Examples of -isometries

Transcription

1 Examples of -isometries Caixing Gu California Polytechnic State University San Luis Obispo, California August, 2014 Thanks to the organizing committee, in particular Professor Woo Young Lee, for the invitation to give a talk

2 Plan of the talk Definition and basic properties Weighted shifts Elementary operators Sum and product of -isometries Composition operators Function of -isometries Related operators

3 Some history Historically, there were first studies of -symmetric operators (with connection to Sturm-Liouville conjugate point theory) by Helton (1972,74), Agler (1980, 1992), Ball- Helton (1980), Bunce (1983), Rodman and McCullough (1996, 1997, 1998). The study of -isometries was started by Agler and Stankus "-isometric transformations of Hilbert space, I, II, III" (1995). For two-isometries, there was another approach by Richter, a representation theorem for cyclic analytic two-isometries (1991), and Olofsson, A von Neumann-Wold decomposition of two-isometries (2004). Hellings, Two-isometries on Pontryagin spaces (2008). For 3-isometries, McCullough (1987, 89).

4 Definition Definition Agler and Stankus, on a Hilbert space is an -isometry if ( ) = ( 1) ( ) X = ( 1) ³ =0 =0 1 ( ) = h 1 ( ) i = kk 2 kk 2 2 ( ) = h 2 ( ) i = kk 2 + kk 2 3 ( ) = h 3 ( ) i = kk 2 kk 2 Throughout the talk, will be a fixed positive integer.

5 Iteration formula ( 1) +1 = ( 1) ( 1) +1 ( ) = ( ) ( ) Thus if is an -isometry, then is an -isometry for Spectrum ( ) If ( ) let be a sequence of unit vectors such that ( ) 0 then D E X ( ) = ( 1) ³ =0 ( 2 1) =0 2 Thus either ( )= or ( ) is injective and has close range.

6 Reproducing formula for = 1 X =0 ³ ( ) = () ( )=( 1+1) ( ) = X ( 1) ( ) = =0 X ³ X ( )= 1 ³ =0 =0 ( ) last equality holds because ( )=0for Covariance operator 1 ( ) 0 1 ( ) = 1 X ³ =0 = 1 ( ) 0 1 ( 1 ( )) is the largest invariant subspace 0 such that 0 is an -isometry.

7 Definition We say is a strict -isometry if is an -isometry but not an ( 1)-isometry. If is an invertible strict -isometry, then is odd Proof. If is an invertible -isometry, then 1 ( ) 0 Note also ( 1 )=( 1) ( ) =0 so 1 is also an -isometry and 1 ( 1 ) 0 But if is even, then 1 ( ) = ( 1) ( 1 ) 1 = 1 1 ( 1 ) 1 0 thus 1 ( )=0and is an ( 1)-isometry. If is a finitely cyclic -isometry and is even, then 1 ( ) is compact. An example of a cyclic 3-isometry where 2 ( ) is not compact is given by Agler and Stankus.

8 On Banach space Recall on is an -isometry if for all h ( ) i = X =0 ( 1) ³ 2 =0 Two authors use this formulation to define -isometries on Banach space. Ahmed, -isometric operators on Banach spaces (2010), Botelho, On the existence of -isometries on spaces (2009). Botelho: There is no (2 2)-isometric weighted shift or composition operator on for 6= 2

9 Bayart, -isometries on Banach spaces (2011). Definition on a Banach space is an ( )-isometry if () ():= X =0 for all When =2 (2) ()= ( 1) ³ =0 2 2 kk + kk =0 Iteration formula (+1) ()= () () () () Spectrum ( ) "Covariance operator" 1 () 0 for all Reproducing formula (two slides down)

10 Hoffman, Mackey, and Searcóid, On the second parameter of an ( )-isometry (2011) Let = 0 is an ( )-isometry if and only if, for each there exists a polynomial () of degree 1 such that () = Proof If is an ( )-isometry, then X =0 ( 1) ³ + =0 0 The result follows from studying this difference equation.

11 Reproducing formulas If is an ( )-isometry, then for all (1) k k = 1 X =0 ( 1) 1 ³ ³ 1 1 (2) k k = 1 X =0 ³ () (3) k k = 1 X =0 () (1) by Bermúdez, Martinón, and Negrín (2009). (2) by Bayart (2011). (3) by Gu.

12 Proof. HMS. If is an ( )-isometry, then ( ) N = ³ N is interpolated by P 1 must also be the unique Lagrange polynomial interpolates {( ) =0 1 1} Using the normal form of the Lagrange polynomial to calculate () yields (1), and using the barrycentric form we get (2). For(3),ifwewrite as () = 1 P =0 then () = becomes 1 X =0 = for =0 1 1 () h i = h i where is a Vandermonde matrix.

13 Weighted shift examples Unilateral (bilateral) weighted unilateral shift = +1 N ( Z) Athavale (1991) If 2 = N then is an -isometry. Chō, Ôta, and Tanahashi (2013) For odd if 2 ( +1)( +2) ( + 1) + = Z ( +1) ( + 2) + where is a positive constant, then is an invertible -isometry

14 Bermúdez, Martinón, and Negrín, Weighted Shift Operators Which are m-isometries (2009). Partial results by Faghih-Ahmadi and Hedayatian (2013). A unilateral weighted shift is an -isometry if and only if 1 2 is given by (for 1) 1 P ( 1) 1 ³ ³ +1 1 =0 1 P ( 1) 1 ³ ³ 1 1 = When =2 is a 2-isometry 0 1 When =3 is a 3-isometry infinite inequalities on 0 1 for 1 ( 1) ( 2)2 0 + ( 1)( 2) 2 ( 1)( 2) ( 1)( 3)2 0 + ( 2)( 3) 2 0

15 X = W 0 2, Y = W 1 2

16 ( )-isometry on Gu, on is a strict ( )-isometry () of degree 1 such that, () 0 for N and ( ) ( +1) = for N () For the bilateral shifts, change " N" to" Z". This gives a transparent view of examples. Athavale (1991) 2 = = ( +2)( +3) ( +1+ 1) ( +1)( +2) ( +1+ 2) = ( +1) 0 () () = ( +1)( +2) ( +1+ 2)

17 Chō, Ôta, and Tanahashi (2013), is odd, 2 = ( +1)( +2) ( + 1) + ( +1) ( + 2) + = ( +1) Z () () = ( +1) ( + 2) + Other simple example of bilateral weighted shifts 2 = ( +1) Z +1 This result also gives a way to check if will generate an ( )-isometry. For =3given 0 1 find a quadratic () such that 0 = (1) (0) 1 = (2) (1) Need to verify that () 0 for N, infinite inequalities again? No! we check the roots of the quadratic ()

18 Here is a way to check if () 0 for N ( Z) by looking at its roots. The polynomial () of degree 1 such that () 0 for N if and only if 1= () = Q1 =1 Q2 =1 ( )( ) ( 2 1 )( 2 ) Q3 =1 ( + ) for some complex numbers = 1 1, positive numbers such that [ 2 1 ]=[ 2 ] = 1 2 and 0 =1 2.All are not integers, but could be integers. A similar result holds for () such that () 0 for Z. By choosing roots of polynomials, we can generate all ( )-isometric shifts.

19 ( )-isometric weighted shifts for 6= Our result is inspired by the following two results. 1. Botelho (2010) There is no strict (2 2)-isometric shift on for 6= 2 The proof uses delicate algebra and the equality conditios in norm inequalities. 2. Hoffman, Mackey, and Searcóid (2011) If is an ( 0 0 )-isometry, then is also an ( )-isometry for n ( ) =((0 1) ) N +o Gu, Let be an ( )-isometric shift on or two sided for 2 and (0 ). Thenthereexist 0 2 and 1 such that ( ) =(( 0 1) + 1) and is an ( 0 )-isometry on or two sided Themainpartoftheproofistoargue = for some integer

20 Gu, There are no ( )-isometric shifts on or for 2 and (0 ). Question ( )-isometric shifts on or? Question Any ( )-isometry on for 6=? Berger-Shaw type result. Assume is a (2)-isometric shift on.then 1 ( )=(1) thus for 1 X h 1 ( ) i Assume is an ( )-isometric shift on where 3 Then 1 ( )=(1 1 ), X 1 ( ) On 2 if is an -isometry for 3 then 1 ( ) is trace class. If is a 2-isometry, then 1 ( ) is not trace class.

21 Elementary Operator For () and let 2 () be the ideal of Hilbert- Schmidt operators. The elementary operator ( ) is defined by ( )() = 2 () When is ( ) on 2 () an -isometry? Botelho, Jamison (2010), Botelho, Jamison and Zheng, (2012) answered this question for =2and =3 They conjectured that, if is a -isometry and is a -isometry, then ( ) on 2 () is a ( + 1)- isometry. They proved for =2or =2 Duggal, Tensor product of -isometries (2012), Yes to the conjecture. ( ) = Gu, Complete the story (2014), ( ) on 2 () is an strict -isometry if and only if for some constant is astrict-isometry and is a strict -isometry with = + 1

22 Derivation ( ) on 2 () is defined by ( )() = 2 () Gu, If is a strict -isometry and is a nilpotent operator of order, then( ) is a (+2 2)-isometry. The converse is almost true excerpt when () =± ± for some [0 2) We construct ( ), " # 0 0 = " = # s = where is a nilpotent operator of order Then ( ) is a strict (2 1)-isometry with () = n ± ±o

23 Note that () ={0} () = n o By a theorem of Rosenblum, () =() (), () = () () = n o = n ± ±o Question ( )( ) on () ()? on ()? When is finite dimensional, is an -isometry if and only if = + where is an unitary and is a nilpotent operator such that = (Agler-Helton- Stankus, Classification of hereditary matrices, 1998). Examples suggest ( ) on () or () for 6= 2 can not be an -isometry unless be an isometry.

24 Sum Bermúdez, Martinón and Noda, An isometry plus a nilpotent operator is an -isometry. Applications (2013) Gu and Stankus (2014) Assume () is commuting and is an -isometry and is a nilpotent operator of order Then + is a (+2 2)-isometry. Proof. Let = +2 2 ( + ) = X =0 X =0 ³ ³ ( + ) () If or then =0or =0. If and then = +2 2 and () =0 Therefore ( + ) =0. This result does not extend to Banach space.

25 Product Bermúdez, Martinón, Noda (2013) Assume () is commuting and is an -isometry and is an - isometry Then is a ( + 1)-isometry We give a proof on Hilbert space with an extra condition. Gu and Stankus (2014) Assume () is double commuting and is an -isometry and is an -isometry. Then is a ( + 1)-isometry Proof. Let = + 1 () = X =0 ³ () () Note that if then () =0and if then = + 1 and () =0 Therefore () =0 This does give a quick proof the result of Duggal because =( )( ) and ( ) ( ) is double commuting. A recent preprint of Trieu Le proves similar results.

26 Composition operators Patton and Robins (2005) No -isometry for composition operator on various Hilbert spaces of analytic functions Botelho (2009) No (2 2)-isometry for composition operator on ( 6= 2)or on () Examples of composition operators on 2 Let : N N = {1 2 3 } () = n () = o () = Cardinality of () on 2, ( n o )= n () o is an ( 2)-isometry if and only if is surjective and X =0 ( 1) ³ () =0 N

27 Atinymodification (Gu), on, ( n o )= n () o is an ( )-isometry if and only if is surjective and X =0 ( 1) ³ () =0 N Example (Botelho), () = denote by ( )=( ) ( ) = ( )

28 Powers of -isometryonbanachspace If () is an ( )-isometry, then is also an ( )-isometry. Jablonski, Complete hyperexpansivity, subnormality...(2002) Bermúdez, Mendoza and Martinón, Powers of -isometries (2013) on Banach space. Gu, A quick proof. Let be an operator on Then for 1 () ( ) = P 1 + = Ã 1 () ( ( ( 1) ) ) Thus if () ( ( ( 1) ) ) = 0 then () ( )=0 Bermúdez, Mendoza and Martinón also offered an interesting converse: if both and are ( )-isometries for two coprime positive integers and then is an ( )-isometry.!

29 Inner function of -isometry Gu, Let be an -isometry on with ( ) D Let () and () be two coprime inner functions such that both () ( ) and (1) ( ) are empty. Then ( ) 1 ( ) is an -isometry. When is a finite Blaschke product, (( )) = P 1 + = Ã 1 ( 1 ) ( )( 1 ) for some ( 1 ) () Therefore if ( )=0 then (( ))! The proof for the general inner function goes by a limit to infinite Blaschke product and by Frostman s Theorem (a mobius transform of an inner function is often an infinite Blaschke product) to singular inner function. This result does not extend to Banach space.

30 Hyperexpansion and hypercontraction on Banach space For () or () ( ) = () () = X =0 X =0 ( 1) ³ ( 1) ³ is ( )-contractive, ( )-hypercontractive, completely -hypercontractive, if ( 1) () () 0 ( 1) () () 0 1 ( 1) () () 0 1 is ( )-expansive, ( )-hyperexpansive, completely -hyperexpansive if 0 is ( )-alternatingly expansive if () () 0

31 On Hilbert space Agler, Hypercontractions and subnormality (1985) Athavale, On completely hyperexpansive operators (1996) Sholapurkar and Athavale, Completely and alternatingly hyperexpansive operators (2000) Exner, Bong Jung and Chunji Li, On -hyperexpansive operators (2006) Olofsson, An operator-valued Berezin transform and the class of -hypercontraction (2007) Exner, Bong Jung and Sang Soo Park, On -contractive and -hypercontractive operators II (2008) Chavan and R.E. Curto, Operators Cauchy dual to 2- hyperexpansive operators: the multivariable case (2012) Agler: () is an -hypercontraction for all 1 if and only if is a subnormal contraction Definition () is a -subnormal operator if k k is completely -hyercontractive.