THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E. MCCARTHY, SANDRA POTT, AND BRETT D. WICK


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1 THIN SEQUENCES AND THE GRAM MATRIX PAMELA GORKIN, JOHN E MCCARTHY, SANDRA POTT, AND BRETT D WICK Abstract We provide a ew proof of Volberg s Theorem characterizig thi iterpolatig sequeces as those for which the Gram matrix associated to the ormalized reproducig kerels is a compact perturbatio of the idetity I the same paper, Volberg characterized sequeces for which the Gram matrix is a compact perturbatio of a uitary as well as those for which the Gram matrix is a Schatte class perturbatio of a uitary operator We exted this characterizatio from to p, where p 1 Itroductio Let D deote the ope uit disk ad T the uit circle Give {α j }, a Blaschke sequece of poits i D, we let B deote the correspodig Blaschke product ad B deote the Blaschke product with the zero α removed Further, we let δ j = B j (α j ), k j = 1 1 α j deote the z Szegő kerel (the reproducig kerel for H ) at α j, g j = k j / k j the H ormalized kerel, ad G the Gram matrix with etries G ij = g j, g i I the secod part of [10, Theorem ], Volberg s goal was to develop a coditio esurig that {g } is ear a orthogoal basis; by this, oe meas that there exist U uitary ad K compact such that g = (U + K)e, where {e } is the stadard orthogoal basis for l By [7, Sectio 3] or [4, Propositio 3], this is equivalet to the Gram matrix defiig a bouded operator of the form I + K with K compact Followig Volberg ad aticipatig the coectio to the Schattep classes, we call such bases U + S bases Volberg showed that {g } is a U + S basis if ad oly if lim δ = 1; i other words, if ad oly if {α } is a thi sequece Assumig {g } is a U +S basis, it is ot difficult to show that the sequece {α } must be thi But Volberg s proof of the coverse is more difficult ad depeds o the mai lemma of a paper of Axler, Chag ad Saraso [, Lemma 5], estimatig the orm of a certai product of Hakel operators as well as a factorizatio theorem for Blaschke products The lemma i [] uses maximal fuctios ad a certai distributio fuctio iequality A more direct proof of Volberg s result is desirable, ad we provide a simpler proof of this result i Theorem 35 of this paper I a secod theorem, lettig S deote the class of HilbertSchmidt operators, Volberg showed (see [10, Theorem 3]) that {g } is a U +S basis if ad oly if =1 δ coverges We are iterested i estimates for the ibetwee cases We provide a ew proof of Volberg s theorem for p = ad prove the followig theorem Theorem 11 For p <, the operator G I S p if ad oly if (1 δ ) p/ < Volberg s theorem covered the cases p = ad p =, but our proofs differ i the followig way: Istead of usig the results of [] ad theorems about Hakel operators, we use the PG Partially supported by Simos Foudatio Grat JM Partially supported by Natioal Sciece Foudatio Grat DMS BDW Partially supported by Natioal Sciece Foudatio Grat DMS
2 P GORKIN, JEMCCARTHY, S POTT, AND BDWICK relatioship betwee growth estimates of fuctios that do iterpolatio o thi sequeces (see [5], [6]) ad the orm of the Gram matrix This simplifies previous proofs ad provides the best estimates available Prelimiaries ad otatio Let {α j } be a sequece i D with correspodig Blaschke product B, ad B j be the Blaschke product with zeroes at every poit i the sequece except α j ad δ j = B j (α j ) The separatio costat δ is defied to be δ := if j δ j Carleso s iterpolatio theorem says that the sequece {α j } is iterpolatig if ad oly if δ > 0, [3] The sequece {α j } is said to be thi if lim j δ j = 1 Give a thi sequece we may arrage the δ j i icreasig order ad rearrage the zeros of the Blaschke product accordigly Recall that if T is a operator o a Hilbert space H ad λ is the th sigular value of T, the give p with 1 p < the Schattep class, S p, is defied to be the space of all compact operators with correspodig sigular sequece i l p, the space of psummable sequeces The S p is a Baach space with orm T p = ( λ p ) 1/p For p =, we let S deote the space of compact operators Recall that k j deotes the Szegő kerel, g j = k j / k j, ad G the Gram matrix with etries G ij = g j, g i (The Gram matrix depeds of course o the sequece {α j }, but we suppress this i the otatio) For {α j } iterpolatig, we let D be the diagoal matrix with etries 1/B j (α j ) It is kow (see, for example, formula (6) of [8]) that (1) G 1 = D G t D For a give sequece {α j }, the iterpolatio costat is the ifimum of those M such that for ay sequece {a j } i l, oe ca fid a fuctio f i H with f(α j ) = a j ad f M a l We shall let M(δ) deote the supremum of the iterpolatio costats over all sequeces {α j } with separatio costat δ The followig result is due essetially to A Shields ad H Shapiro [9] See [1, Propositio 95] for a proof of this versio Propositio 1 Let {α j } be a iterpolatig sequece i D (i) If the iterpolatio costat is M, the both G ad G 1 are bouded by M (ii) If G = C 1 ad G 1 = C, the the iterpolatio costat is bouded by C 1 C We shall use the followig estimate of JP Earl (see [5] or [6]) to obtai our results Theorem (Earl s Theorem) The iterpolatio costat M(δ) satisfies ( ) δ M(δ) δ 3 Schattep classes I this sectio we provide estimates o the Schattep orm of G I We will eed the theorem ad lemma below
3 THIN SEQUENCES AND THE GRAM MATRIX 3 Theorem 31 (see eg [11, Theorem 133]) Let T be a operator o a separable Hilbert space, H If 0 < p the { } T p S p = if T e p : {e } is ay orthoormal basis i H ad if p < { } T p S p = sup T e p : {e } is ay orthoormal basis i H We say that two sequeces {x } ad {y } of positive umbers are equivalet if there exist costats c ad C, idepedet of, such that cy x Cy for all We write x y We will also write A B to idicate that there exists a costat C such that A CB Lemma 3 Let {e } deote the stadard orthoormal basis for l ad {α j } be a iterpolatig sequece i D with correspodig δ j The Proof We have (G I)e 1 δ (G I)e = (G I)(G I)e, e g, g 1 g, g 1 g =, g 1 g,, g g, g +1 = g, g j g, g j g, g +1 = 1 αj 1 α 1 α α j = 1 α j α 1 α α j But log x 1 x for x > 0 ad log x < c(1 x) for x < 1 bouded away from 0 ad some costat c idepedet of x, so log x 1 x for x bouded away from 0 Cosequetly, (G I)e log α j α 1 α α j = log α j α 1 α α j = log δ 1 δ
4 4 P GORKIN, JEMCCARTHY, S POTT, AND BDWICK Note that the costats ivolved do ot deped o Combiig the lemma with Theorem 31, we obtai the followig theorem Theorem 33 The followig estimates hold: If p < the If 0 < p the If p = the (1 δ ) p G I p G I p S p S p ; (1 δ ) p ; (1 δ ) G I S Lemma 34 Let {α j } be a iterpolatig sequece ad G the correspodig Gram matrix Let C = G 1 The G I C 1 Proof By (1), we have G CI, ad as G is a positive operator, we have G (1/C)I Therefore ( ) 1 C 1 I G I (C 1)I, ad as C > 1, we get G I C 1 I what follows, for a positive iteger N, we let G N deote the lower righthad corer of the Gram matrix obtaied by deletig the first N rows ad colums of G Thus, G N = 1 g N+1, g N g N+j, g N g N, g N+1 1 g N+j, g N+1 ad λ 0 deotes the th sigular value of G I, where the sigular values are arraged i decreasig order We are ow ready to provide our simpler proof of Volberg s result [10, Theorem, p 15] Theorem 35 The sequece {α } is a thi sequece if ad oly if the Gram matrix G is the idetity plus a compact operator Proof ( ) Suppose {α } is thi By discardig fiitely may poits i the sequece (α ), we ca assume that the sequece has a positive separatio costat, ad hece is iterpolatig Let G N be the Gram matrix of {g j } for j N We shall let δ N,j deote the δ j defied for G N (that is, correspodig to the Blaschke sequece {α j } j N ) Note that δ N,j δ N+j for j = 0, 1,,, ad so we have that δ(n) := if j δ N,j if j δ N+j = δ N By Theorem ad Propositio 1, G 1 N (M(δ(N)) ( 1 + ) 4 1 δ(n) δ(n) 4 ( 1 + ) 4 1 δn δ 4 N
5 Applyig Lemma 34 THIN SEQUENCES AND THE GRAM MATRIX 5 G N I N ( 1 + ) 4 1 δn δ 4 N 1 C 1 δ N, where C is a costat idepedet of N Sice 1 δ N 0 as N, we coclude that G I is compact ( ) From (1), we have (31) G 1 I = D (G t I)D + [D D I] If G I is compact, the so are G t I ad G 1 I = G 1 (I G) Therefore from (31), we have D D I is compact, which meas lim j δj = 1 Cosequetly, the sequece is thi Theorem 36 For p <, the operator G I S p if ad oly if (1 δ ) p/ < Proof By Theorem 33, if G I S p, the the sum is fiite Now suppose the sum is fiite Usig Lemma 34 as i Theorem 35, we have where C is idepedet of N By [11, Theorem 1411], G N I N C 1 δ N, λ N+1 if{ (G I) F : F F N }, where F N is the set of all operators of rak less tha or equal to N Therefore, takig F to be the matrix with the same first N rows ad colums as G I, which is of rak at most N, we have λ N+1 G N+1 I N+1 C 1 δ N+1, by our computatio above Therefore λ N+1 p C p (1 δ N+1 ) p/ Sice the sigular values are arraged i decreasig order, λ +1 > λ for each Thus, if N (1 δ N) p/ <, the λ p λ +1 p < ad we coclude that G I S p We coclude by remarkig that it is possible to trace through the proofs above to determie costats c ad C, which deped oly o δ = if δ, such that for p, (3) c 1 δ l p G I Sp C 1 δ l p I particular, by choosig δ close eough to 1, oe ca choose c ad C i (3) arbitrarily close to ad 4 ( 1/p ), respectively Questio 37 Is Theorem 36 true for p <? Ackowledgemet We thak the referee for his or her careful readig of this mauscript as well as helpful commets
6 6 P GORKIN, JEMCCARTHY, S POTT, AND BDWICK Refereces [1] J Agler ad JE McCarthy, Pick Iterpolatio ad Hilbert Fuctio Spaces, America Mathematical Society, Providece, 00 [] Sheldo Axler, SuYug A Chag, ad Doald Saraso, Products of Toeplitz operators, Itegral Equatios Operator Theory 1 (1978), o 3, , [3] L Carleso, A iterpolatio problem for bouded aalytic fuctios, America J Math 80 (1958), [4] I Chaledar, E Fricai, ad D Timoti, Fuctioal models ad asymptotically orthoormal sequeces, A Ist Fourier (Greoble) 53 (003), o 5, [5] J P Earl, O the iterpolatio of bouded sequeces by bouded fuctios, J Lodo Math Soc (1970), [6] J P Earl, A ote o bouded iterpolatio i the uit disc, J Lodo Math Soc () 13 (1976), o 3, [7] Emmauel Fricai, Bases of reproducig kerels i model spaces, J Operator Theory 46 (001), o 3, suppl, [8] Paul Koosis, Carleso s iterpolatio theorem deduced from a result of Pick, Complex aalysis, operators, ad related topics, Oper Theory Adv Appl, 113, Birkhäuser, Basel, 000, pp [9] HS Shapiro ad AL Shields, O some iterpolatio problems for aalytic fuctios, America J Math 83 (1961), [10] A L Vol berg, Two remarks cocerig the theorem of S Axler, SY A Chag ad D Saraso, J Operator Theory 7 (198), o, , 4 [11] Kehe Zhu, Operator theory i fuctio spaces, Secod, Mathematical Surveys ad Moographs, vol 138, America Mathematical Society, Providece, RI, 007 3, 5 Pamela Gorki, Departmet of Mathematics, Buckell Uiversity, Lewisburg, PA USA address: Joh E McCarthy, Dept of Mathematics, Washigto Uiversity i St Louis, St Louis, MO address: Sadra Pott, Faculty of Sciece, Cetre for Mathematical Scieces, Lud Uiversity, 100 Lud, Swede address: Brett D Wick, School of Mathematics, Georgia Istitute of Techology, 686 Cherry Street, Atlata, GA USA address:
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