Introduction
Properties
Unilateral weighted shift operator
Examples of m-isometries
Teresa Bermúdez, Antonio Martinón, Juan Agustín Noda
Workshop on Functional Analysis
On the occasion of the 60th birthday of José Bonet
WFAV15
Valencia, 15th to the 19th of June 2015
Departamento de Análisis Matemático
Universidad de la Laguna
Teresa Bermúdez, Antonio Martinón, Juan Agustín Noda
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Examples of m-isometries
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Examples of m-isometries
[T. B. , A. Martinón, J. A. Noda, Weighted shift and composition
operator on `p (N) which are (m, q)-isometries, in preparation]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
WFAV15-José Bonet
Definitions
History and motivation
m-isometries on Banach spaces
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Denote
H ! complex Hilbert spaces.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Denote
H ! complex Hilbert spaces.
L(H) := {T : H → H, bounded and linear operator}
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Denote
H ! complex Hilbert spaces.
L(H) := {T : H → H, bounded and linear operator}
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Denote
H ! complex Hilbert spaces.
L(H) := {T : H → H, bounded and linear operator}
1
T : H −→ H is an isometry if kTxk = kxk for all x ∈ H.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Denote
H ! complex Hilbert spaces.
L(H) := {T : H → H, bounded and linear operator}
1
T : H −→ H is an isometry if kTxk = kxk for all x ∈ H.
2
T ∈ L(H) is an isometry ⇔ T ∗ T = I.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries






















WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries ⇔ T|Ker (T )⊥ is an isometry






















WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries ⇔ T|Ker (T )⊥ is an isometry










 Quasi-isometries











WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries ⇔ T|Ker (T )⊥ is an isometry










 Quasi-isometries ⇔ T|R(T ) is an isometry











WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries ⇔ T|Ker (T )⊥ is an isometry










 Quasi-isometries ⇔ T|R(T ) is an isometry



A-isometries








WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries ⇔ T|Ker (T )⊥ is an isometry










 Quasi-isometries ⇔ T|R(T ) is an isometry



A-isometries ⇔ kTxkA = kxkA , hx, y iA := hAx, y i








WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries ⇔ T|Ker (T )⊥ is an isometry










 Quasi-isometries ⇔ T|R(T ) is an isometry



A-isometries ⇔ kTxkA = kxkA , hx, y iA := hAx, y i








···
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Generalizations
Isometry

Partial isometries ⇔ T|Ker (T )⊥ is an isometry










 Quasi-isometries ⇔ T|R(T ) is an isometry



A-isometries ⇔ kTxkA = kxkA , hx, y iA := hAx, y i








···
m-isometries
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Some m-isometries
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Some m-isometries
1-isometry ! T ∗ T − I = 0
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Some m-isometries
1-isometry ! T ∗ T − I = 0
2-isometry ! T ∗2 T 2 − 2T ∗ T + I = 0
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Some m-isometries
1-isometry ! T ∗ T − I = 0
2-isometry ! T ∗2 T 2 − 2T ∗ T + I = 0
3-isometry ! T ∗3 T 3 − 3T ∗2 T 2 + 3T ∗ T − I = 0
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Some m-isometries
1-isometry ! T ∗ T − I = 0
2-isometry ! T ∗2 T 2 − 2T ∗ T + I = 0
3-isometry ! T ∗3 T 3 − 3T ∗2 T 2 + 3T ∗ T − I = 0
Definition
T ∈ L(H) is an m-isometry if
m
m−k
∑ (−1)
k =0
m
T ∗k T k = 0 ,
k
WFAV15-José Bonet
Examples of m-isometries
(1.1)
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Some m-isometries
1-isometry ! T ∗ T − I = 0
2-isometry ! T ∗2 T 2 − 2T ∗ T + I = 0
3-isometry ! T ∗3 T 3 − 3T ∗2 T 2 + 3T ∗ T − I = 0
Definition
T ∈ L(H) is an m-isometry if
m
m−k
∑ (−1)
k =0
m
T ∗k T k = 0 ,
k
Properties
WFAV15-José Bonet
Examples of m-isometries
(1.1)
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Some m-isometries
1-isometry ! T ∗ T − I = 0
2-isometry ! T ∗2 T 2 − 2T ∗ T + I = 0
3-isometry ! T ∗3 T 3 − 3T ∗2 T 2 + 3T ∗ T − I = 0
Definition
T ∈ L(H) is an m-isometry if
m
m−k
∑ (−1)
k =0
m
T ∗k T k = 0 ,
k
(1.1)
Properties
T is an m-isometry
m
⇔
m−k
∑ (−1)
k =0
m
kT k xk2 = 0 , ∀x ∈ H
k
WFAV15-José Bonet
Examples of m-isometries
(1.2)
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Bibliography
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Bibliography
J. Agler, Amer. J. Math. 1990
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Bibliography
J. Agler, Amer. J. Math. 1990
J. Agler, M. Stankus, Integral Equations Operator Theory,
1995
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Bibliography
J. Agler, Amer. J. Math. 1990
J. Agler, M. Stankus, Integral Equations Operator Theory,
1995
J. Agler, M. Stankus, Integral Equations Operator Theory,
1995
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Bibliography
J. Agler, Amer. J. Math. 1990
J. Agler, M. Stankus, Integral Equations Operator Theory,
1995
J. Agler, M. Stankus, Integral Equations Operator Theory,
1995
J. Agler, M. Stankus, Integral Equations Operator Theory,
1996
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Connections















m-isometries














WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Connections















m-isometries














WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Connections

Toeplitz operators














m-isometries














WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Connections

Toeplitz operators









 Nonstationary stochastic processes


















WFAV15-José Bonet
Examples of m-isometries
m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Connections

Toeplitz operators









 Nonstationary stochastic processes




m-isometries
Invariant subspaces of the shift operator














WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Connections

Toeplitz operators









 Nonstationary stochastic processes




m-isometries
Invariant subspaces of the shift operator










···




WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Definition
T is a strict m-isometry if T is an m-isometry and not
(m − 1)-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Definition
T is a strict m-isometry if T is an m-isometry and not
(m − 1)-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Definition
T is a strict m-isometry if T is an m-isometry and not
(m − 1)-isometry.
Im = Im (H) := {T ∈ L(H) : T is an m-isometry} .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Definition
T is a strict m-isometry if T is an m-isometry and not
(m − 1)-isometry.
Im = Im (H) := {T ∈ L(H) : T is an m-isometry} .
Properties
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Definition
T is a strict m-isometry if T is an m-isometry and not
(m − 1)-isometry.
Im = Im (H) := {T ∈ L(H) : T is an m-isometry} .
Properties
1
T is an m-isometry ⇒ T is an (m + k )-isometry for any
k ≥ 0.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
Definition
T is a strict m-isometry if T is an m-isometry and not
(m − 1)-isometry.
Im = Im (H) := {T ∈ L(H) : T is an m-isometry} .
Properties
1
2
T is an m-isometry ⇒ T is an (m + k )-isometry for any
k ≥ 0.
[Athavale, 1991]
I1 ( I2 ( · · · ( Im ( Im+1 .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
m-isometries on Banach spaces
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
m-isometries on Banach spaces
A linear map T : X −→ X is an (m, q)-isometry if for all
x ∈ X,
m
m−k m
kT k xkq = 0 .
(1.3)
∑ (−1)
k
k =0
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
m-isometries on Banach spaces
A linear map T : X −→ X is an (m, q)-isometry if for all
x ∈ X,
m
m−k m
kT k xkq = 0 .
(1.3)
∑ (−1)
k
k =0
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
m-isometries on Banach spaces
A linear map T : X −→ X is an (m, q)-isometry if for all
x ∈ X,
m
m−k m
kT k xkq = 0 .
(1.3)
∑ (−1)
k
k =0
[F. Bayart, Math. Nachr., 2011.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
m-isometries on Banach spaces
A linear map T : X −→ X is an (m, q)-isometry if for all
x ∈ X,
m
m−k m
kT k xkq = 0 .
(1.3)
∑ (−1)
k
k =0
[F. Bayart, Math. Nachr., 2011.]
[P. Hoffmann, M. Mackey, and M. Searcóid, IEOT, 2011.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Definitions
History and motivation
m-isometries on Banach spaces
m-isometries
m
∑ (−1)
k =0
m−k
m
kT k xk2 = 0 , ∀x ∈ H
k
[F. Botelho, Acta Sci. Math. (Szeged), 2010.]
[O. Sid Ahmed, AEJM,2010.]
m-isometries on Banach spaces
A linear map T : X −→ X is an (m, q)-isometry if for all
x ∈ X,
m
m−k m
kT k xkq = 0 .
(1.3)
∑ (−1)
k
k =0
[F. Bayart, Math. Nachr., 2011.]
[P. Hoffmann, M. Mackey, and M. Searcóid, IEOT, 2011.]
[T. B., A. Martinón, and V.M. Müller, JOT, 2014.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[J. Agler, J. Helton, M. Stankus, LAA, 1998.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[J. Agler, J. Helton, M. Stankus, LAA, 1998.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[J. Agler, J. Helton, M. Stankus, LAA, 1998.]
Let H be a finite dimensional Hilbert space and an odd
integer m.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[J. Agler, J. Helton, M. Stankus, LAA, 1998.]
Let H be a finite dimensional Hilbert space and an odd
integer m.

∃U unitary






Q nilpotent
T ∈ L(H) is a strict m-isometry ⇔





 UQ = QU
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[J. Agler, J. Helton, M. Stankus, LAA, 1998.]
Let H be a finite dimensional Hilbert space and an odd
integer m.

∃U unitary






Q nilpotent
T ∈ L(H) is a strict m-isometry ⇔





 UQ = QU
WFAV15-José Bonet
Examples of m-isometries
T = U+Q
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[J. Agler, J. Helton, M. Stankus, LAA, 1998.]
Let H be a finite dimensional Hilbert space and an odd
integer m.

∃U unitary






Q nilpotent
T ∈ L(H) is a strict m-isometry ⇔





 UQ = QU
WFAV15-José Bonet
Examples of m-isometries
T = U+Q
n=
m+1
2
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[J. Agler, J. Helton, M. Stankus, LAA, 1998.]
Let H be a finite dimensional Hilbert space and an odd
integer m.

∃U unitary






Q nilpotent
T ∈ L(H) is a strict m-isometry ⇔





 UQ = QU
WFAV15-José Bonet
Examples of m-isometries
T = U+Q
n=
m+1
2
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Finite dimensional space
dim(H) = n
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Finite dimensional space
dim(H) = n
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Finite dimensional space
dim(H) = n
Im = Im (H) := {T ∈ L(H) : T is an m-isometry} .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Finite dimensional space
dim(H) = n
Im = Im (H) := {T ∈ L(H) : T is an m-isometry} .
I1 = I2 ( I3 = I4 ( · · · ( I2n−1 = I2n = · · · .
Return
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.
A an isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.
A an isometry
Q nilpotent operator
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.
A an isometry
Q nilpotent operator
Q n = 0, AQ = QA
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.


A an isometry



Q nilpotent operator
⇒ A+Q




Q n = 0, AQ = QA
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.


A an isometry



Q nilpotent operator
⇒ A + Q is a strict (2n − 1)-isometry.




Q n = 0, AQ = QA
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.


A an isometry



Q nilpotent operator
⇒ A + Q is a strict (2n − 1)-isometry.




Q n = 0, AQ = QA
Example
Given an odd m = 2n − 1.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.


A an isometry



Q nilpotent operator
⇒ A + Q is a strict (2n − 1)-isometry.




Q n = 0, AQ = QA
Example
Given an odd m = 2n − 1.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.


A an isometry



Q nilpotent operator
⇒ A + Q is a strict (2n − 1)-isometry.




Q n = 0, AQ = QA
Example
Given an odd m = 2n − 1.
There exists T ∈ L(`2 (N)) that is an m-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.


A an isometry



Q nilpotent operator
⇒ A + Q is a strict (2n − 1)-isometry.




Q n = 0, AQ = QA
Example
Given an odd m = 2n − 1.
There exists T ∈ L(`2 (N)) that is an m-isometry.
Q(x1 , x2 , · · · ) := (x2 , x3 , · · · , xn , 0, 0, · · · ) ! nilpotent
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, J. Noda, JMAA, 2013]
Let H be a Hilbert space.


A an isometry



Q nilpotent operator
⇒ A + Q is a strict (2n − 1)-isometry.




Q n = 0, AQ = QA
Example
Given an odd m = 2n − 1.
There exists T ∈ L(`2 (N)) that is an m-isometry.
Q(x1 , x2 , · · · ) := (x2 , x3 , · · · , xn , 0, 0, · · · ) ! nilpotent
T = I +Q
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.
A an m-isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.
A an m-isometry
Q nilpotent operator
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.
A an m-isometry
Q nilpotent operator
Q n = 0, AQ = QA
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.


A an m-isometry



Q nilpotent operator
⇒ A+Q




Q n = 0, AQ = QA
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.


A an m-isometry



Q nilpotent operator
⇒ A + Q is a (2n + m − 2)-isometry.




Q n = 0, AQ = QA
Return
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.


A an m-isometry



Q nilpotent operator
⇒ A + Q is a (2n + m − 2)-isometry.




Q n = 0, AQ = QA
Return
References
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.


A an m-isometry



Q nilpotent operator
⇒ A + Q is a (2n + m − 2)-isometry.




Q n = 0, AQ = QA
Return
References
[C. Gu, M. Stankus, Linear Algebra Appl. 2015.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
[T. B, A. Martinón, V. Müller, J. Noda, AAA, 2014]
Let H be a Hilbert space.


A an m-isometry



Q nilpotent operator
⇒ A + Q is a (2n + m − 2)-isometry.




Q n = 0, AQ = QA
Return
References
[C. Gu, M. Stankus, Linear Algebra Appl. 2015.]
[T. Le, JMAA , 2015.]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
WFAV15-José Bonet
Examples of m-isometries
Finite Hilbert space
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
Q : C2 −→ C2
(x, y ) −→ Q(x, y ) := (y , 0)
The following assertions hold:
WFAV15-José Bonet
Examples of m-isometries
Finite Hilbert space
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
Finite Hilbert space
Q : C2 −→ C2
(x, y ) −→ Q(x, y ) := (y , 0)
The following assertions hold:
1
I + Q is not a (3, p)-isometry on `2p for any 1 ≤ p < ∞ and
p 6= 2.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
Finite Hilbert space
Q : C2 −→ C2
(x, y ) −→ Q(x, y ) := (y , 0)
The following assertions hold:
1
2
I + Q is not a (3, p)-isometry on `2p for any 1 ≤ p < ∞ and
p 6= 2.
I + Q is a strict (2k + 1, 2k )-isometry on (C2 , k.k2k ) for any
k = 1, 2, 3, . . .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
Finite Hilbert space
Q : C2 −→ C2
(x, y ) −→ Q(x, y ) := (y , 0)
The following assertions hold:
1
2
I + Q is not a (3, p)-isometry on `2p for any 1 ≤ p < ∞ and
p 6= 2.
I + Q is a strict (2k + 1, 2k )-isometry on (C2 , k.k2k ) for any
k = 1, 2, 3, . . .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
Infinite dimensional Banach space.
WFAV15-José Bonet
Examples of m-isometries
Finite Hilbert space
Infinite Hilbert space
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
Infinite dimensional Banach space.
Finite Hilbert space
Infinite Hilbert space
X := {f : [0, 1] → R continuous such that f (1) = 0}

 f (t + 12 ) if 0 ≤ t ≤ 21
(Qf )(t) :=

0
if 12 < t ≤ 1 .
I + Q is not an (m, q)-isometry for any m and any q > 0.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Finite dimensional Hilbert space
Infinite dimensional Hilbert space
Banach space
Problem
What it happens in Banach space?
Finite dimensional Banach space.
Infinite dimensional Banach space.
Answer: No
WFAV15-José Bonet
Examples of m-isometries
Finite Hilbert space
Infinite Hilbert space
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Unilateral weighted forward shift
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Unilateral weighted forward shift
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Unilateral weighted forward shift
Given w := (wn )n∈N
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Unilateral weighted forward shift
Given w := (wn )n∈N
Sw
: `2 (N)
−→ `2 (N)
x := (x1 , x2 , · · · ) −→ Sw x = (0, w1 x1 , w2 x2 , · · · )
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Unilateral weighted forward shift
Given w := (wn )n∈N
Sw
−→ `2 (N)
: `2 (N)
x := (x1 , x2 , · · · ) −→ Sw x = (0, w1 x1 , w2 x2 , · · · )
Sw an isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Unilateral weighted forward shift
Given w := (wn )n∈N
Sw
−→ `2 (N)
: `2 (N)
x := (x1 , x2 , · · · ) −→ Sw x = (0, w1 x1 , w2 x2 , · · · )
Sw an isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Unilateral weighted forward shift
Given w := (wn )n∈N
Sw
: `2 (N)
−→ `2 (N)
x := (x1 , x2 , · · · ) −→ Sw x = (0, w1 x1 , w2 x2 , · · · )
Sw an isometry ⇔ |wn | = 1.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B. A. Martinón, E. Negrín, IEOT, 2010]
Sw is an m-isometry ⇔
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B. A. Martinón, E. Negrín, IEOT, 2010]
Sw is an m-isometry ⇔
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B. A. Martinón, E. Negrín, IEOT, 2010]
Sw is an m-isometry ⇔
|wn |2 =
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B. A. Martinón, E. Negrín, IEOT, 2010]
Sw is an m-isometry ⇔
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−1
k !(m − k − 1)!
|w0 · · · wk |2
z }| {
m−k −1 (n − 1)... (n − 1 − k ) ...(n − m)
∑ (−1)
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
WFAV15-José Bonet
Examples of m-isometries
>0,
|w0 · · · wk |2
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B. A. Martinón, E. Negrín, IEOT, 2010]
Sw is an m-isometry ⇔
z }| {
m−1
∑ (−1)
|wn |2 =
m−k −1 n... (n − k ) ...(n − m + 1)
k =0
m−1
k !(m − k − 1)!
|w0 · · · wk |2
z }| {
m−k −1 (n − 1)... (n − 1 − k ) ...(n − m)
∑ (−1)
k !(m − k − 1)!
k =0
⇔ |wn |2 =
>0,
|w0 · · · wk |2
p(n + 1)
, p(n) > 0 for all n > 0
p(n)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
WFAV15-José Bonet
p(n+1)
p(n) ,
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.
WFAV15-José Bonet
p(n+1)
p(n) ,
p a positive
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.
p(n+1)
p(n) ,
p a positive
a1 , a2 , . . . , ar ∈ C \ {0}
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.
p(n+1)
p(n) ,
p a positive
a1 , a2 , . . . , ar ∈ C \ {0}
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.
p(n+1)
p(n) ,
p a positive
a1 , a2 , . . . , ar ∈ C \ {0}
m ≥ r +2
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.

a1 , a2 , . . . , ar ∈ C \ {0} 
m ≥ r +2
p(n+1)
p(n) ,
p a positive
⇒ ∃Sw m-isometry.

WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.

a1 , a2 , . . . , ar ∈ C \ {0} 
m ≥ r +2
p(n+1)
p(n) ,
p a positive
⇒ ∃Sw m-isometry.

Sw is an m-isometric (non-isometric)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.

a1 , a2 , . . . , ar ∈ C \ {0} 
m ≥ r +2
p(n+1)
p(n) ,
p a positive
⇒ ∃Sw m-isometry.

Sw is an m-isometric (non-isometric)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
The structure of m-isometric weighted shift operators (Hilbert space)
Theorem
[B. Abdullah, T. Le, preprint]
Sw is an m-isometry ⇔ |wn |2 =
polynomial of degree m − 1.

a1 , a2 , . . . , ar ∈ C \ {0} 
m ≥ r +2
p(n+1)
p(n) ,
p a positive
⇒ ∃Sw m-isometry.

Sw is an m-isometric (non-isometric) ⇒ Sw is of Hadamard
product of strictly 2 and 3-isometric.
Hadamard product
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
|wn |2 =
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
|wn |2 =
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
An m-isometry is totally determined by the first m − 1
weights.
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
An m-isometry is totally determined by the first m − 1
weights.
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
An m-isometry is totally determined by the first m − 1
weights.
w1 , w2 , . . . , wm−1
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
An m-isometry is totally determined by the first m − 1
weights.
w1 , w2 , . . . , wm−1
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
An m-isometry is totally determined by the first m − 1
weights.
w1 , w2 , . . . , wm−1
Problem
Which are the admissible weights of an m-isometry on `2 (N)?
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
An m-isometry is totally determined by the first m − 1
weights.
w1 , w2 , . . . , wm−1
Problem
Which are the admissible weights of an m-isometry on `2 (N)?
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Properties
z }| {
m−1
∑ (−1)
|wn |2 =
k =0
m−k −1 n... (n − k ) ...(n − m + 1)
k !(m − k − 1)!
z
m−1
}|
|w0 · · · wk |2
{
(n − 1)... (n − 1 − k ) ...(n − m)
|w0 · · · wk |2
∑ (−1)m−k −1
k
!(m
−
k
−
1)!
k =0
An m-isometry is totally determined by the first m − 1
weights.
w1 , w2 , . . . , wm−1
Problem
Which are the admissible weights of an m-isometry on `2 (N)?
WFAV15-José Bonet
Examples of m-isometries
>0,
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
⇔
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
⇔
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
n|w1 |2 − (n − 1)
> 0 ∀n ≥ 1
⇔ |wn |2 =
(n − 1)|w1 |2 − (n − 2)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
n|w1 |2 − (n − 1)
> 0 ∀n ≥ 1 ⇒ |w1 | ≥ 1
⇔ |wn |2 =
(n − 1)|w1 |2 − (n − 2)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
n|w1 |2 − (n − 1)
> 0 ∀n ≥ 1 ⇒ |w1 | ≥ 1
⇔ |wn |2 =
(n − 1)|w1 |2 − (n − 2)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
1
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
n|w1 |2 − (n − 1)
> 0 ∀n ≥ 1 ⇒ |w1 | ≥ 1
⇔ |wn |2 =
(n − 1)|w1 |2 − (n − 2)
√
w1 = 2 ⇒ Sw (Dirichlet shift) is a 2-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
1
Hilbert space
Banach space
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
n|w1 |2 − (n − 1)
> 0 ∀n ≥ 1 ⇒ |w1 | ≥ 1
⇔ |wn |2 =
(n − 1)|w1 |2 − (n − 2)
√
w1 = 2 ⇒ Sw (Dirichlet shift) is a 2-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Theorem
[T. B., A. Martinón, E. Negrín, IEOT,2010]
Sw is a 2-isometry
n|w1 |2 − (n − 1)
> 0 ∀n ≥ 1 ⇒ |w1 | ≥ 1
⇔ |wn |2 =
(n − 1)|w1 |2 − (n − 2)
√
w1 = 2 ⇒ Sw (Dirichlet shift) is a 2-isometry.
Case of an isometry
Case of a 2-isometry
1.4
1.3
w22
1
Hilbert space
Banach space
1.2
w22 = 2 − w12
1
1.1
1
1
1.2
1.4
1.6
1.8
w12
Figure: Graphical representation of the suitable weights to be an isometry or 2-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
wn > 0,
PW ,3 (n) := 12 n(n − 1)w12 w22 − n(n − 2)w12 + 21 (n − 1)(n − 2)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
wn > 0,
PW ,3 (n) := 12 n(n − 1)w12 w22 − n(n − 2)w12 + 21 (n − 1)(n − 2)
Theorem
[T. B., A. Martinón, J. Noda, preprint]
The following statements are equivalent:
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
wn > 0,
PW ,3 (n) := 12 n(n − 1)w12 w22 − n(n − 2)w12 + 21 (n − 1)(n − 2)
Theorem
[T. B., A. Martinón, J. Noda, preprint]
The following statements are equivalent:
1
Sw is a strict 3-isometry with weighted sequence (wn )n≥1 .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
wn > 0,
PW ,3 (n) := 12 n(n − 1)w12 w22 − n(n − 2)w12 + 21 (n − 1)(n − 2)
Theorem
[T. B., A. Martinón, J. Noda, preprint]
The following statements are equivalent:
1
2
Sw is a strict 3-isometry with weighted sequence (wn )n≥1 .
!
2(n − 2) n − 2
2
PW ,3 (n) = Wn and w2 > sup
−
.
n−1
nw12
n≥2
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
wn > 0,
PW ,3 (n) := 12 n(n − 1)w12 w22 − n(n − 2)w12 + 21 (n − 1)(n − 2)
Theorem
[T. B., A. Martinón, J. Noda, preprint]
The following statements are equivalent:
1
2
3
Sw is a strict 3-isometry with weighted sequence (wn )n≥1 .
!
2(n − 2) n − 2
2
PW ,3 (n) = Wn and w2 > sup
−
.
n−1
nw12
n≥2
PW ,3 (n) = Wn and

2(n − 2) n − 2



 n − 1 − nw 2
1
w22 >

1


 2− 2
w1
WFAV15-José Bonet
if
n−2
n
< w12 <
if 1 ≤ w12 .
Examples of m-isometries
n−1
n+1
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
3-isometries
2
1
0 1/3
1
2
Figure: Region of the admissible weights w1 and w2 of a strict 3-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
wn > 0,
Hilbert space
Banach space
PW ,4 (n) := ∑3k =0 (−1)3−k
4-isometries
WFAV15-José Bonet
Examples of m-isometries
n
k
n−k −1
3−k
Wk
Introduction
Properties
Unilateral weighted shift operator
wn > 0,
Hilbert space
Banach space
PW ,4 (n) := ∑3k =0 (−1)3−k
4-isometries
Theorem
The following statements are equivalent:
WFAV15-José Bonet
Examples of m-isometries
n
k
n−k −1
3−k
Wk
Introduction
Properties
Unilateral weighted shift operator
wn > 0,
Hilbert space
Banach space
PW ,4 (n) := ∑3k =0 (−1)3−k
n
k
n−k −1
3−k
Wk
4-isometries
Theorem
The following statements are equivalent:
1
Sw is a strict 4-isometry with weighted sequence (wn )n≥1 .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
wn > 0,
Hilbert space
Banach space
PW ,4 (n) := ∑3k =0 (−1)3−k
n
k
n−k −1
3−k
Wk
4-isometries
Theorem
The following statements are equivalent:
1
Sw is a strict 4-isometry with weighted sequence (wn )n≥1 .
2
PW ,4 (n) = Wn and
!
3(n
−
3)
3(n
−
3)
n
−
3
.
w32 > sup
−
+
n−2
nw12 w22 (n − 1)w22
n≥3
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
4-isometries
w32 > sup
n≥3
n−3
3(n − 3)
3(n − 3)
−
+
2
2
2
n−2
nw1 w2 (n − 1)w2
WFAV15-José Bonet
Examples of m-isometries
!
.
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
4-isometries
w32 > sup
n≥3
n−3
3(n − 3)
3(n − 3)
−
+
2
2
2
n−2
nw1 w2 (n − 1)w2
WFAV15-José Bonet
Examples of m-isometries
!
.
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
4-isometries
w32 > sup
n≥3
n−3
3(n − 3)
3(n − 3)
−
+
2
2
2
n−2
nw1 w2 (n − 1)w2
Q(n, w1 , w2 ) :=
!
.
3(n − 3)
3(n − 3)
n−3
−
+
.
2
2
2
n−2
nw1 w2 (n − 1)w2
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
4-isometries
w32 > sup
n≥3
n−3
3(n − 3)
3(n − 3)
−
+
2
2
2
n−2
nw1 w2 (n − 1)w2
Q(n, w1 , w2 ) :=
!
.
3(n − 3)
3(n − 3)
n−3
−
+
.
2
2
2
n−2
nw1 w2 (n − 1)w2
e 3 : = {(w1 , w2 ) : Q(3, w1 , w2 ) ≥ Q(n, w1 , w2 ) for all n ≥ 3}
R
e 4 : = {(w1 , w2 ) : Q(4, w1 , w2 ) ≥ Q(n, w1 , w2 ) for all n ≥ 4}
R
···
e 3 , R4 := R
e4 \ R
e3, · · · · · ·
R3 := R
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
1
1
L3(w1 )
l3 (3, w1 )
l3 (4, w1 )
0.8
L3(w1 )
l3 (3, w1 )
0.8
0.6
w22
w22
0.6
0.4
0.4
0.2
0.2
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4
0
R3
0
0.2 0.4 0.6 0.8
w12
w12
Figure: Graphical representation of the region R3 .
WFAV15-José Bonet
Examples of m-isometries
1
1.2 1.4
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
1
1
L3(w1 )
l3 (3, w1 )
l3 (4, w1 )
0.8
L3(w1 )
l3 (3, w1 )
0.8
0.6
w22
w22
0.6
0.4
0.4
0.2
0.2
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4
0
R3
0
0.2 0.4 0.6 0.8
w12
1
1.2 1.4
w12
Figure: Graphical representation of the region R3 .
(
R3 := (w1 , w2 ) :
w22
1
2
4
< 1−
if < w12 ≤ and w22 <
9
3w12 9
WFAV15-José Bonet
Examples of m-isometries
9w12 −2
18w12
)
if
w12
>
4
9
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
1
1
L3(w1 )
l3 (3, w1 )
l3 (4, w1 )
0.8
L3(w1 )
l3 (3, w1 )
0.8
0.6
w22
w22
0.6
0.4
0.4
0.2
0.2
0
0
0.2 0.4 0.6 0.8
1
1.2 1.4
0
R3
0
0.2 0.4 0.6 0.8
w12
1
1.2 1.4
w12
Figure: Graphical representation of the region R3 .
(
R3 := (w1 , w2 ) :
w22
1
2
4
< 1−
if < w12 ≤ and w22 <
9
3w12 9
(w1 , w2 ) ∈ R3 ⇒ w3 > 0 = supn≥3 Q(n, w1 , w2 ) .
WFAV15-José Bonet
Examples of m-isometries
9w12 −2
18w12
)
if
w12
>
4
9
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
R4
w22
w22
f4
R
w12
R3
w12
f4 , R3 and R4 .
Figure: Graphical representation of the regions R
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
R4
w22
w22
f4
R
R3
w12
w12
f4 , R3 and R4 .
Figure: Graphical representation of the regions R




R4 = (w1 , w2 ) :



9w12 −2
18w12
9w12 −2
18w12
< w22 <
8
6
< w2 <
32w12 −9
36w12
WFAV15-José Bonet
− 2w1 2 if
1
7
15
< w12 ≤
if w12 >
Examples of m-isometries
9
16
9
16







.
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
R4
w22
w22
f4
R
R3
w12
w12
f4 , R3 and R4 .
Figure: Graphical representation of the regions R




R4 = (w1 , w2 ) :



(w1 , w2 ) ∈ R4 ⇒ w32 >
9w12 −2
18w12
9w12 −2
18w12
1
4w12 w22
< w22 <
8
6
− 2w1 2 if
< w2 <
32w12 −9
36w12
1
7
15
< w12 ≤
if w12 >
9
16
9
16
− w12 + 32 = sup Q(n, w1 , w2 ) .
WFAV15-José Bonet
2
n≥4
Examples of m-isometries







.
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
R5
f5
R
w22
w2p
R4
w12
R3
w1p
Figure: Graphical representation of the regions Rf5 , R3 , R4 and R5 .
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Bothelho’s result
Hilbert space
Banach space
[Acta Sci. Math. 2010]
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
Sw is a (2, 2)-isometry ⇒ p = 2.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
Sw is a (2, 2)-isometry ⇒ p = 2.
Theorem
Let Sw be a non isometric on `p (N).
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
Sw is a (2, 2)-isometry ⇒ p = 2.
Theorem
Let Sw be a non isometric on `p (N).
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
Sw is a (2, 2)-isometry ⇒ p = 2.
Theorem
Let Sw be a non isometric on `p (N).
Sw is a (2, q)-isometry ⇒ p = q.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
Sw is a (2, 2)-isometry ⇒ p = 2.
Theorem
Let Sw be a non isometric on `p (N).
Sw is a (2, q)-isometry ⇒ p = q.
Example
Sw on `p (N) with wn :=
n+2
n+1
1/p
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
Sw is a (2, 2)-isometry ⇒ p = 2.
Theorem
Let Sw be a non isometric on `p (N).
Sw is a (2, q)-isometry ⇒ p = q.
Example
Sw on `p (N) with wn :=
n+2
n+1
1/p
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Bothelho’s result
[Acta Sci. Math. 2010]
Sw is a non isometric on `p (N)
Sw is a (2, 2)-isometry ⇒ p = 2.
Theorem
Let Sw be a non isometric on `p (N).
Sw is a (2, q)-isometry ⇒ p = q.
Example
Sw on `p (N) with wn :=
n+2
n+1
1/p
Sw is a (3, 2p)-isometry.
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
"It doesn’t work canonical basis "
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
"It doesn’t work canonical basis "
Theorem
[Gu, 2015]
Sw strict (m, q)-isometric on `p (N). Then
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
"It doesn’t work canonical basis "
Theorem
[Gu, 2015]
Sw strict (m, q)-isometric on `p (N). Then
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
"It doesn’t work canonical basis "
Theorem
[Gu, 2015]
Sw strict (m, q)-isometric on `p (N). Then
1
Sw strict (m0 , p)-isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
"It doesn’t work canonical basis "
Theorem
[Gu, 2015]
Sw strict (m, q)-isometric on `p (N). Then
1
Sw strict (m0 , p)-isometry
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
(m, q)-isometries on `p (N)
Sw : `p (N) −→ `p (N) an (m, p)-isometry
“canonical basis"
Sw : `p (N) −→ `p (N) an (m, q)-isometry
"It doesn’t work canonical basis "
Theorem
[Gu, 2015]
Sw strict (m, q)-isometric on `p (N). Then
1
Sw strict (m0 , p)-isometry
2
(m, q) = (k (m0 − 1) + 1, kp), for some positive integer k
WFAV15-José Bonet
Examples of m-isometries
Introduction
Properties
Unilateral weighted shift operator
Hilbert space
Banach space
Thank
you!
WFAV15-José Bonet
Examples of m-isometries