sup |ϕ(z)|>ρ

μ

(

z

)

g

(

z

)

f

ϕ

(

z

)

−

f

rl

ϕ

(

z

)

g

H∞μ sup fB1 sup |ϕ(z)|ρ

f

ϕ

(

z

)

−

f

rl

ϕ

(

z

)

(29)

+

sup |ϕ(z)|>ρ

μ

(

z

)

g

(

z

)

1 2ln 1

+ |

ϕ

ϕ(z)

(

rl

ϕ

(

z

))

|

1

− |

ϕ

ϕ(z)

(

rl

ϕ

(

z

))

|

.

(30) Since

ϕ

ϕ(z)

rl

ϕ

(

z

)

=

ϕ

(

z

)

−

Pϕ(z)

(

rl

ϕ

(

z

))

−

sqQϕ(z)

(

rl

ϕ

(

z

))

1

−

rl

ϕ

(

z

),

ϕ

(

z

)

=

|

ϕ

(

z

)

|

(

1

−

rl) 1

−

rl

|

ϕ

(

z

)

|

2

ϕ

(

z

),

and since the function

h

(

x

)

=

ln1

+

x

1

−

x (31)

is increasing on the interval

[

0

,

1

),

we obtain sup |ϕ(z)|>ρ

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

ϕ(z)

(

rl

ϕ

(

z

))

|

1

− |

ϕ

ϕ(z)

(

rl

ϕ

(

z

))

|

sup |ϕ(z)|>ρ

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(32)

Now we estimate the quantity in (29). Let

Il

:=

sup

fB1 sup

|ϕ(z)|ρ

f

ϕ

(

z

)

−

f

rl

ϕ

(

z

).

By using the mean value theorem and the definition of the Bloch space, we obtain

Il

sup fB1 sup |ϕ(z)|ρ

(

1

−

rl

)

ϕ

(

z

)

sup |w|ρ

∇

f

(

w

)

ρ

1

−

rl 1

−

ρ

2 sup fB1

f

_B

=

Cρ

(

1

−

rl)

→

0 asl

→ ∞

.

(33)

Lettingl

→ ∞

in (29) and (30), using (32) and (33), and then letting

ρ

→

1 we obtain the inequality

Pϕg

e,B→Bμ

_|lim sup ϕ(z)|→1 1 2

μ

(

z

)

g

(

z

)

ln 1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(34)

From (27), (34) and since

Pϕg

_e,B→Bμ

Pϕg

_e,B0→Bμ

,

both equalities in (16) follow.

Now assume

ϕ

∞

<

1, then similar to operators in (28) it is proved that the operator Pϕg

:

B

(

or

B

0

)

→

B

μis compact, which is equivalent with (17), finishing the proof of the theorem.

2

The following result regarding the compactness of the operator P_ϕg

:

B

(

or

B

0

)

→

B

μ is a direct consequence of Theo-rem 3.

Corollary 2.Assume g

∈

H

(

B

)

, g

(

0

)

=

0,

μ

is normal,

ϕ

is a holomorphic self-map of

B

such that

ϕ

∞

=

1, and the operator Pϕg

:

B

(

or

B

0

)

→

B

μis bounded. Then the operator Pϕg

:

B

(

or

B

0

)

→

B

μis compact if and only if

lim

|ϕ(z)|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

=

0

.

(35)

5. Essential norm of the operatorPϕg

:

B

(or

B

0

)

→

B

μ,0

Here we calculate the essential norm of the operator Pϕg

:

B

(

or

B

0

)

→

B

μ,0.

Theorem 4.Assume g

∈

H

(

B

)

, g

(

0

)

=

0,

μ

is normal,

ϕ

is a holomorphic self-map of

B

and the operator Pϕg

:

B

(

or

B

0

)

→

B

μ,0is

bounded. Then

Pϕg

_e,B→Bμ,0

=

Pϕg

_e,B0→Bμ,0

=

1 2lim sup_|z|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(36)

Proof. Since Pϕg

:

B

(

or

B

0

)

→

B

μ,0 is bounded, then for the test function f0

(

z

)

≡

1

∈

B

0, we obtain thatg

∈

H_μ∞_,0

.

First assume

ϕ

∞

<

1

.

Then, similar to operators in (28) it can be proved that Pϕg

:

B

(

or

B

0

)

→

B

μ,0is compact. Hence

Pϕg

e,_B(or_B0)→Bμ,0

=

0

.

On the other hand, since

ϕ

∞

<

1, andg

∈

H∞_μ,0

,

it follows that

lim sup |z|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

ln1

+

ϕ

∞ 1

−

ϕ

∞

lim |z|→1

μ

(

z

)

g

(

z

)

=

0

,

from which (36) follows in this case.

Now assume

ϕ

∞

=

1

.

It is clear that lim sup |z|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

lim sup_|ϕ(z)|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

(37)

Assume that

(

zk)k_∈N is such a sequence that lim sup |z|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

=

klim→∞

μ

(

zk

)

g

(

zk

)

ln 1

+ |

ϕ

(

zk)

|

1

− |

ϕ

(

zk)

|

.

If sup_k∈N

|

ϕ

(

zk)

|

<

1, then in view of the fact g

∈

H∞_μ,0

,

the last limit is zero and consequently the second limit in (37) is also zero. Otherwise, there is a subsequence

(

ϕ

(

zkl

))l

∈N such that

|

ϕ

(

zkl

)

| →

1 as l

→ ∞

, so that both limits in (37) are equal, that is,

lim sup |z|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

=

lim sup_|ϕ(z)|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

.

From this and by Theorem 3 the result follows in this case, finishing the proof of the theorem.

2

Corollary 3.Assume g

∈

H

(

B

)

, g

(

0

)

=

0,

μ

is normal,

ϕ

is a holomorphic self-map of

B

and the operator Pϕg

:

B

(

or

B

0

)

→

B

μ,0is

bounded. Then the operator P_ϕg

:

B

(

or

B

0

)

→

B

μ,0is compact if and only if lim

|z|→1

μ

(

z

)

g

(

z

)

ln1

+ |

ϕ

(

z

)

|

1

− |

ϕ

(

z

)

|

=

0

.

(38)

Acknowledgment

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