GENERALIZED COMPOSITION OPERATORS ON WEIGHTED FOCK SPACES Текст научной статьи по специальности «Математика»

ISSN 2074-1871 Ycfri/iMCKi/m MaTeMaTi/mecKi/m wypHa^. TOM 16. № 2 (2024). C. 105-117.

GENERALIZED COMPOSITION OPERATORS ON WEIGHTED

FOCK SPACES

M. WORKU, L.T. WESEN

Abstract. The generalized composition operators J^ and Cf, induced bv analytic functions g and # on the complex plane C, are defined by

J?(f )(*) = j f'($(u))g(u)dumd Cf(f )(z) = j f'(u)g(u)du. 0 0

In this paper, we consider these operators on weighted Fock spaces Jf, consisting of entire functions, which are £p(C)-integrable with respect to the measure d\(z) = (z\lA(z), where dA is the usual Lebesgue area measure in C. We assume that the weight function № in the spaces satisfies certain smoothness conditions, in particular, this weight function

\z I2

grows faster than the Gaussian weight defining the classical Fock spaces.

We first consider bounded and compact properties of Jf and Cf, and characterize these properties in terms function theory of inducing functions g and given by

($(z))-&(z)

JVlg .- ..tv,/ n e

1 + V (z)

Our characterization is simpler to use than the Berezin type integral transform characterization. In some cases, our result shows that these operators experience poorer boundedness and compactness structures when acting between such spaces than the classical Fock spaces. For instance, for <P(z) = z, there is no nontrivial bounded J^ and C® on weighted Fock spaces. In the case of classical Fock spaces, they are bounded if and only if g is constant.

In the second part of this paper, we apply our simpler characterization of boundedness and compactness to further study the Sehatten-elass membership of these operators. In particular, we express the Schatten SP(J![) class membership property in terms of Cp(C, A&dA)-integrability of M^.

Keywords: weighted Fock spaces, generalized composition operator, Sehatten-elass, boundedness, compactness.

Mathematics Subject Classification: 47B32, 30H20, 46E22, 46E20, 47B33

1. Introduction

Given analytic functions g and ^ on the complex plane C, generalized composition operators J^ and C^, induced by g and are defined by

jf(f )(*) = J f ($(u;))g(u;)dw*adCf(f )(z) = J f (u)g(u)dw. (1.1)

00

M. Worku, L.T. Wesen, Generalized composition operators on weighted Fock spaces. © Worku M., Wesen L. T. 2024. Submitted June 27, 2023.

Specifically, for the case ^ is an identity function, that is, <P(z) = z,

z

JtU)(z) = C*U)(z) = j f(v)9(u>)du>

0

is the well-known Volterra companion operator denoted by Jg. Operators in are first introduced by S, Li and S, Stevic |2|, |3|, on some spaces of analytic functions defined on a unit disc, and then considered by several authors on different function spaces. In particular, on classical Fock spaces, T, Mengestie |4|, |5| characterized bounded, compact and Schatten-class membership properties in terms of the properties of inducing symbols g and Later in [ ], together with the first author of this paper, they studied some topological properties of these operators. In |6|, T, Mengestie and S, Ueki studied boundedness and compactness of the particular operator Jg, on weighted Fock spaces T^ (defined below), with a weight function W satisfying the following conditions (as in |1|):

(1) W : [0, to) ^ R+ is twice continuously differentiate function and W (z) = W (|z|), C,

(2) The Laplacian of W is positive and there exists a function u(z) obeying1

f 1, 0 ^ |z| < 1

( ) 1 (AW(|z|))-1, |z| ^ 1,

has the following properties;

(I) v is a radial positive differentiate function and decreases to zero as | z| ^ to, (II) lim u'(r) = 0.

(III) either there exists a constant a > 0 such that u(r)ra increases for large r or lim v'(r) log ¿r =

Throughout the manuscript we assume that the weight function W satisfies the above conditions,

|Z|2

We note that this kind of weight function grows faster than the Gaussian weight defining the classical Fock spaces.

Next we define a corresponding weighted Fock space. Let d\(z) = e (z^dA(z), where dA is the Lebesgue area measure in C, Then, for 0 < p < to, the weighted Fock space Tf is space of analytic functions on C, which are CP(C, dA)-integrable, that is,

\\f\\Pj* := / |Wfe-P*{z)dA(z) < to. (l2)

c

The growth type weighted Fock space TX consists of analytic functions on C such that

\\fWj* := sup |/(z)|e< to. (1,3)

zee v '

Recently, in |8|, Z, Yang and Z, Zhou characterized boundedness and compactness of the generalized composition operator J^, acting between the weighted Fock spaces Tf and Tf, for 0 < p,q < to, in terms of the Berezin type integral transform,

J (1(++W^r ^-qnz)dA(z),

e

where k(Wt&) is normalized kernel function of Tf- Motivated by this research, the purpose of the present work is to further characterize boundedness and compactness of J^ on weighted

1 The notation U(z) x V(z) means both U(z) < V(z) and V(z) < U(z), where U(z) < V(z) (or V(z) > U(z)) means that there exists a constant C such that U(z) < CV(z) holds.

Fock spaces in terms of a simpler function

JVl9 (Z)- 1 + y, (z) e '

by using the notion of embedding map. We also do the same for the similar operator Cf. More-

*>p(T2

over, applying the simplified condition, we characterize the Schatten class SP(T2> ) membership

property of Jf and Cf

2. Preliminaries

We begin the section with some preliminary results that will be used in the proof of our main results. In |1|, |6|, weighted Fock spaces were described in terms of a derivative, which expresses (1,2) and (1,3) as

11/11^ - 1/(0)ip + / (1 e-WdAM (2.1)

c

for finite p and

||/x|/(0)| + e-*(Z). (2-2)

This type of estimate is usually called Littlewood-Paley type estimate, and it plays an important role specially in studying integral operators. Let D(a, r) be a disc with a center a and a radius r. By Lemma 7 in [ ], for 0 < p < ro and subharmonic functions & and f we have the pointwise estimate

if(z)lpe-pf(z) < i if(w)lpe-pf(w)dA(w) < u(z)-2Hf , (2.3)

a2u (z)2 J

D(z,av(z))

for a small positive number a, which shows that the space Tf is a reproducing kernel Hilbert space. At the same time, an explicit formula for the kernel K(w,f) is not known vet. However, if {en} is an orthonormal basis of Tf, then

d_

)^n(w) emu (w,W ) / y K^n\

n n

Moreover, by Corollary 8 and Lemma 22 in |1| we have

HKwfMl* = ^ KHi2 x ^(w)-2e2f(w) (2.4)

n

£ K(w)i2 x !K(w^(V(w))2 x (W(w))2V(w)-2e2f(w). (2.5)

K(w,w)(z)— en(z)en(w) and g=jK(.w,v)(z) = en(z)e'n(w).

9 K

In |1|, there were constructed test functions, which played the role of kernel function, as it is stated by the following lemma.

Lemma 2.1. For a large number R there exists a number rj(R) such, that for any w E C with |w| > Tj(R), there exists an entire test function F(w,r) in Tf with

^F^r^t* x v(w)p for 0 < p < ro and ||F(w,R) x 1. In particular, for z E D(w, Ru(w)),

\F[w,R)(Z) 1 + W (z )

■e-w(z) x 1. (2.6)

We also need the following statement called covering lemma to prove our main results; this lemma was proved in |1|,

Lemma 2.2. Assume that t : C ^ (0, to) is a continuous function such that

z — w

|t(z) — t(w^ ^ |—-—^, z,w e C, lim t(z) = 0. 4

Then there exists a sequence {zn} such, that

(i) Zi e D(Zn, t(Zn)) for i = n, and Un^iD(Zn, t(Zn)) = C;

(ii) Uze d(z„,t(zn))D(z, t(z)) C D(zn, 3t(zn)) and a sequence {D(zn, 3t(zn))} is a covering of C of finite multiplicity.

3. BOUNDEDNESS AND COMPACTNESS

As it has been noted in the Introduction, the boundedness and compactness of the Volterra Jg

Jg

comparison with the case of the classical Fock spaces. Our result in this paper shows that this is also the case for the generalized composition operator Jf in general.

Before proving our main results, we state the next lemma, which gives the form of <P whenever the function Mf(z) or Mff) (z) is uniformly bounded over C,

Lemma 3.1. Let g and <P be nonconstant entire functions. If there exist a positive constant C such that Mf(z) ^ C or Mff) (z) ^ C for alI z e C, then <P(z) = az + b for some a,b e C with |a| ^ 1.

Proof. We first assume that Mf is uniformly bounded over C, Then there exists a positive C

19« C(){e-(f(z')-^(z))-1,

and since g is nonconstant, we have W(<P(z)) — W(z) < 0. This implies

lim sup W($(z)) — W(z) ^ 0. (3.1)

Since $ has its own power series expansion, by ( ) we conclude that $ reads as <P(z) = az + b for some a,b e C with |a| ^ 1. The case when Mff) is uniformly bounded can be treated in the same way. The proof is complete, □

Theorem 3.1. Let 0 < p ^ q ^ to, and let g and <P be nonconstant entire functions. Then i) Jf : T* ^ Tf is

(a) bounded if and only if supzeC A W(z)sMf(z) < to;

(b) compact if and only if lim|z|^.x AW(z)sMf(z) = 0, where

p ^ q < to

=

—

pq , 1

-, p < q = to 0, p = q = to.

(ii) Cf : Tf ^ T? is

(a) bounded if and only if supzeC A W (z)s Mff)(z) < to;

(b) compact if and only if lim|z|^x AW(z)sMf(f)(z) = 0.

Proof, (i) The proof for the case p ^ q = to was given in Theorem 2,1 in [ ], Here we study the case p ^ q < to. For f e T^, an application of the Littlewood-Palev type formula in (2,1) to Jff gives

WJff Wk - J U' ^e-mz)dK(z) = J

C C

where d/i(q>f>w) is a pull-back measure given by

f-1(B)

for every Borel subset B of C, Then it is easy to observe that the operator Jf : T^ ^ T' is bounded (respectively, compact) if and only if the embedding operator E : 0(Pt&) ^ T' is bounded (respectively, compact), where O^^) is space of entire functions such that

I f(z)I*

(1 + W (Z))P

(z

)dk(z)

C

is finite. But, by proposition 3,8 of [ ], E is bounded if and only if for some 5 > 0

sup—^ f (W (z))q eq*(z)dii(qf*)(z) (3.2)

w^C u(w) P J

K ' D(w,Su(w))

E

Hm i (W(z))qeq*(z)di(qfm(z) = 0. (3.3)

i/(w) p J

v ' D(w,Su(w))

Substituting back i(q) into (3,2) and (3.3), we obtain that Jf is bounded if and only if

1

(Mf)q(z)dA(z) < to, (3.4)

D(w,&v(w))

2L I w^c jj(w) p

and the operator is compact if and only if

Hm i (Mf)q(z)dA(z) = 0. (3.5)

y(w) p J

v ' D(w,Su(w))

Our next step to simplify these conditions.

q — p _

(a) First assume supzeC AW(z)^Mf (z) is finite. From (3.4), using the fact that u(w) x u(z) for z e D(w,5f(w)), we obtain

sup —i (Mf)q (z)dA(z)

w^C u(w) P J

y ' D(w,S v(w))

1 C 1

^ (supAW(w)i—M<f(w))(sup-2q -2(P—q)dA(z)) ^

Wc a Awec u(w)V J (u(z)) p J { '

y ' D(w,6v(w)) VV^;;

1 f 1

< suP 2q -Kp—^dA(z) < to.

w^C u(w) P j (u(z)) P

K ' D(w,Su (w))

On the other hand, if (3,4) holds, then using the estimate in (2,3), the subharmonicity of g$r'(<p)e*and the fact that 1 + W(w) x 1 + W(z) for w e D(z, bv(z)), we get

sup AW (z)(Mf)q(z)

.zee

— 2 q

< sup(, U(zj * i \g(w)\q(W($(w)))qe4*Ww))-q*(w)dA(w))

JcK(1 + W(z))i J ^ v v v ;;; v )) ^

D(z ,Sv(z))

sup—I (Mf)q(w)dA(w) < m.

zee v(z)2q J

K ' D(z,Sv(z))

q-p

These relations and (3,6) yield that Jf is bounded if and only if the function AW(z) ^Mf(z) is uniformly bounded over C,

(b) From ( ), ( ) and ( ), it is easy to see that the operator is compact if and only if

q — p _

the function AW(z)^Mf(z) goes to zero as \z\ ^ m.

(ii) The proof of this part is very similar to the proof of part (i) above and this is why we omit it. The proof is complete, □

The above theorem shows that, if p is strictly less than q and Jf : Tf ^ Tf (or Cf : Tf ^ Tf), then the unboundedness of the L aplacian of W forces the operator to have poorer structure compared with the classical Fock spaces case, cf. [7], In particular, the operator Jg : Tf ^ Tf is bounded if and onlv if

!

q—p .

AW(z) pq \g(z)\, q < m AW(z)p \g(z)\, q = m

is bounded. This holds true if and only if g is the zero function and hence there is no nontrivial bounded Jg in this case. But, in the classical Fock spaces case the operator is bounded if and only if g is constant (see [ ]). We may now proceed to the case 0 < q < p ^ m, in this case, our next result shows that boundedness and compactness of the operator Jf : Tf ^ Tf (respectively, Cf : Tf ^ Tf) are equivalent.

Theorem 3.2. Let 0 < q < p ^ m, and Iet g and <P be nonconstant entire functions. Then

if : -pf g : Tp

(i) Jf : Tf ^ Tf is bounded or compact if and only if f (Mf)r(z)dA(z) is finite, where

e

pg

{f ■ *_< m (3.8)

q, p = m.

(ii) Cf : Tf ^ Tf is bounded or compact if and only if f (Mf$))r(z)dA(z) is finite, where r

e

is as defined above.

Proof, (i) As it has been shown in the proof of Theorem 3,1, Jf : Tf ^ Tf is bounded (respectively, compact) if and only if the embedding operator E : ^ Tf is bounded

(respectively, compact). But, by Proposition 3,8 in [ ], the boundedness and compactness of E are equivalent to the condition that the function

T(z) := i (Mf)q(z)dA(z),

V (w)2 J a

D(w,&v(w))

r

belongs to Cq (C, dA) for some 5 > 0, Now we consider two different cases.

r

Case 1: p < to. Suppose T is in Cq (C, dA), where r = Then using the estimate in

( ) and the fact that 1 + W(z) x 1 + W(w), for z e D(w, 8v(w)), we obtain

'(Mf)r (z)dA(z) = f I9(Z1 e^W^z)

cc

<[ {zni\w, {z))q J 19 (wW (V W Z))f eq*(f(^-q^ dA(w)^j dA(z)

v (z)2(1 + V (z))i

C D(z,Sv(z))

<f( ^ i b ^ (W f( f)))q eq*(f(w))-q*(w)dA(w)] P—qdA(z) < to.

~ J \v(z)2 J (1 + W(w))q V 7 W

C D(z ,8u(z))

On the other hand, if Mf is in C(C, dA), r = then (2,1) and the Holder inequality give

y p q ^

vft wr. - *(z)W

II yJ Hjqr J (1 + yi (z))q ^ >

<(i , 11:1!'' e ^ ^^ dMz)) p 4 >(*(:))r e WW) dA(z))

~\J (l + V(<P(z))P v7 \J (l + V(z)Y v7

c

<( [ \f '($(z))\P e(*(Z)) dA(z\ < || f||i

<\J (l + V($(z))'e d (Z)J

(1 + W (<P(Z))P yjJ

C

where in the latter estimate we have employed a change of variable and the identity <P(z) = az+b with 0 < ^ ^ 1 due to Lemma 3,1, Therefore, Jf is bounded and hence the function T belongs to Ci(C, dA).

Case 2: p= to. Suppose T is in C(C,dA). Proceeding then as in Case 1, we get

fiMtnz)dMz) = j ^ Wy (f(z))-• ^Hz)

C C

< / u{zyq{z))q J ti(w)^(W($(z)))qeq*(f(w))-q*(w)dA(w)dA(z)

C D(z,6v(z))

< i -71r i ^^(W^*)))qeq*(f(w))-q*(w)dA(w)dA(z) < to.

J v(z)2q J (1 + W(w))q y J w

C D(z ,S v(z))

On the other hand, if Mf e Cq(C,dA), then Mf is bounded and hence @(z) = az + b with 0 < |a| ^ 1 (see Lemma ), Using this and the estimate in ( ), we find:

<} sun | r w.u)dK(z)\

\ z€C (l+W ($(Z))" )\J (1 + 1" M)" )

< 11/11?,»•

^ oo

Therefore, Jf is bounded and hence T is in C(C,dA). In view of Cases 1 and 2 we conclude that the operator is bounded or compact if and only if f (Mf)r(z)dA(z) is finite,

C

(ii) Here proof is very similar to the above proof and we omit it. The proof is complete, □

4. SCHATTEN-CLASS MEMBERSHIP

The singular values of a compact operator T on a Hilbert space H are the square roots of the positive eigenvalues of the operator T*T, where T* denotes the adjoint of T. Given 0 < p < <x, the Schatten class of a Hilbert space 'denoted bv SP(H), is the space of all compact operators T on H with its singular value sequence {¡3n} belonging to the sequence space lv. The space SP(H) is Banach space for 1 ^ p < <x with the norm

\\T k = ( £ IPnlP)P

In particular, Si(H) is called the trace class and S2(H) is called the Hilbert-Schmidt class. We next characterize the Schatten class membership of Jf and Cf on . Our next theorem gives necessary condition for these operators to belong to Sp(Tf ) and sufficient condition will be provided later.

Theorem 4.1. Let 0 < p < œ and (g, <) be pair of nonconstant entire functions. If Jf (respectively, Cf) is in the class Sp(Tf ), then Mf (respectively, Mff)) belongs to Cp(C, AWdA).

Proof. We will prove the statement for the operator Jf, and a similar procedure works for the operator Cf. Firs, we define a scalar product {•, •)* on by

<f, g)* := f (0)g(0) + f e-2*(z)dA(z), (4.1)

C

which by ( ) gives an equivalent norm on Tf, and divide the proof into two cases.

Case 1: 0 < p < 2. Since the operator J^ is in Sp), then the operator (Jf)*(Jf) is in Sf ) and has a canonical decomposition:

( Jt)* (Jf)f = £&< en)*e n

where {en} is an orthonormal basis in Tf and {ffn} is the sequence of singular values of a Jf)*(Jf

positive operator (Jf)*(Jf). Moreover,

\\( Jf)*( Jf)\\l £ = £ \fn\p.

Using the estimate in (2.5) and the Holder's inequality, we then obtain

J(MtY(w)№(w)dA(w) = j ^^(^p)r(w)dA(w)

CC

r \g(w)nv($(w))y-2II^Kt^mw))^

(l + W (W))P w)))p-2 le 'n(<

(l + W (W))P ~ (4.2)

v- rig(w)\p(w'(<(w)))p-2K(«(w))I2

p(p-2(f(w))-pf(w) j\(w)

2-^i (i + w(w))p ( )

p

<V / f \9(w)l2i e'n(<(w))\2 № (w) dA(w)\ 2

(l + w (W))2 6 (W)

n v C '

2-p

e'n(<(w^2 e-2f(f(w))dA(wW) 2

(W (<(W)))2

C

)

Since Jf is compact, by Theorem 3.1 and Lemma 3.1 the function $ has the form <$(z) = az + b for some a,b e C with 0 < |a| ^ 1. Then a change of variable gives

i -2*(f(w))dA(w) < i J^Mw^L-2*(f(w))dA(w)

J (W'(<*(w)))2 y J~J (1 + W($(w)))2 y J

CC

1 f W^e-2nC)dA(() xlM^ = 1,

I«\2J (i + v(0)2

and

p

E ( fe-^dAw) = BtfrtfK,ol

n \ c ' n

(i + FW))2 -e--(w)dA(w)] = EKwr-

£\ ^ = W(J!)'(J!)»Sp ■ — i

By (4.2) and the above two estimates we obtain

J (Mff(w)AW(w)dA(w) < ||(Jf)*(< to

C

and this is the desired inequality.

Case 2: 2 ^p < to. Let {zn} be the sequence as in Lemma and {en} be an orthonormal basis in Tf, Let T be an operator taking en(z) to f( zn,R) (z) = F(z„z)(z ■ By Proposition 9 in [ ], T is bounded operator and JfT'm in Sp(Tf), and by Theorem 1.33 of [ ],

£ WJf Azn,K)WlPjf = E WJfTenW^ < TO. (4.3)

n n

Using ( ) and the estimate u(z) x u(zn) for z e D(zn,5u(zn)), we get

J (Mfy(z)AW (z)dA(z) = £ J ^ ¿*(*(z))-*(z)AV (z)dA(z)

C n D(zn,5v(zn))

< E»(Zn)-P I (H(*))pv(z)-2dA(z),

n

D( zn,S v( z„))

where

^(W' ($(w)" 2 (1 + W' (w))2

D(z ,Sv(z))

Using the estimate in (2.6), we proceed as follows:

H (z) = i ^ (W)\2(V (*(W))2 e2*(*(-))-2^)dA(w).

p

E « :n)-*( j \9(f+V^)2 ^mZ))-2* OdA^Y

n D( zn,S v( zn)) '

. . p e( / 1 * t ^^<- )<"(*)' <Eii-tt,^

<EI / (er^e e-2'(l)<E "*

D(zn,&v( z„))

which, by (4.3), is finite, and therefore, Mf belongs to C(C, AW dA). The proof is complete. □

Theorem 4.2. Let 1 < p < to and (g,$) be a pair of nonconstant entire functions. If Mf (respectively, Mf(f)) is in CP(C, AW(<P)dA), then Jf (respectively, Cf) is in the class Sp(T2f).

Proof. The proof of the theorem for the two operators is very similar. This is why we provide the proof only for the operator Jf. We consider two cases.

Case 1: 1 < p < 2. Let {en} be an orthonormal basis of ■ Then by Theorem 1,27 in [ ], Jf is in the class Sp(Tf) if and only if

y] \< JgZ n, en)*\P <

n

where <■, ■) * was defined in ( ), Since p > 1, the Holder inequality yields

Z\<Jf<n,OT <£ (/ dA<z))

n n V if /

fP P \ \p < V i [ \g(z)en($(z))e'n(z)\p-2f(z

(1 + V (z))2

C

^J {1 + v)Pz))2PWn(*(*))\pK(z)\2-*e-2f(z)dA(Z) " C

e'n(z)\ -2f (z)i\(\\P 1

J (1 + V(z))2

C

\g (z)\p

)dA(z)y

e 2f(z)dA(z)

xj ^-i- nz))2p (E \-'n(^(z))\P\e'n(z)\2-p)e-2f(z)dA(z). c n

Since p < 2, applying the Holder inequality, using ( ) and the estimate

^{z))2(1-p) < -(z)2-P

u($(z))2-P'

f 2 —f

£Kmz))\pK(z)\2-p ^ (E^m2)E\e'n(z)\2)2

n n

d d = 11 ^ZK(f(z),f) ^ 11 -^K(z,f)

xHK{Hz)f )IIPTi (V (<P(z)))PIlK(z,f )II2-(V> (z)f _ (V (<P(z)))p(V (z))2-pepf(t(z))+(2-^)f(z) X v(<P(z))pv (z)2-p

(V (<P( z)))p(V (z))P epf (t(z))+(2-p)f (z) < v($(z))2 .

Substituting this estimate into (4,4), we get

Vl(Jfe e ) |P < i \g(Z)\P^((P(Z)))P(V(Z))PePf(t(z))-Pf(z)dA( ) Jge^ en )*\ <J (1 + v (z))2p U($(Z))2 e dA(z)

n C

x I (m*(z))(<P(z))dA(z) < ro.

(4.4)

Case 2: 2 ^ p < ro. Similarly, let {en} be an orthonormal basis of , By Theorem 1,33 [9], it is sufficient to show that IIJfen IIP* < ro, Fo r p = 2, by estimates (2,1) and (2,5)

n 2

we find

E Vt«^ x ^ff^ e-»^MA W n n C

_ r b km*' ($(z)))2 e 2#(f(z))-2#(z) dA( z)

X J (1 + W' (Z))2u (<P(z))2 ^^ (4'5)

C

= J (Mf(z))PAW($(z))dA(z).

C

For p > 2, by the estimate in ( ), the Holder's inequality and the norm estimate in ( ) we obtain

E x e^^A«) P

n c

<

„ -*' (z)+(*-2)' (f(z)) d A ( \

¿A/ (l + W' (z))2 6 d (Z))

)) d ( )

w I K(*(z))\2\g(Z)\*(l + W(<P(z))*-2„

-*' (z)+(*-2)' (f(z)) d A (-) \

l (l + W'(z))2 ( ') (4.6)

e'n($(z))\2 ^ -

(l + W' ($(z)))2<

x , / ■ -i.;'/-e-№(f(z))dA(z^ 2

E|e'n(«(z))\2) lg(z)\P(W'(<(z))p-2

< I _J1_p-pf(z)+(p-2)f

<J (1 + W' (z))2 e dA(Z)

C

^ r \g(z)\p(w'(<(z))p p-^z+vrnz))^,

~ J (1 + W'(z))2v(<(z))2 aK(Z)

C

= J (m*(z))PAW(<(z))dA(z) <J (m^(z))PAW(z)dA(z).

CC

By the above relations and (4.5) we conclude that Jf G SP(Tf ). The proof is complete. □

Our next proposition gives another characterization of Schatten class membership of Jf and Cf on Tf, It holds for any compact operator T on Tf.

Proposition 4.3. Let 0 < p < to an d (g ,<) be a pair of nonconstant entire functions such that Jf (respectively, Cf ) is compact on Tf. Then

(i) if 2 ^ p < to and Jf (respectively, Cf) is in SP(Tf ), then the f une tion \\Jf k(w>f)\\^^ (respectively, \\Cfk(w,f)\\^) belongs to CP(C, AWdA).

(ii) if 0 < p < 2 and the function \\ Jfk(w>f)\\r* (respectively, \\Cfk(w>f)\\r*) belongs to CP(C, AW dA), then Jf (respective ly, Cf) is in Sp(Tf ). 2

Proof. We again prove for the operator Jf only. Since Jf is compact, it has a canonical decomposition,

Jfî = f, en)en, (4.7)

n

where {¡3n} is a sequence of real numbers tending to 0 as n goes to infinity and {en} is an orthonormal basis of (see, Theorem 1,20 in [9]), Moreover,

IIJXP = £ \Pn\p.

n

Applying ( ) to the normalized kernel function k(w,z), we obtain

IIJfhw^t* = II K(w,f)II-* £ \Pn\2\\en(w)\2 X u(w)2e-2Z(w £ ^H!en(w)\2. (4.8)

2

n

(i) If p= 2, then using the estimate in ( ), we find

g

C

IIJfkwmIIP* AV (w)dA(w) x e-2Z(w)^ \Pn\2\\en(w)\2)dA(w)

= £ № J \en(w)\2e-2Z (w) dA(w) = £ \/5n\2 = I I Jf 11 % < ro.

n C n

If p > 2, then again using the estimate in ( ), applying Holder's inequality and using

£ \ en(w)\2 = K(w,Z)(w) = I IK(w,f)II2r*

n

together with estimate (2.4), we get

I I Jfk{w,f)I IP* AV (w)dA(w) x u(w)p-2e-pZ(w) ^ \0n\2\\en(w)\2)f dA(w)

p

k,. ,^IIP ,AV(w)dA(w) X I u(wn)p-2P-pZ(w)i\ \2\\p.Jm)\2\

C C n

^ u(w)p-2e-pf(w\Y, \Pn\p\K(w)\2){J2 K(w)\2) 2 dA(w)

C

x £ \/3n\p \en(w)\2e-2Z(w)dA(w)

n C

= £ \Pn \P =I I Jf 111 < ro.

n

(ii) Using the estimate in ( ), applying Holder's inequality and the estimate in ( ), we obtain

I I Jf I I I = £ \Pn\P = £ \Pn\p 11 en 11 p* = £ \Pn\p i \ en(z)\2e-2Z(z^ dA(z)

n

(' I I K(zf) 11

£ \^n\P J \ en(z)\2 u(z)2p2 dA(z)

^ (,)\2\ ^^

V(z)2

J [2^\Pn\2\en(z)\2) 2 dA(z)

< f&\Pn\2\en(z)\2) f (E\en(,)\2) " IIKi0LdA(z) C n n ^ '

II Jfk(z,z)IP*AV(z)dA(z) < ro,

'g '"(z,< C

from which the conclusion follows. The proof is complete. □

Acknowledgments

The authors would like to thank Prof, Tesfa Mengestie for discussions on the subject matter,

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Mafuz Worku,

Department of Mathematics, Jimma University

378, Jimma, Ethiopia

E-mail: mafuzhumer@gmail.com

Legessa Tekatel Wesen,

Jimma University,

Department of Mathematics,

378, Jimma, Ethiopia

E-mail: wesen08@gmail. com