Compactness of Composition Operators from the p-bloch Space to the q-bloch Space on the Classical Bounded Symmetric Domains

Transcription

1 Advaces i Pure Matheatics, 4, 4, Published Olie Deceber 4 i SciRes htt://wwwscirorg/oural/a htt://dxdoiorg/436/a4474 Coactess o Coositio Oerators ro the -Bloch Sace to the -Bloch Sace o the Classical Bouded Syetric Doais Jiabig Su *, Huiua Li, Xigxig Miao, Rui Wag School o Matheatics ad Statistics, Jiagsu Noral Uiversity, Xuzhou, Chia Eail: * sub@sueduc Received 3 October 4; revised 3 Noveber 4; acceted Deceber 4 Coyright 4 by authors ad Scietiic Research Publishig c his work is licesed uder the Creative Coos Attributio teratioal Licese (CC BY htt://creativecoosorg/liceses/by/4/ Abstract this aer, we itroduce the weighted Bloch saces ( (, bouded syetric doais (, R o the irst tye o classical R, ad rove the euivalece o the ors ad,, ( Furtherore, we study the coactess o coositio oerator C ro, R,, ad obtai a suiciet ad ecessary coditio or ( R ( to ( ( : ( (, ( (, C R R to be coact Keywords Bloch Sace, Classical Bouded Syetric Doais, Coositio Oerators, Coactess, Berga Metric troductio Let Ω be a bouded hoogeeous doai i he class o all holoorhic uctios o Ω will be Ω, the coositio is deoted by C, ad C is called the coositio oerator with sybol he coositio oerators as well as related oerators kow as the weighted coositio oerators betwee deoted by H ( Ω For a holoorhic sel-a o Ω ad H( * Corresodig author How to cite this aer: Su, JB, Li, HJ, Miao, XX ad Wag, R (4 Coactess o Coositio Oerators ro the -Bloch Sace to the -Bloch Sace o the Classical Bouded Syetric Doais Advaces i Pure Matheatics, 4, htt://dxdoiorg/436/a4474

2 J B Su et al the weighted Bloch saces were ivestigated i [] [] i the case o the uit disk, ad i [3]-[7] or the case o the uit ball he study o the weighted coositio oerators ro the Bloch sace to the Hardy sace H was carried out i [8] [9] or the uit ball Characterizatios o the boudedess ad the coactess o the coositio oerators ad the weighted oes betwee the Bloch saces were give i []-[] or the olydisc case, ad i [3]-[8] or the case o the bouded syetric doais Furtherore, we will give soe results about the coositio oerators or the case o the weighted Bloch sace o the bouded syetric doais 93s all irreducible bouded syetric doais were divided ito six tyes by E Carta he irst our tyes o irreducible doais are called the classical bouded syetric doais, the other two tyes, called excetioal doais, cosist o oe doai each (a 6 ad 7 diesioal doai he irst three tyes o classical bouded syetric doais ca be exressed as ollows [9]: (, { : is a colex atrix, } R >, where ad is the idetity atrix, is the trasose o ; Let A ( ai atrix C ( c i ( { : is a syetric atrix,, } ( { } R > R : is a atisyetric atrix,, > B b he Kroecker roduct A B o A ad B is deied as the s t s t such that the eleet at the ik -th row ad l -th colu ci ab i [9] he the Berg- R, is as ollows (see [9]: ad ( a etric o ( where u ( u,, u,, u,, u is a colex vector, z R, ( i ( ( ( ( ( H u, u u u, ( u is the cougate trasose o u, ad Followig ioey s aroach (see [8], a holoorhic uctio is i the Bloch sace ( R (, su Q ( < R (, Now we deie a holoorhic uctio to be i the -Bloch sace ( (, where R (, ( ( R, i su det Q <, ( ( u ( uu, Q ( su : u { }, H ( (,, (,, (,, ( z z z z We ca rove that ( R (, is a Baach sace with or ( α with the case o ( B be a holoorhic sel-a o R (, Let ( i : ( (, ( (, C R R will be a coact oerator Let ( d d,, i which is siilar We are cocered here with the uestio o whe diag,, deote a diagoal atrix with diagoal eleets d,, d this work,we shall deote by C a ositive costat, ot ecessarily the sae o each occurrece Sectio, we rove the euivalece o the ors deied i this aer ad i [] Sectio 3, we state several auxiliary results ost o which will be used i the roos o the ai results Fially, i Sectio 4, we establish the ai result o the aer We give a suiciet ad ecessary coditio or the coositio oerator C ro the -Bloch sace ( R (, to the -Bloch sace ( R (, to be coact, where ad Seciically,we rove the ollowig result: 65

3 J B Su et al heore Let be a holoorhic sel-a o R (, he C : ( R (, ( R (, is coact i ad oly i, or every ε >, there exists a δ > such that ( det H ( ( J( u, J( u H ( ( ( ( uu, det or all u { } wheever dist ( (, R < δ, R (, he coactess o the coositio oerators or the weighted Bloch sace o the bouded syetric, R, ; we oit the details doais o R ( R ( is siilar with the case o ( he Euivalece o the Nors Deote [] ( ( ( su det ad,,, R (, < ε, Lea (Blooield-Watso [] Let A ( a i det ( B AB where B is ay atrix ad satisies BB heore ad,, ( are euivalet R, is Proo he etric atrix o ( For ay R (,, let ( be a Heritia atrix he (3 deta ( ( ( (, ( ϕ Aut ( R, with ϕ ( he ( ϕ ( ( ϕ ( ( (( ϕ ( ( ϕ ( ( ( (, J, J J J Deote ϕ ψ the ψ (, Jψ ( Jϕ ad (, ( Jϕ ( (( Jϕ ( ( Jψ ( ( Jψ ( hus ( ( ( ( ( ( ( ( ( ( (, Jψ ( ( Jψ ψ Hece ( ψ ( u det ( Q ψ ( det ( su : { }, u u H ( uu, det ( su{ ( ψ ( u : u { }, u } Furtherore, Sice { } ( ( ( ( ( det, R (, R (, ( ( su det Q { } ( ( ( ( ( su det, ( (, ( ( ( det (, ( ( ( det 65

4 J B Su et al hus For, ( u det ( Q ( det ( su : u { }, u { u (, u } the we have Cobiig ( ad (3, Next, ad hereore, the roo is coleted ( { ( } ( det det, det ( (,,, ( ( ;,,, ( ( (,,, ( (3 3 Soe Leas Here we state several auxiliary results ost o which will be used i the roo o the ai result N Lea 3 [8] Let be a bouded hoogeeous doai he there exists a costat C, deedig oly o, such that ( ( (, ( (, z z H J z u J z u CH u u (3 or each z wheever holoorhically as ito itsel Here Hz (, l ( z o, J ( z deotes the Jacobia atrix o z k lk, N uu deotes the Berga etric Lea 3 Let be a holoorhic sel-a o R (, ad K a coact subset o R (, there exists a costat C > such that det det ( ( ( ( ( ( (, ( H ( uu, H J u J u or all u { } wheever ( K δ Proo For δ (,, let EW { W R ( ( W R δ} For ay coact K R (,, there exists a costat (, M (, such that det ( ( > M, wheever ( :, : dist, ( C δ such that K E δ W he there exists K he (3 65

5 J B Su et al hus ( det < M det ( ( ( Cobiig Lea 3 with (33 shows that (3 holds Lea 33 (Hadaard [] Let A ( a i be a Heritia atrix he (33 deta a (34 ii i ad euality holds i ad oly i A is a diagoal atrix z R, he Lea 34 Let ( ( Proo For ay R (, hus we have, we have zi, i,,, < < ( ( zii det (35 i > δ st zs zt st, t ollows ro Lea 33 that ( zi ( zii det i i zz, deote its etric atrix, R, i ad oly i Lea 35 Let R (, be a classical bouded syetric doai, ad ( he a holoorhic uctio o R ( is i ( ( (36 holds, the su R (, { } ( ( ( ( det, < (36 R (, { } ( ( ( ( su det, (37 Proo We ca get the coclusio by the rocess o the roo o heore Lea 36 [8] Let where ( ( λ λ λ P U diag,,,, V R, Q U U diag,,,, λ λ λ diag,,,,,,, R V V λ λ λ U ad V are uitary atrices ad λ λ λ < Deote ( ( ( Φ P Q P P R, R (, ( ΦP( Aut ( R ( ( ( Φ Φ ; P P, ; (3 ( P ad ( P Φ Φ ; ; P P he (4 d ( Qd R ad d ( Q Φ Φ dr or (, P P P R ; 653

6 J B Su et al (5 ( ( ( Φ Q P P R or R (, ; P (6 ( ( ( ( ( P Q Q P Φ Φ or (, P P R Lea 37 C : ( R (, ( R (, is coact i ad oly i C k ay bouded seuece { k } i ( R (, that coverges to uiorly o coact subsets o (, Proo he roo is trial by usig the oral ethods 4 Proo o heore Proo Let { k } be a bouded seuece i ( (, coact subsets o R (, Suose (3 holds he or ay δ > as k or R R with k C, ad k uiorly o ε >, there exists a, such that ( det H ( ( J( u, J( u ε < H ( ( ( ( uu, C det or all u { } wheever dist ( (, R (, < δ ad R (, By the chai rule, we have ( k ( ( k ( ( J( u { } ad J ( u, the we get ( ( u u { } ad J ( u the ( k ( ( t ollows ro (4 ad (4 that ( k ( ( J( u ( ( (, ( k ( ( (, ( ( H J u J u u H uu, H J u J u H uu, ( ( H ( uu, k u det ( QC ( su det (, u { }, J ( u k (4, (4 det ( H ( ( J( u, J( u k su, u { } (43 H ( ( ( ( uu, det ε < C ε, (44 C wheever dist ( (, R (, < δ ad (, R O the other had, there exists a costat > such that i { HW ( uu, : u, dist ( W, R (, δ} So i dist ( W, R (, δ, the ( ( ( WW ( WW WW HW ( uu, HW ( u u, u u We assue that { k } coverges to uiorly o coact subsets o R (, it is easy to see that { } coverges to uiorly o coact subsets o (, W u W W ( k k k det det det there exists k large eough such that k det ( ( ( ( k ( ( ( H ( ( J( u, J ( u ( By Weierstrass heore, R hus, or give ε >, J u ε < (45 C 654

7 or ay u { }, J ( u wheever dist ( (, R (, δ ad R (, eualities (43 ad (45 ad Lea 3, it ollows that, or k large eough, C k wheever dist ( (, R (, δ ad (, Cobiig (44 ad (46 shows that C k J B Su et al he by i- < ε (46 R < ε as k large eough So : ( R (, ( R (, is coact For the coverse, arguig by cotradictio, suose C : ( (, ( (, ε >, a seuece { } i R (, with, u i { }, such that C the coditio (3 ails he there exist a ( R ( as ad a seuece { } or all,, det ( ( ( det Now we will costruct a seuece o uctios { } ( { } is a bouded seuece i ( R (, ; ( { } R R is coact ad ( H J u J u ( (, ( H ( u, u teds to uiorly o ay coact subset o (, ( C as ε satisyig the ollowig three coditios : R ; he existece o this seuece will cotradict the coactess o C We will costruct the seuece o uctios { } accordig to the ollowig our arts: A - D Part A: Suose that ( re,,, where E is the atrix whose eleet at the kth row ad lth colu is ad the other eleets are Sice as R (, ito itsel, < r < ad r as J u w w,, w, w,, w,, w,, w Usig orula (, we have Deote ( by ( ( ( ( ( the Deote ( (( H w, w w diag r,,, diag r,,, w ( w ( w w w ( r r l k l k k, l w l k r r l k k, l A, B w w, C w, ( ( ( ( (47 H w, w A B C (48 We costruct the seuece o uctios { } accordig to the ollowig three dieret cases Case or soe, ax ( B, C A the set ( (49 ( e z ( z a( r, (4 655

8 J B Su et al where a is ay ositive uber Case or soe, the set ( A C ax, B (4 ( z l z k ( e z iθ l iθk e e (4 l k ( z a( r where θ argw, i or soe l, w or or soe k, w, relace the corresodig ter e z by (the sae below Case 3 or soe, ax A, B C (43 the set ( iθ ( e z (44 k, l ( z a( r ( e z { } Next, we will rove that the seueces o uctios ( the coditios (, ( ad ( o begi with, we will rove the seuece o uctios ( ditios We ca get that { } ( ( ( ( det, iθ deied by (4, (4 ad (44 all satisy { } deied by (4 satisies the three co- a( r e ( det ( z z l k ( z a( r ( e z ( ( z ( z ( z ( t ollows ro Lea 35 that C ( l k { } deied by (4 satisies coditio ( his roves that the seuece o uctios ( Let E be ay coact subset o R (, he there exists a (, z E By (4, we have or ay ( Sice ( z ρ such that ρ (45 a( r a( r e z e z a( r z ( z ( e z ρ ( a( r e z z a( r a( r e e z z ρ 656

9 J B Su et al But { } a( r e as hus, a( r e z z coverges to uiorly o E hereore, deied by (4 { } coverges to uiorly o E as hus, the seuece o uctios ( satisies the coditio ( Now (48 ad (49 ea that Cobiig (47 ad (46, we have Sice H w, w, A B C 3, A (46 ( ( ( ( ( ( C det his roves that C by (4 satisies coditio ( ( ( ( ( ( H ( u, u ( a r ( r e a( r ( e r ( re w ( ( ( re w ε r z ( r HrE ( w, w 3( w ( r 3 J u ε We ca rove that the seuece o uctios ( - ( by usig the aalogous ethod as above Part B: Now we assue that ( ( t is clear that r r as, where λ ig coditios (-( Usig orula (, we have ε ( a r ( r e li a( r a ( e r { } as, which eas that the seuece o uctios ( { } ( ( ( deied deied by (4 or (44 satisies the coditios ( r E r E,,, > ad or ( (, R we ca assue that r ( ad λ, we ca use the sae ethods as i Part A to costruct a seuece o uctios ( ( ( ( ( ( ( ( r { } ( λ satisy- ( ( ( ( H w, w w diag r, r,,, diag r, r,,, w Deote w w w w ( ( ( ( w l wk ( ( 3 3 ( ( r ( ( ( l k r r r r wl wk w ( 3 3 3,3 ( r l k k l 657

10 J B Su et al he, w w w w ( ( ( ( ( r ( r ( r ( r A, B, C, D w w, E w w, F w l k l k ( 3 3 ( 3 3 3,3 ( r l k ( l k k l r ( ( H, ( ( w w A B C D E F We costruct the seuece o uctios { } Case or soe, the set Case or soe, ( (47 accordig to the ollowig six dieret cases ( ax B, C, D, E, F A ( ( z a r e z ( ax A, C, D, E, F B, the set iθ iθ ( ( e z e z ( ( ( z ( z a r a r e z e z Case 3 or soe, ax A, B, D, E, F C the set Case 4 or soe, the set Case 5 or soe, the set ( ( ( ( z a r e z ( ax A, B, C, E, F D ( z z (48 (49 (4 iθ l iθk e e (4 ( l3 k3 ( z a r l k e z ( ax A, B, C, D, F E, 658

11 J B Su et al ( z z iθl iθk e e (4 ( l3 k3 ( z a r l k e z Case 6 or soe, ax ( A, B, C, D, E F the set iθ ( e z (43 ( 3 k,3 l ( z a r e z By usig the sae ethods as i Part A, we ca rove the seueces o uctios { ( } deied by (48-(43 satisyig coditios ( - ( Now, as a exale,we will rove that the seuece o uctios { ( } deied by (49 satisyig the coditios ( - ( R,, we have For ay ( hus ii ii z < z, z < z, i, (44 ( ( k, l z ( a r iθ e iθ ( e z e z 3 3 ( ( ( z ( z a r a r e z e z ( a r iθ iθ e ( e z e z 3 3 ( ( ( z ( z a r a r e z e z ( ( z ( z a r ( a r e z e z 4 < ( z z ( z ( z ( z ( z ( z ( z 4 4 z z z z z z 3 3 ( ( ( ( ( ( 659

12 J B Su et al By Lea, we have { } ( ( ( ( det, ( ( { ( } det det, ( z ( z 4 4 ( z ( z ( z ( z ( z ( z 4 (45 { } t ollows ro Lea 35 ad (45 that C deied by (49 satisy the coditio ( Let E be ay coact subset o R (, Sice there exists a ρ (, such that zii ρ >, i, hus ( ( a r a r e z e z ( ( ρ ( z ( z Sice ( i a r ( i ( i a z r a r ii z ii his roves that the seuece o uctios ( e e e, i, zii zii ρ ( ( a r a r e ( i a r z e z So e as hus, ( z ( z coverges to ui- coverges to uiorly o E as hus, the seuece { } deied by (49 satisies the coditio ( For case,, H w w 6 B (46 orly o E hereore,the seuece o { } o uctios ( Cobiig (47 ad (46, we have ( ( ( ( ( ( ( ( ( C r r ( ( ( ( J u ( ( ( ( det ε H u, u H w, w r E r E ( ( ε ( ( r r ( ( 6( a r a r ( ( e r e r Sice r E r E w ( ( ( 66

13 J B Su et al ( ( ( r ( r li ( ( a r a r ( ( a e r e r his roves that C by (49 satisies coditio ( λ <, the by Lea 36, there exist Φ ( ( ad ( we deote { } as, which eas that the seuece o uctios ( r E r E ( ( Φ i (, r E ( ( ( ( ( r E r E r E R such that ( ( Φ r E r E ad Φ r E,, Ψ Φ Φ ( ( ( E ( ( r r E r E ( (, the ( ( ( ( ( Ψ ( ( Ψ r E r E r E re, where r r ( Deote g Ψ, where the seuece o uctios { } ( ( ( ( ( ( Ψ where w J ( u ad v J ( ( ( w ( g ( ( w ( ( re v ( ( H w, w HrE ( v, v ad ( ( (, Ψ Aut R ad deied is the seuece obtaied i Part A We have ( ( ( ( ( re ( H w, w H J Ψ w, J Ψ w H v, v, Ψ Now (47 ilies that C g det ( li ( r λ ( ( ( g ( J u ( H ( u, u ( ( ( g w det ( ( ( ( H, w w ( ( ( ( re v ( r ( r HrE ( v, v ε ε t is clear that, ad cobiig the discussio i Part A,we ca get that Cg as ; that eas the seuece o uctios { g } satisies coditio ( We rove that the seuece o uctios { g } is a bouded seuece i ( R (, Ψ Aut R,, Sice ( ( ( So g is bouded Next we rove that { } ( ( ( ( ( Q ( ( g Q Ψ ( det det g coverges to uiorly o ay coact subset E o R (, ( ( ( (, lk, Ψ Ψ the by the deiitio o that l k Let (47 Ψ ad Lea 36, we ca get a calculatio directly 66

14 J B Su et al t is clear that ( z r ( r z ( ( Ψ z z ( Ψ ( coverges uiorly to Ψ ( z λ z z λ z i (, R Sice λ < ad λe λe R (,, there siilarly exist Ψ ( i Aut ( (, Ψ ( λe λe λe, ad the irst cooet o Ψ ( is Ψ ( t is clear that ( orhic o R (, Let M su Ψ ( Ψ ( or E For ( Aut ( (, M ( ( ( M E R such that Ψ is holo- Ψ R, we kow Ψ < We ay choose M > such that M < M < hus, or large eough, Ψ < ad ro this it ollows that { } by the deiitio o (, ( Hece ( ( ( Ψ > M > ( ( g Ψ coverges to uiorly o E { g } satisies coditios (--(, ad this cotradicts the coactess o C Part C: Assue that ( k ( r E,,, k ( ( ( where > r r r R, we ay assue that ( ( r, r λ, ( r λ as, where λk, k,3,, Just as i Part B, we ca use the sae ethods to rove the coclusio Ad or λ k, k,3,,, we ay oly show the seuece o uctios { ( } which satisy the coditios ( - ( here Usig orula (, we have the, Deote For ( ( ( ( ( ( kk ( ( ( ( ( ( ( ( ( ( ( H w, w w diag r, r,,r ( k diag r, r,, r,,,w ( wkk w wlk k ( k k< l ( k l ( r r r wkk w ( w lk ( k ( k l ( r r r w ( k l ( k r ( ( k ( ( k r ( ( ( l A, k,,, B, k < l, C, k,, H ( ( w ( ( ( (, w A k B C k k k< l k w (48 We costruct the seuece o uctios { } accordig to the ollowig three dieret cases Case or soe, ( ( ( ( i st i ( k ax A, B, C A, ik,,,, s< t, 66

15 J B Su et al the set ( k (,,, k (49 ( k ( zkk a r e zkk Case or soe, the set ( ( ( ( i st i ( ax A, B, C B, i,,, s < t, k < l, ( iθ iθlk ( ( e z e zlk, ( k ( l ( zkk ( zll a r a r e zkk e zll k < l Case 3 or soe, the set ( ( ( ( i st i ( k ax A, B, C C, i, k,,, s < t, (43 ( k iθ ( e z, k,, (43 ( k l ( zkk a r e zkk { } deied by (49-(43 satisyig coditios ( - ( Part D: the geeral situatio For ( R (,, there exist a uitary atrix P ad a uitary atrix Q such that Usig the sae ethods as i Part A ad Part B, we ca rove the seueces o uctios ( We ay assue that P P ad Q Q that as or ay k, l (, ( k ( ( P Q r E kk k as Let P ( ad P ( Let Ψ ( P Q ad ( ; P Ψ PQ or R O course, P is a uitary atrix, Q is a uitary atrix, ad verges uiorly to Ψ ( o R (, Let g ( ( Ψ (,,, where the seuece o { } Fro the sae discussio as that i Part B, we kow that { ( } coact subset E R (,, Ψ ( E is also a coact subset o R (, ( D o R (, such that Ψ( E D D R ( Sice ( set, P eas ( { ( } Ψ co- are the uctios obtaied i Part C g satisies coditios ( ad ( For the, so we ca choose a oe sub- { } Ψ ( o R (,, it ollows that Ψ ( E D as Sice ( we kow ( { g } teds to uiorly o E hus, { } Ψ coverges uiorly to { } teds to uiorly o D, g satisies coditio ( 663

16 J B Su et al Ackowledgeets We thak the Editor ad the reeree or their coets Research is uded by the Natioal Natural Sciece Foudatio o Chia (Grat No 785 ad the Postgraduate ovatio Proect o Jiagsu Provice o Chia (CXLX-98 Reereces [] Raos-Ferádez, JC ( Coositio Oerators betwee μ-bloch Saces Extracta Matheaticae, 6, [] Wol, E ( Weighted Coositio Oerators betwee Weighted Bloch ye Saces Bulleti de la Société Royale des Scieces de Liège, 8, [3] Dai, JN ( Coact Coositio Oerators o the Bloch Sace o the Uit Ball Joural o Matheatical Aalysis ad Alicatios, 386, htt://dxdoiorg/6/aa767 [4] Li, JF ( Coositio Oerators ro -Bloch Saces to Little -Bloch Saces o the Uit Ball o C Acta Matheatica Siica (Series B, 3, 3- [5] hag, M ad Xu, W (7 Coositio Oerators o α-bloch Saces o the Uit Ball Acta Matheatica Siica (Series B, 3, 99- [6] hag, XJ ad Li, JX (9 Weighted Coositio Oerators Betwee μ-bloch Saces over the Uit Ball o C (Chiese Acta Matheatica Siica (Series A, 9, [7] hou, H ad eg, HG (3 Coositio Oerators betwee -Bloch Sace ad -Bloch Sace i the Uit Ball Progr Progress i Natural Sciece (Egl Ed, 3, [8] Stevo, S (8 Nor o Weighted Coositio Oerators ro Bloch Sace to H µ o the Uit Ball Ars Cobiatoria, 88, 5-7 [9] ag, XM ad hag, RJ (3 Weighted Coositio Oerator ro Bloch-ye Sace to H Sace o the Uit Ball Matheatical eualities ad Alicatios, 6, htt://dxdoiorg/753/ia-6- [] Li, SX ad hu, XL (4 Essetial Nor o Weighted Coositio Oerator betwee α-bloch Sace ad -Bloch Sace i Polydiscs teratioal Joural o Matheatics ad Matheatical Scieces, 69-7, [] hou, H ad Shi, JH ( Coact Coositio Oerators o the Bloch Sace i Polydiscs Sciece i Chia Series A, 44, 86-9 htt://dxdoiorg/7/bf87878 [] hou, H ad Wei, Q (5 Weighted Coositio Oerators o the Bloch Sace i Polydiscs (Chiese Joural o Matheatics (Wuha Uiversity, 5, [3] Alle, RF ad Coloa, F ( Weighted Coositio Oerators o the Bloch Sace o a Bouded Hoogeeous Doai : Oerator heory: Advaces ad Alicatios, Volue Oerators, Matrices ad Aalytic Fuctios, Oer heory Adv Al,, Birkhäuser Verlag, Basel, -37 [4] Alle, RF ad Coloa, F (9 Multilicatio Oerators o the Bloch Sace o Bouded Hoogeeous Doais Coutatioal Methods ad Fuctio heory, 9, htt://dxdoiorg/7/bf3375 [5] Deg, FW ad Ouyag, CH (6 Bloch Saces o Bouded Syetric Doais i Colex Baach Saces Sciece i Chia Series A, 49, htt://dxdoiorg/7/s [6] Shi, JH ad Luo, L ( Coositio Oerators o the Bloch Sace o Several Colex Variables (Chiese Acta Matheatica Siica (Chiese Series, 44, - [7] ioy, RM (98 Bloch Fuctios i Several Colex Variables, Bulleti o the Lodo Matheatical Society,, 4-67 htt://dxdoiorg//bls/44 [8] hou, H ad Shi, JH ( Coactess o Coositio Oerators o the Bloch Sace i Classical Bouded Syetric Doais Michiga Matheatical Joural, 5, htt://dxdoiorg/37// [9] Lu, QK (963 he Classical Maiolds ad the Classical Doais (Chiese Shaghai Scietiic ad echical Publishers, Shaghai [] Pa, WQ ( Coositio Oerators betwee -Bloch Sace ad -Bloch Sace i the First Classical Bouded Syetric Doai Master s hesis, Jiagsu Noral Uiversity, Xuzhou [] Kuag, JC (4 Alied eualities (Chiese 3d Editio, Shadog Sciece ad echology Press, Shadog 664