Scientiae Mathematicae Japonicae Online, Vol.7 (2002), IN CSL-ALGEBRA ALGL

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1 Scietiae Mathematicae Japoicae Olie, Vol , SELF-ADJOINT INTERPOLATION PROBLEMS IN CSL-ALGEBRA ALGL Youg Soo Jo ad Joo Ho Kag Received December 10, 2001 Abstract. Give vectors x ad y i a Hilbert space, a iterpolatig operator is a bouded operator T such that Tx = y. A iterpolatig operator for N vectors satisfies the equatio Txi = yi, fori = 1; 2; ;. I this article, we ivestigate self-adjoit iterpolatio problems i CSL-Algebra AlgL. 1. Itroductio Let C be a collectio of operators actig o a Hilbert space H ad let x ad y be vectors o H. A iterpolatio questio for C ass for which x ad y is there a bouded operator T 2 C such that Tx = y. A variatio, the `N-vector iterpolatio problem', ass for a operator T such thattx i = y i for fixed fiite collectios fx1;x2; ;x g ad fy1;y2; ;y g.the N-vector iterpolatio problem was cosidered for a C Λ -algebra U by Kadiso[9]. I case U is a est algebra, the oe-vector iterpolatio problem was solved by Lace[10]: his result was exteded by Hopewasser[4] to the case that U is a CSL-algebra. Much[11] obtaied coditios for iterpolatio i case T is required to lie i the ideal of Hilbert- Schdt operators i a est algebra. Hopewasser[5] oce agai exteded the iterpolatio coditio to the ideal of Hilbert-Schdt operators i a CSL-algebra. Hopewasser's paper also cotais a suiciet coditio attributed to S. Power for iterpolatio N-vectors, although ecessity was ot proved i that paper. I this article, we ivestigate the self-adjoit iterpolatio problems i CSL-Algebra AlgL: Give vectors x ad y i a Hilbert space ad a commutative subspace lattice L o H, whe is there a self-adjoit operator A i AlgL such thatax = y? First, we establish some otatios ad covetios. A commutative subspace lattice L, or CSL L is a strogly closed lattice of pairwise-commutig projectios actig o a Hilbert space H. We assume that the projectios 0 ad I lie i L. We usually idetify projectios ad their rages, so that it maes sese to spea of a operator as leavig a projectio ivariat. If L is CSL, AlgL is called a CSL-algebra. The symbol AlgL is the algebra of all bouded liear operators o H that leave ivariat all the projectios i L. Let x ad y be 2000 Mathematics Subject Classificatio ; 47L35 Key words ad phrases ; Self-Adjoit Iterpolatio Problem, Subspace Lattice, CSL-Algebra AlgL.

2 452 Y.-S. JOO AND J.-H. KANG vectors i a Hilbert space. The <x;y>meas the ier product of vectors x ad y. I this paper, we use the covetio 0 =0, whe ecessary. 2. Results Let H be a Hilbert space ad L be a commutative subspace lattice of orthogoal projectios actig o H cotaiig 0 ad I. The AlgL is the algebra of all bouded liear operators o H that leave ivariat all the projectios i L. Let M be a subset of a Hilbert space H. The M meas the closure of M ad M? the orthogoal complemet ofm. Let N be the set of all atural umbers ad let C be the set of all complex umbers. Defiitio. Let H be a Hilbert space ad let A be a operator actig o H. The A is called a self-adjoit operator if A Λ = A. Theorem 1. Let H be a Hilbert space ad let L be a subspace lattice o H. Let x ad y be vectors i H. If there isaoperator A i AlgL such that Ax = y, A is self-adjoit ad every E i L reduces A, the supρ P ie i y ie i x : 2N; i2 C ad E i 2L for every E i L. <Ex;y>=< Ey;x> Proof. We ca get the first result by Theorem 1 [8] uder the give hypothesis. So we eed to show that <Ex;y>=< Ey;x>for every E i L wheever A Λ = A. Sice AE = EA, A Λ E = EA Λ for every E i L. SiceAx = A Λ x = y, A Λ Ex = AEx = Ey for every E i L. Hece <Ey;x>=< A Λ Ex;x >=< Ex; Ax >=< Ex;y>for every E i L. Let x ad y be vectors of a Hilbert space H. Let M = M 1 = i E i x : 2 N; i 2 C i E i y : 2 N; i 2 C ad E i 2L ad E i 2L : ad Theorem 2. Let H be a Hilbert space ad let L be a commutative subspace lattice o H. supρ P ie i y ie i x : 2 N; i 2 C ad E i 2L Let x ad y be vectors i H. Assume that M 1 ρ M. If <Ex;y>=<Ey;x> for every E i L, the there isaoperator A i AlgL such that y = Ax, A Λ = A ad every E i L reduces A. Proof. We ca get results except that A Λ = A by Theorem 1 [8] uder the give hypothesis. So we eed to prove thatif<ex;y>=< Ey;x>for every E i L, the A Λ = A. Sice

3 SELF-ADJOINT INTERPOLATION PROBLEMS IN CSL-ALGEBRA ALGL 453 <Ex;y>=< Ey;x>for every E i L, <A i E i x;x>=< i E i Ax; x > =< i E i y; x > =< i E i x;y>: Sice y 2 M, A Λ x = y. Sice EA = AE, EA Λ = A Λ E for every E i L. So A Λ i E i x= i A Λ E i x = i E i A Λ x = i E i y: Sice M 1 ρ M, A Λ f = 0 for every f i M?.HeceA Λ = A. Corollary 3. Let H be a Hilbert space ad let L be a commutative subspace lattice o H. supρ P ie i y ie i x : 2 N; i 2 C ad E i 2L Let x ad y be vectors i H. Assume that M is dese i H. If <Ex;y>=<Ey;x> for every E i L, the there isaoperator A i AlgL such that y = Ax, A Λ = A ad every E i L reduces A. If we summarize Theorems 1, 2 ad Corollary 3, we ca get the followig theorem. Theorem 4. Let H be a Hilbert space ad let L be a commutative subspace lattice o H. Let x ad y be vectors i H. Assume that M 1 ρ M or M is dese i H. The the followig statemets are equivalet. 1 There exists a operator A i AlgL such that Ax = y, A Λ = A ad every E i L 2 supρ P ie i y ie i x : 2 N; i 2 C ad E i 2L reduces A. <Ex;y>=< Ey;x>for every E i L. Theorem 5. Let H be ahilbert space adl be a subspace lattice oh. Let x1;x2; ;x ad y1;y2; ;y bevectors i H. If there isaoperator A i AlgL such that Ax p = y p for all p =1; 2; ;, A Λ = A ad every E i L reduces A,

4 454 Y.-S. JOO AND J.-H. KANG the sup =1 ;ie ;i y i P m i =1 ;ie ;i x i : m i2n; l» ; E ;i 2L ad ;i 2 C <Ex p ;y j >=< Ey p ;x j > for every E i L ad all p; j =1; 2; ;. Proof. By Theorem 2 [8], we ow that P sup =1 ;ie ;i y i m i =1 ;ie ;i x i : m i2 N;l» ; E ;i 2Lad ;i 2 C < 1. Sice A Λ = A, <Ey p ;x j >=<EAx p ;x j >=<AEx p ;x j >=<Ex p ;A Λ x j >=<Ex p ;y j > for every E i L ad all p; j =1; 2; ;. Theorem 6. Let H be a Hilbert space ad L be a commutative subspace lattice o H. Let x1;x2; ;x ad y1;y2; ;y bevectors i H. Let K = K 1 = =1 =1 ;i E ;i x i : m i 2 N; l» ; E ;i 2Lad ;i 2 C ;i E ;i y i : m i 2 N;l» ; E ;i 2Lad ;i 2 C Assume that K 1 ρ K. P If sup =1 ;ie ;i y i m i =1 ;ie ;i x i : m i2 N;l» ; E ;i 2Lad ;i 2 C : ad <Ex p ;y j >=< Ey p ;x j > for every E i L ad all p; j =1; 2; ;, the there exists a operator A i AlgL such that Ax p = y p for all p =1; 2; ;, A Λ = A ad every E i L reduces A. Proof. By Theorem 2 [8], there exists a operator A i AlgL such thatax p = y p for all p =1; 2; ;ad every E i L reduces A. WewattoshowthatA Λ = A if <Ex p ;y j >=< Ey p ;x j > for every E i L ad all p; j =1; 2; ;. First, we willshowthata Λ x p = y p for all p =1; 2; ;. Sice <Ex p ;y j >=< Ey p ;x j > for all E i L ad all p; j =1; 2; ;, m i <A =1 m i ;i E ;i x i ;x j > =< ;i E ;i Ax i ;x j > =1 m i =< ;i E ;i y i ;x j > =1 m i =< ;i E ;i x i ;y j >: =1 Sice fy1;y2; ;y gρk, y j = A Λ x j for all j =1; 2; ;. Sice K 1 ρ K, A Λ f = 0 for every f i K?.HeceA Λ = A.

5 SELF-ADJOINT INTERPOLATION PROBLEMS IN CSL-ALGEBRA ALGL 455 Corollary 7. Let H be a Hilbert space ad L be a commutative subspace lattice o H. Let x1;x2; ;x ad y1;y2; ;y bevectors i H. Assume that K = sup ;i E ;i x i : m i 2 N;l» ; E ;i 2Lad ;i 2 C =1 =1 ;ie ;i y i P m i =1 ;ie ;i x i : m i 2 N;l» ; E ;i 2Lad ;i 2 C is dese i H. If <Ex q ;y j >=< Ey q ;x j > for every E i L ad all q; j =1; 2; ;, the there exists a operator A i AlgL such that Ax p = y p for all p =1; 2; ;, A Λ = A ad every E i L reduces A. If we summarize Theorems 5, 6 ad Corollary 7, we ca get the followig theorem. Theorem 8. Let H be a Hilbert space ad L be a commutative subspace lattice o H. Let x1;x2; ;x ad y1;y2; ;y bevectors i H. Assume that K 1 ρ K or K is dese i H. The the followig statemets are equivalet. 1 There exists a operator A i AlgL such that Ax p = y p for all p =1; ;, A Λ = A ad every E i L reduces A. 2 sup =1 ;ie ;i y i P m i =1 ;ie ;i x i : m i 2 N;l» ; E ;i 2Lad ;i 2 C ad <Ex p ;y j >=< Ey p ;x j > for every E i L ad all p; j =1; 2; ;. < 1 If we modify proofs of Theorems 5, 6, 7 ad 8 a little bit, we ca prove the followig theorems. So we will ot their proofs. Theorem 9. Let H be a Hilbert space ad let L be a subspace lattice oh. Let fx g ad fy g be two ifiite sequeces of vectors i H. If there isaoperator A i AlgL such that Ax = y for all =1; 2;, A Λ = A ad every E i L P reduces A, the sup =1 ;ie ;i y i m i =1 ;ie ;i x i :m i;l2n;e ;i 2L ad ;i 2 C 1 ad <Ex p ;y j >=< Ey p ;x j > for every E i L ad all p; j =1; 2;. Theorem 10. Let H be a Hilbert space ad let L be a commutative subspace lattice o H. Let fx g ad fy g be two ifiite sequeces of vectors i H. Let U = U 1 = ;i E ;i x i : m i ;l2 N; E ;i 2Lad ;i 2 C =1 ;i E ;i y i : m i ;l2n;e ;i 2L ad ;i 2 C : =1 Assume that U 1 ρ U. ad <

6 456 Y.-S. JOO AND J.-H. KANG If sup =1 ;ie ;i y i P m i =1 ;ie ;i x i : m i;l2 N; E ;i 2Lad ;i 2 C < 1 ad <Ex p ;y j >=< Ey p ;x j > for every E i L ad all p; j, thethereisaoperator A i AlgL such that Ax = y for all =1; 2;, A Λ = A ad every E i L reduces A. Corollary 11. Let H be a Hilbert space ad let L be a commutative subspace lattice o H. Let fx g ad fy g be two ifiite sequeces of vectors i H. Let U= =1 i H. If sup ;i E ;i x i : m i ;l2n;e ;i 2L ad ;i 2 C P m i =1 =1 ;ie ;i y i. Assume that U is dese ;ie ;i x i : m i;l2 N;E ;i 2Lad ;i 2 C < 1 ad <Ex p ;y j >=<Ey p ;x j > for every E i L ad all p; j, thethereisaoperator A i AlgL such that Ax =y for all =1;2;, A Λ =A ad every E i L reduces A. Theorem 12. Let H be a Hilbert space ad let L be a commutative subspace lattice o H. Let fx g ad fy g be two ifiite sequeces of vectors i H. Let U = U 1 = ;i E ;i x i : m i ;l2 N;E ;i 2Lad ;i 2 C =1 ;i E ;i y i : m i ;l2 N;E ;i 2Lad ;i 2 C =1 or U is dese i H. The the followig statemets are equivalet. every E i L reduces A. 2 sup =1 ;ie ;i y i P m i =1 ad let ;ie ;i x i :m i;l2n;e ;i 2L ad ;i 2 C. Assume that U 1 ρ U 1 There isaoperator A i AlgL such that Ax j = y j for all j =1; 2;, A Λ = A ad <1 ad <Ex p ;y j >=< Ey p ;x j > for every E i L ad all p; j. From Theorem 2, we ca get the followig theorem. Theorem 13. Let H be a Hilbert space ad let L be a commutative subspace lattice o H. Let x1; ;x ad y be vectors i H. If sup ρ P m ie i y P P m =1 ie i x : m 2 N; E i 2Lad i 2 C <Ex p ;y >=< Ey;x p > for every E i L ad all p =1; 2; ;, the there areoperators A1; ;A i AlgL such that y = P =1 A x, A Λ l = A l ad every E i L reduces A l for all l =1; 2; ;.

7 SELF-ADJOINT INTERPOLATION PROBLEMS IN CSL-ALGEBRA ALGL 457 Refereces 1. Arveso, W. B., Iterpolatio problems i est algebras, J. Fuctioal Aalysis, , Douglas, R. G., O majorizatio, factorizatio, ad rage iclusio of operators o Hilbert space, Proc. Amer. Math. Soc., , Gilfeather, F. ad Larso, D., Commutats modulo the compact operators of certai CSL algebras, Operator Theory: Adv. Appl. 2 Birhauser, Basel, 1981, Hopewasser, A., The equatio Tx= y i a reflexive operator algebra, Idiaa Uiversity Math. J , Hopewasser, A., Hilbert-Schdt iterpolatio i CSL algebras, Illiois J. Math. 4, , Jo, Y. S., Isometries of Tridiagoal Algebras, Pacific Joural of Mathematics 140, No , Jo, Y. S. ad Choi, T. Y., Isomorphisms of AlgL ad AlgL1, Michiga Math. J , Jo, Y. S. ad Kag J. H., Iterpolatio problems i CSL-Algebras AlgL, to appear i Rocy Moutai Joural of Math. 9. Kadiso, R., Irreducible Operator Algebras, Proc. Nat. Acad. Sci. U.S.A. 1957, Lace, E. C., Some properties of est algebras, Proc. Lodo Math. Soc., 3, , Much, N., Compact causal data iterpolatio, Aarhus Uiversity reprit series Youg Soo Jo Dept. of Math., Keimyug Uiversity Taegu, Korea ysjo@mu.ac.r Joo Ho Kag Dept. of Math., Taegu Uiversity Taegu, Korea jhag@taegu.ac.r