Bounded composition operators on weighted function spaces in the unit disk

Let H be a space of holomorphic functions on some domain G of the complex plane C . For a holomorphic self-map у of G , the composition operator C ^ is defined as

C ^ f = f ◦ v, f e H .

Researchers are interested in the relation between the operator properties of C _v and the functional properties of the symbol ϕ . Questions about boundedness, compactness, compact difference, essential norm, cyclicity, complex symmetry and so on, have been considered. These topics are interesting and have received great attention during the past several decades. We refer the reader to the excellent monographs [1–3] and references therein for more detailed information.

Many studies have been done on composition operators on various different spaces of holomorphic functions on the unit disk or the unit ball, such as Hardy spaces, Bergman spaces, Dirichlet spaces, spaces of bounded holomorphic function and weighted Banach spaces with sup-norm (see, e.g., [4–9]).

Recently, in [10] a general class of weighted Hardy spaces of entire functions, which contains the Fock and Fock-type spaces as well as several other well-known spaces, has been considered and bounded composition operators on these spaces have been studied. It is interesting that the characterization for boundedness of such operators depends closely on the behavior of the weight sequence.

The aim of this paper is to introduce a general class of weighted spaces of holomorphic functions in the unit disk and study two special forms of bounded composition operators on these spaces. This newly introduced class of spaces includes classical Hardy, Bergman and Dirichlet spaces.

So far as we know, this class of spaces has not been treated before. We hope that our approach may inspire the reader to investigate further properties and obtain more general results in the future.

The structure of the paper is as follows. In section 2, the class of weighted spaces H ( в ) of holomorphic functions in the unit disk is performed and a short but essential comparison with the case H( в, E) of entire functions is discussed. Section 3 deals with boundedness of composition operators C ϕ induced by affine symbols ϕ , while in Section 4, the monomial symbols are of our interest.

2.    Preliminaries

In 1974 Shields [11] introduced the following weighted spaces. Let a domain G contain the origin and в = {в _п } П ==о be a sequence of positive numbers. The weighted space H ( в ) is defined as follows:

_∞            1 / 2

I ^ I an|2вП)    < ro n=0

∞

H( в ) = f ( z ) = ^a n z n G O( G ) : \\ f \ = n =0

where O( G ) is the space of all holomorphic functions on G . This space H ( в ) is an inner product space with

∞

( ^f, g) = ^2^а п ^ь п ^в П , n =0

for every f ( z ) = ^ ^ =0 a n z n and g ( z ) = ^ П =о b n z n . Note that H ( в ) is not necessarily a complete normed space.

Denote by r the radius of the smallest open disk centered at the origin which contains G . It is well known that H ( в ) is a Hilbert space if and only if liminf _n ,^ в П/n ^ r In particular, in case lim n ^^ в п/n = ro , we have a Hilbert space of entire functions H ( в, E ) . It is, however, worth to note that since the case of entire functions H ( в, E ) is essentially different from the one of holomorphic functions H ( в ) and moreover, it is already considered in [10], we eliminate those weights for which lim n ,^ в П/n = ro .

So throughout this paper, we pay our attention to a model case when G is the open unit disk D = { z G C : |z| <  1 } and so the condition

1 ^ liminf в П/n < ro                                 (1)

n→∞ is supposed to hold.

We note that for en = 1 , 3n = ( n + 1) - 2 , and 3n = ( n + 1) 2 , the space H ( в ) becomes respectively the classical Hardy space H 2 (D) , Bergman space A ² (D) , and Dirichlet space D ² (D) in the unit disk.

Note also that h_n ( z ) = в - 1 z ⁿ ( n = 0 , 1 , 2 ,... ) form an orthonormal basis for the Hilbert space H ( в ) .

We can easily verify that H ( в ) is a functional Hilbert space, and hence C , is bounded on H ( в ) if and only if it maps H ( в ) into itself.

It is important to have a remark that in the case of weighted spaces of entire functions H ( в, E ) , bounded composition operators C , can be induced only by affine functions ^ ( z ) = az + b . However, in the case of weighted spaces H ( в ) in the unit disk, the situation is much more complicated. It is not true anymore, as we can see in some examples below.

Example 2.1. For the symbol ^ ( z ) = z ² , the composition operator C , is bounded on each of the classical Hardy, Bergman, and Dirichlet spaces; however C , is unbounded in H ( в ) in the case в _п = A n with A >  1 .

<1 When ^ ( z ) = z ² , for every f ( z ) = ^ П =₀ a n z n E H ( в ) , we have

∞ llC,fllH2(D) = E |an|2 = IlfIlH2(D), n=0

IIC^fII a2(d) = E |an| 2n + 1 ^ E |an| n + 1 = Hf IIa2(d), n=0

∞∞

Il Cf liW) = E K| ² (2n. + 1) <  2 £ |a „ | ² ( n + 1) = 2 f I D 2 ( D ) .

n=0

For вп = An with A > 1, we take a function f (z) = ^^=1 ^n^zn E O(D). Then ∞∞∞

I f I ² = E      = E " E . < -,

A2n n2

n=0          n=1

which shows that f (z) E H(в). However, ^          ^^ ic,/12      ■■ =e     A4n=£ a_, n=0           n=1

which diverges since A >  1 . >

Remark 2.2. Replacing ^ ( z ) = z ² by monomials ^ ( z ) = z ^m with some integer number m >  1 , by the same techniques, we can obtain similar results.

3.    The Case when ^(z) is Affine

In this section, we study boundedness of composition operators on weighted spaces H ( в ) induced by affine symbols. This study is inspired by the case of entire functions [10].

Let the symbol ^ ( z ) = az + b ( a, b E C ). For C , to be well-defined, ^ ( z ) needs to be a self-map of D .

We have the following simple result.

Lemma 3.1. An affine function ^ ( z ) = az + b ( a, b E C ) is a self-map of D if and only if either of the following conditions holds:

• (i)    a = 0 and |b| <  1 ,

• (ii)    0 < |a| <  1 and |b| <  1 — |a| .

• <1 Suppose that ^ ( z ) = az + b is a self-map of D . If a = 0 , then |^ ( z ) | = |b| <  1 and hence we get (i). If a = 0 , then ^ (D) is an open disk centered at the point b of radius |a| . Since ^ is a self map of D , ^ (D) C D . In order for the image disk to be fully contained in the unit disk, we have |a| C 1 and 1 — |b| * |a| , which gives (ii).

Conversely, suppose ^ ( z ) satisfies (i) or (ii). If (i) holds, then |^ ( z ) | = |b| <  1 . In case (ii) holds, |^ ( z ) | = |az + b| C |az| + |b| < |a| + |b| C 1 for all z E D . In both cases ^ is a self-map of D . ▻

From Lemma 3.1, we notice that there are two trivial cases of affine symbols which always induce bounded composition operators. The first case is ^ ( z ) = b with |b| <  1 . The second case is ^ ( z ) = az with 0 < |a| C 1 . In particular, when |a| = 1 , this is a unitary operator.

As a result, throughout this section, we are interested only in the symbol ^ ( z ) = az + b satisfying

0 < |a| C 1 — |b| <  1 .                                          (2)

With h _n ( z ) = 71- z ⁿ , we get

β _n

^C^) = ± (^ b^{) n} = f^ ak b -

Thus,

IICAII ² = Ё ((n)|“| k |b| ^{n -} k I " У .                        (3)

and

) =    ', a ' b ” ^{- ke} k - n * ^k                          (4)

I 0,                  k > n.

In the sequel, we consider separately necessary conditions and sufficient conditions for boundedness of C _v on H ( в ) . As it will be seen, several of necessary conditions are “very close” to sufficient ones. We note that some proofs are similar to those of the case of entire functions in [10].

• 3.1.    Necessary conditions. We have following necessary conditions for boundedness

of C ^ on H ( в ) , which are similar to those appearing in [10] and proved analogously.

Proposition 3.2. Let ^ ( z ) = az + b satisfy (2) . Consider three statements:

• (i)    C ^ is a bounded composition operator on H ( в ) .

• (ii)    {”C ^ h _n ”} _n > o is a bounded sequence.

• (iii)    The following two inequalities hold:

  |a| limsup k →∞

  
  

  (md (‘      At’’)

  
  

  C 1 ,

  
  

  and

  |b| limsup^sup |f‘ ⁺ m^|a| ^k -^k- |     C 1 .

  — >^ \ k*0 ( \ ^k /     ^e k + m

  
  
  
  

Then the implications (i) = ^ (ii) = ^ (iii) hold.

< (i) = ^ (ii). Obviously,

WCphn” < lollop for all n * 0, where «C^«op is the operator norm of C,.

• (ii)    = ^ (iii). By (4) and the Cauchy-Schwarz inequality,

• 3.2.    Sufficient Conditions. The following lemma is adapted from [10, Lemmas 12 and 13] for the case of entire functions. We sketch the proof for the sake of completeness.

(k + m |a|k|b|m= Шhk+m,hk>| C \C^hk+m\ \ k /        ek+m for all k,m p 0. From this inequality and boundedness of {\C^hn\}n>0, both (5) and (6) follow. >

Lemma 3.3. Let pz) = az + b satisfy (2). Suppose that the sequence {\C^hn\}n>0 is bounded by some M > 0. Then there exists an integer s E N such that

∞∞

£ £ МЛ )| ² < ».

k =0 n =( s +1) k

<1 First, let n ^ k ^ 0 . By (3) and (4), we have

(1 + | a | ) ^{n - k} I a |^k | t C , h „ ,h _b ) | = ^£ (n _l k)| a | ¹ | a | ^k (^)| a | ^k |b ^{n - k} '

= E (k Г) 1=0 -        '

I a| k |b| ^l A

P k + l

n

V k +

|a|

k + l _{| b |} n - k - l

P k + l β n

C £ H CA + illl C K\\ C (n — k + 1) M 2 . l =0

This implies that

( n - k + 1) M

KCiphn, hk )| C 7m for all n ^ k ^ 0 .

^ ⁿ , k/i (1 + |a| ) ^{n - k} | а | ^k

Next, let s E N satisfy (1 + |a| ) ^s |a| >  1 . Then putting I = n — ( s + 1) k and using the above inequality, we get

∞∞

k =0 n =( s +1) k

c

∞∞

M 4EE k=0 l=0

( sk + 1 + 1) ²

(1 + |a| ) ^2(sk+l) |a| ^2k

C

4 £    ( sk + 1) ²    £ ^ ( I +1) ²

{ (1 + |a| ) ^s |a|} ^{2 k}£^ (1 + |a| ) ²¹ ’ k =0                   l =0

where the first series converges by the choice of s , while the convergence of the second series is obvious. >

Now we are ready to state and prove sufficient conditions for boundedness of C ^ on H ( P ) .

Proposition 3.4. Let pz) = az + b satisfy (2). Suppose the following condition holds sup n>1

n

E k=1

k|(C ^ h n ,h k >|

)

< TO.

Then C ^ is bounded on H ( P ) .

<1 By (7), there is a number M >  0 such that

^E k I{ C _v h _n ,h k >| ² <  M ² for every n >  1 .

k =1

Then the sequence {hC ^ h n h} n > 0 is also bounded by M , because

n

Hence, by Lemma 3.3, there exists a number s E N such that

∞∞

Take and fix an arbitrary function f ( z ) = ]E^ =o c n h n ( z ) in H ( в ) . First we compute { C ^ f, h k } in two cases with respect to k .

If k ^ 1 , then, by (4),

∞

( s +1) k - 1

^c n ^{{ C} ^ ^h n , h k > +

∞

E

^c n C v ^h n . h k >.

n =( s +1) k

which implies that

/ ( s +1) k - 1                 \ ²     / ^                    \ ²

2|{Cy>f, hk)|2 < I E   |cn{C^hn,hk>I) + I E |cn{C^hn’hk>I I n=k                            n=(s+1)k

• < sk ^E C^h.>| ² + ( ² J  |c n | ² ) (  52 |{C^h k ■

n = k                          n =( s +1) k           n =( s +1) k

( s +1) k - 1                           /    ^                  \

n =( s +1) k

Next taking summation over k ^ 0 and using the calculations above, we obtain

I              ^ ( s +1) k - 1                          ^      ^

k =1     n = k                            k =0 n =( s +1) k

∞ n

• < s E Ic n l ² E k I{ C ^ h n , h k >| ² + A h f h ² <  sM 2 h f h 2 + A h f h 2 -

• n=1     k=1

From this we conclude that C _v is bounded on H ( в ) . >

We can modify sufficient condition (7) by a slightly stronger condition.

Corollary 3.5. In Proposition 3 . 4 , condition (7) can be replaced by the following: There exists a function h : { 0 , 1 , 2 ,...} ^ R such that

∞ kh2(k) < to and k=1

sup m ^ 0

m } ^ h ( k ) , P k+m

for every k ^ 0 .

< For any n ^ 1 , by (4), we have

^k I k ) ^ 52 kh ^{2 ( k )}<  ^то ,

which implies (7). Thus, by Proposition 3 . 4 , C . is bounded on H ( в ) . >

We can further develop condition (8).

Corollary 3.6. In Corollary 3 . 5 , condition (8) can be further replaced by the following inequality

|a| limsup ( sup (fk + m^ k >^ у m^0 ( \ ^k /

_m β _k ^в к+т

Ik) <1

which is very closed to necessary condition (5) in Proposition 3 . 2 .

<1 Suppose condition (9) is satisfied. Then there exists r E (0,1) such that k1

> I < r <  1 .

|a| limsup ( sup (fk + mk| k -u у m>0 I V k / m βk вк+т

For this r, by the properties of limsup, we can find a constant M > 0 (depending only on r) so that sup f f+ m |a|k|b|m Qвк | < Mrk for all k > 0.

m ^ 0 ( \ ^k )        ^e k+m J

Taking h ( k ) = Mr k , we can easily verify that ^ k =i kh ² ( k ) <  to . Thus we obtain condition (8) and so C _v is bounded on H ( | ) . >

Corollary 3.6 shows that the strict version of (5) is a sufficient condition for boundedness of C _v on H ( в ) . Using some ideas in the proof of [10, Lemma 18], we prove that the same result is true for the strict version of (6).

Proposition 3.7. Let ^ ( z ) = az + b satisfy (2) such that

|b| limsup (sup (fk + mK | m >^ у k > 0 I V ^k /

k βk вк+т

Im) < '•

Then C . is bounded on H ( в ) .

< Similarly to the proof of Corollary 3.6, there exist M > 0 and r E (0,1) such that sup f f + m^ |a|k|b|m двк | < Mrm for all m ^ 0. k>0 I V k / A+m J

From this and (4), it follows that

\^ C _v h k+m ,h k )| = fk + m |a| k \ b \ mk C Mr m for all m,k >  0 .         (11)

\ ^k /         P k+m

Take and fix an arbitrary polynomial f ( z ) = ^2^=0 c _n h _n ( z ) € H ( в ) , where only finite many c _n are nonzero. By (4), we get

∞

∞∞

∞∞

C ^ f = ^Cf, h k >k = E E C k+m < C ^ h k+m , h k >k = E Eck + m ^C ^ h k + m , h k ^ '

k =0

k =0 m =0

m =0 k =0

This and (11) imply that

∞

IC / л < E m=0

∞

52 C k+m ( C ^ h k+m ,h k>k k =0

∞∞

= EE m=0 k=0

2                     \2

\ ^c k + m \ \(C V h k + m ^{, h} k ) \ I

∞ c m л/iiE rm m=0

M

1 — r

I f h.

From this and the density of the set of all polynomials in H ( в ) , the assertion follows. >

Combining Proposition 3.2, Corollary 3.6 and Proposition 3.7, we have the following important summary for boundedness of C _v on H ( в ) .

Theorem 3.8. Let ^ ( z ) = az + b satisfy (2) .

• (1)    If the operator C ^ is bounded on H ( в ) , then the following two inequalities hold:

• ⁽ⁱ⁾    \a\ limsup f sup {( ^k+m )\ ^b \ ^me^e km } k ) ^C 1 ,

k ^ro \m> 0

• (ii)    \b\ limsup f sup {( ^kt^m )\ ^a \ ^kefkE } m ) <  ¹ m ^ro \k > 0 ¹          k k+m    j

• (2)    Conversely, if one of the following conditions hold:

• (iii)    \a\limsup fsup {(^k+m)\^b\^me^ekm}^k) < 1, k^ro \m>0             ⁺

• ⁽ⁱv)    \b\ limsup fsup {(^k+m)\^a\^ke^ekE} m) < ^1,

4. The Case when ^(z) are Monomials

• 4.1.    Monomial symbols. Let the symbol ^(z) = cz^m(c € C \ {0}, m € N \ {1}). As in the previous section, to define well Cϕ the symbol should be a self-map of D. We can easily check that it is so if and only if 0 < \c\ C 1, i.e. c € D \ {0}. Thus throughout this section, such a symbol ϕ is supposed to consider.

m^ro \k>0 1            +m } / then the operator C^ is bounded on H(в).

We can easily prove the following criterion for boundedness of C ^ on H ( в ) .

Theorem 4.1. Let c € D \ {0} and m € N \ {1}. The operator C^ induced by the symbol ^(z) = czm is bounded on H(в) if and only if sup n∈N

\ c \ ⁿ 3 mn

β n

< TO.

Moreover, in this case, the above supremum is equal to |C ^ | _op .

< Suppose that Cp is bounded on H(в). Then for every n E N, lollop » BC^hnB = -1 |(czm)n| = tcnlmn, βn             βn which gives

|c|nemn         и sup о C HC^Hop- n∈N  βn

Conversely, suppose that (12) holds. Then for every function f ( z ) = ^2^=0 a n z n E H ( в ) , we have

∞∞

|C . f i ² = if ( cz m) ll ² = E kPH²^,, = E Ы2в.2

n =0                n =0

| c | C sup — n ∈ N

n _{β mn} β n

У ^ | а n | 2 в П = f sup — n =0            ^{n ∈ N}

n _{β mn} β n

^f ■ У β n

У 2 if ii² .

This shows that C _v is bounded on H ( в ) and also

Wl^mn

H C^||op ^{C sup}    о

.

n ∈ N β n

From both necessity and sufficiency it follows that

|с|Пвтп _ и,-, и sup о — HC^Hop- ^ n∈N  βn

The following is a consequence of Theorem 4.1.

Corollary 4.2. Let c E D \ {0} and m E N \ {1}. If limsup fУ " n→∞   βn

>    ’

then Cczm is unbounded on H(в). In particular, if limsup f У n n→∞   βn

= то,

then C cz m is unbounded on H ( в ) for every 0 < |c| C 1 .

<1 By the hypothesis, there exist a sequence (nk) f то and a number а > 0 such that f mmnk у k > а > A. for every k e N.

V в п к J              |c|

Hence,

complicated than on classical Hardy space H 2 (D) and Bergman space A 2 (D) . More precisely, it is well-known that every holomorphic self-map of D with relatively compact image in D induces a bounded composition operator on these spaces. However, by Corollary 4.2, this is not valid for general weighted spaces H ( в ) .

We end this section with the following sufficient conditions for boundedness of the operators C ϕ induced by such holomorphic self-maps of D .

Theorem 4.3. Let ^ be a holomorphic self-map of D such that ^ (D) is compact in D . If

∞

E e n < -, n =0

then C _v is bounded on H ( в ) .

<1 Take a number r G (0 , 1) so that ^ (D) H ( в ) , we have

C r D . Then, for every f ( z ) = ^ П =₀ a n z n G

∞ sup If(z)| C E |an|rn C

|z|Crn

/ ~ T2n\ 2 / ~\

E i 5>M

\n=0 вп / \n=0/

= C r I f II,

where

/ ~  2n \ 2

C r := E >

\n=0 ^вп /

< to , since, by (1), limsup — - C r ² <  1 - n →∞ _β n n

Now writing C ^ f ( z ) = ^2^ =0 b n z n and using the Cauchy integral formula, for every 5 G

(0 , 1) , we get

|b n | =

I ( C ^ f ) n (0) | n !

C — 2 n

J   z n+1

| z =

1                       1                   C„

^C   max^f ^{( z ) | C    s} up^{| f ( z ) | C} ^f ¹

6 n | z =               6 n | z | C r            6 ⁿ

which implies that |b _n | C C r hf II for every n G N .

Consequently,

∞             ¹ 2

E IbnI2en    C Cr If I n=0

i: p^

n =0

From this the assertion follows. >

Remark 4.4. Condition (13) is essential but not sharp for Theorem 4.3. Indeed, for

1 ,

в п = 1 ,

n ⁿ ,

n = 0, n = 1, 2, 3 mod 4, n = 0 mod 4, n = 0,

(13) does not hold and, by Corollary 4.2, C _cz m is unbounded on H ( в ) for every c G D \ { 0 } and m G N \ { 1 } .

On the other hand, for в _п = 1 , (13) also does not hold, however in this case H ( в ) is Hardy space H ² (D) and hence C _v is bounded on H ( в ) for every holomorphic function ^ with ^ (D) C D .