Examples of -isometries

Caixing Gu

California Polytechnic State University

San Luis Obispo, California

August, 2014

Thanks to the organizing committee, in particular Pro-

fessor Woo Young Lee, for the invitation to give a talk

Plan of the talk

• Definition and basic properties

• Weighted shifts

• Elementary operators

• Sum and product of -isometries

• Composition operators

• Function of -isometries

• Related operators

Some history

Historically, there were first studies of -symmetric opera-

tors (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 decomposi-

tion of two-isometries (2004). Hellings, Two-isometries

on Pontryagin spaces (2008). For 3-isometries, McCul-

lough (1987, 89).

Definition

Definition Agler and Stankus, on a Hilbert space

is an -isometry if

( ) = (− 1)( )=

X=0

(−1)−³

´ ∗ = 0

1( ) = ∗ −

h1( ) i = kk2 − kk22( ) = ∗2 2 − 2 ∗ +

h2( ) i =°°° 2°°°2 − 2 kk2 + kk2

3( ) = ∗3 3 − 3 ∗2 2 + 3 ∗ −

h3( ) i =°°° 3°°°2 − 3 °°° 2°°°2 + 3 kk2 − kk2

Throughout the talk, will be a fixed positive integer.

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=0

(−1)−³

´ °°°

°°°2→ (||2 − 1) = 0

Thus either ( ) = − or ( ) ⊆ is injective

and has close range.

Reproducing formula for ≥

∗ =−1X=0

³

´( )

∗ = ()( ) = (− 1 + 1)( )=

X=0

³

´(− 1)( )

=X

=0

³

´( ) =

−1X=0

³

´( )

last equality holds because ( ) = 0 for ≥

Covariance operator −1( ) ≥ 0

−1( ) =

⎛⎝−1X=0

³

´( )

⎞⎠ −1

= ∗

−1≥ 0

(−1( )) is the largest invariant subspace0 suchthat |0 is an -isometry.

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) ≥ 0But if is even, then

−1( ) = (−1)−1 ∗−1−1(−1)−1

= − ∗−1−1(−1)−1 ≤ 0thus −1( ) = 0 and 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.

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

Bayart, -isometries on Banach spaces (2011).

Definition on a Banach space is an ( )-isometry

if

()( ) :=X=0

(−1)−³

´ °°°°°° = 0

for all ∈ When = 2

(2)( ) =°°° 2°°° − 2 kk + kk = 0

Iteration formula

(+1)( ) = ()( )− ()( )

Spectrum ( ) ⊆

"Covariance operator" −1( ) ≥ 0 for all ∈

Reproducing formula (two slides down)

Hoffman, Mackey, and Searcóid, On the second parame-

ter 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.

Reproducing formulas

If is an ( )-isometry, then for all ≥ ∈

(1) kk =−1X=0

(−1)−1−³

´³− 1−

− 1−

´ °°°°°°

(2) kk =−1X=0

³

´( )

(3) kk =−1X=0

( )

(1) by Bermúdez, Martinón, and Negrín (2009).

(2) by Bayart (2011).

(3) by Gu.

Proof. HMS. If is an ( )-isometry, then ()∈N =³°°°°°°´

∈N is interpolated by ∈ P−1

must also be the unique Lagrange polynomial interpo-

lates {( ) = 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), if we write as () =−1P=0

then

() =−1X=0

= for = 0 1 · · · − 1

becomes

()h0 1 · · · −1

i=

h0 1 · · · −1

i

where is a Vandermonde matrix.

Weighted shift examples

Unilateral (bilateral) weighted unilateral shift

= +1 ∈ N ( ∈ Z)

Athavale (1991) If 2 =+1+−1

+1 ∈ N then is

an -isometry.

Cho, Ôta, and Tanahashi (2013) For odd if

2 =(+ 1)(+ 2) · · · (+− 1) +

(+ 1) · · · (+− 2) + ∈ Z

where is a positive constant, then is an invertible

-isometry

Bermúdez, Martinón, and Negrín, Weighted Shift Oper-

ators Which are m-Isometries (2009).

Partial results by Faghih-Ahmadi and Hedayatian (2013).

A unilateral weighted shift is an-isometry if and only

if 2−1 is given by (for ≥ 1)−1P=0

(−1)−1−³+1

´³−

−1−´ ¡01 · · ·−1

¢2−1P=0

(−1)−1−³

´³−1−−1−

´ ¡01 · · ·−1

¢2

When = 2 is a 2-isometry ⇐⇒ 0 1

When = 3 is a 3-isometry⇐⇒ infinite inequalities

on 0 1 for ≥ 1(−1)2

2021 − (− 2)20 +

(−1)(−2)2

(−1)(−2)2 20

21 − (− 1)(− 3)20 +

(−2)(−3)2

0

X = W02, Y = W1

2

( )-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 =+ 1 +− 1

+ 1

=(+ 2)(+ 3) · · · (+ 1 +− 1)(+ 1)(+ 2) · · · (+ 1 +− 2)

= (+ 1)

() ≥ 0

() = (+ 1)(+ 2) · · · (+ 1 +− 2)

Cho, 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)−1 + 1

−1 + 1 ∈ Z

This result also gives a way to check if 0 1 · · ·−2will generate an ( )-isometry.

For = 3 given 0 1 find a quadratic () such

that

0 =

(1)

(0)

1 =

(2)

(1)

Need to verify that () 0 for ∈ N, infinite inequal-ities again? No! we check the roots of the quadratic

()

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 = 21 + 22 +3

() =1Q=1(− )(− )

•2Q=1(− 2−1)(− 2) •

3Q=1(+ )

for some complex numbers = 1 · · · 1, posi-

tive numbers 2−1 2 such that [2−1] = [2] =

1 · · · 2 and 0 = 1 · · · 2. All 2−1 2are 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.

( )-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= 2The 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 forn( ) = ((0 − 1) + 1 0) ∈ N+

o

Gu, Let be an ( )-isometric shift on or two sided

for ≥ 2 and ∈ (0∞). Then there exist 0 ≥ 2and ≥ 1 such that

( ) = ((0 − 1) + 1 )and is an (0 )-isometry on or two sided

The main part of the proof is to argue = for some

integer

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 1X

h1( )

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.

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 = 2 and = 3

They conjectured that, if is a -isometry and ∗ isa -isometry, then () on 2() is a (+ − 1)-isometry. They proved for = 2 or = 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

a strict -isometry and ∗ is a strict -isometry with = + − 1

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

0

# 2 0

=

" + 0

0 − +

#

=

2+

s1− 2

4

where is a nilpotent operator of order Then ()

is a strict (2− 1)-isometry with () =n±±

o

Note that

() = {0 } () =n −

oBy 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= 2can not be an -isometry unless be an isometry.

Sum

Bermúdez, Martinón and Noda, An isometry plus a nilpo-

tent operator is an -isometry. Applications (2013)

Gu and Stankus (2014) Assume ∈ () is com-

muting and is an -isometry and is a nilpotent op-

erator of order Then + is a (+2−2)-isometry.Proof. Let = + 2− 2

(+) =X

=0

−X=0

³

´³−

´(∗ +∗)∗−−()

If or ≥ then ∗ = 0 or = 0. If and

then

− − = + 2− 2− − ≥

and −−() = 0 Therefore (+) = 0.

This result does not extend to Banach space.

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 dou-

ble commuting and is an -isometry and is an

-isometry. Then is a (+ − 1)-isometryProof. Let = + − 1

() =X

=0

³

´∗−()()

Note that if ≥ then () = 0 and if then

− = +−1− ≥ and −() = 0 Therefore() = 0

This does give a quick proof the result of Duggal because

⊗ = (⊗ ) (⊗) and (⊗ ) (⊗) is doublecommuting.

A recent preprint of Trieu Le proves similar results.

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 oper-

ator 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

A tiny modification (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 →

1

. 1

. 2 ← 5 ← 8 ← · · ·- 3 ← 6 ← 9 ← · · ·- 4 ← 7 ← 10 ← · · ·

(1 2 3 · · · ) = (1 1 1 1 2 3 4 5 · · · )

1

. 1

← 2 ← 7 ← 10 ← 13 ← · · ·- 3 ← 9 ← 12 ← 15 ← · · ·

4. 4

- 5 ← 6 ← 8 ← 11 ← · · ·

(1 2 3 · · · )= (1 1 1 4 4 5 2 5 2 3 7 8 9 · · · )

Powers of -isometry on Banach space

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

()( ) =

P1+···=

Ã

1 · · ·

!()(

(0·1+1·2+···(−1)))

Thus if ()( (0·1+1·2+···(−1))) = 0 then

()( ) = 0

Bermúdez, Mendoza and Martinón also offered an inter-

esting converse: if both and are ( )-isometries

for two coprime positive integers and then is an

( )-isometry.

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.

Hyperexpansion and hypercontraction onBanach space

For ∈ () or ∈ ()

( ) =X=0

(−1)−³

´ ∗

()( ) =X=0

(−1)−³

´ °°°°°°

is ( )-contractive, ( )-hypercontractive, com-

pletely -hypercontractive, if

(−1)()( ) ≥ 0

(−1)()( ) ≥ 0 1 ≤ ≤

(−1)()( ) ≥ 0 ≥ 1 is ( )-expansive, ( )-hyperexpansive, completely

-hyperexpansive if

≤ 0 is ( )-alternatingly expansive if

()( ) ≥ 0

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 ≥ 1if and only if is a subnormal contraction

Definition ∈ () is a -subnormal operator if kkis completely -hyercontractive.