# Perturbation ofm-Isometries by Nilpotent Operators

Authors: Teresa Bermúdez; Antonio Martinón; Vladimir Müller; Juan Agustín Noda
Journal: Abstract and Applied Analysis
 (2014)

---

## Abstract

We prove that ifT is an m-isometry on a Hilbert space and Q an n-nilpotent operator commuting with T, then T
+
Q is a 2
n
+
m
-
2-isometry. Moreover, we show that a similar result for m
,
q-isometries on Banach spaces is not true.


---

## Body

## 1. Introduction

The notion ofm-isometric operators on Hilbert spaces was introduced by Agler [1]. See also [2–5]. Recently Sid Ahmed [6] has defined m-isometries on Banach spaces, Bayart [7] introduced (
m
,
q
)-isometries on Banach spaces, and (
m
,
q
)-isometries on metric spaces were considered in [8]. Moreover, Hoffman et al. [9] have studied the role of the second parameter q. Recall the main definitions.A mapT
:
E
→
E (m
≥
1 integer and q
>
0 real), defined on a metric space E with distance d, is called an (
m
,
q
)
-isometry if, for all x
,
y
∈
E,
(1)
∑
k
=
0
m
-
1
m
-
k
m
k
d
T
k
x
,
T
k
y
q
=
0
.
We say that T is astrict
(
m
,
q
)
-isometry if either m
=
1 or T is an (
m
,
q
)-isometry with m
>
1 but is not an (
m
-
1
,
q
)-isometry. Note that (
1
,
q
)-isometries are isometries.The above notion of an(
m
,
q
)-isometry can be adapted to Banach spaces in the following way: a bounded linear operator T
:
X
→
X, where X is a Banach space with norm ·, is an (
m
,
q
)-isometry if and only if, for all x
∈
X,
(2)
∑
k
=
0
m
-
1
m
-
k
m
k
T
k
x
q
=
0
.In the setting of Hilbert spaces, the caseq
=
2 can be expressed in a special way. Agler [1] gives the following definition: a linear bounded operator T
:
H
→
H acting on a Hilbert space H is an (
m
,
2
)-isometry if
(3)
∑
k
=
0
m
-
1
m
-
k
m
k
T
*
k
T
k
=
0
.
(
m
,
2
)-isometries on Hilbert spaces will be called for short m-isometries.The paper is organized as follows. In the next section we collect some results about applications of arithmetic progressions tom-isometric operators.In Section3 we prove that, in the setting of Hilbert spaces, if T is an m-isometry, Q is an n-nilpotent operator, and they commute, and then T
+
Q is a (
2
n
+
m
-
2
)-isometry. This is a partial generalization of the following result obtained in [10, Theorem 2.2]: if T is an isometry and Q is a nilpotent operator of order n commuting with T, then T
+
Q is a strict (
2
n
-
1
)-isometry.In the last section we give some examples of operators on Banach spaces which are of the form identity plus nilpotent, but they are not(
m
,
q
)-isometries, for any positive integer m and any positive real number q.Notation. Throughout this paper H denotes a Hilbert space and B
(
H
) the algebra of all linear bounded operators on H. Given T
∈
B
(
H
), T
* denotes its adjoint. Moreover, m
≥
1 is an integer and q
>
0 a real number.


## 2. Preliminaries: Arithmetic Progressions and(
m
,
q
)-Isometries

In this section we give some basic properties ofm-isometries. We need some preliminaries about arithmetic progressions and their applications to m-isometries. In [11], some results about this topic are recollected.LetG be a commutative group and denote its operation by +. Given a sequence a
=
a
n
n
≥
0 in G, the difference sequence D
a
=
(
D
a
)
n
≥
0 is defined by (
D
a
)
n
≔
a
n
+
1
-
a
n. The powers of D are defined recursively by D
0
a
:
=
a, D
k
+
1
a
=
D
(
D
k
a
). It is easy to show that
(4)
D
k
a
n
=
∑
i
=
0
k
-
1
k
-
i
k
i
a
i
+
n
,
for all k
≥
0 and n
≥
0 integers.A sequencea in a group G is called anarithmetic progression of order h
=
0,1
,
2
…, if D
h
+
1
a
=
0. Equivalently,
(5)
∑
i
=
0
h
+
1
-
1
h
+
1
-
i
h
+
1
i
a
i
+
j
=
0
for j
=
0,1
,
2
,
…. It is well known that the sequence a in G is an arithmetic progression of order h if and only if there exists a polynomial p
(
n
) in n, with coefficients in G and of degree less than or equal to h, such that p
(
n
)
=
a
n, for every n
=
0,1
,
2
…; that is, there are γ
h
,
γ
h
-
1
,
…
,
γ
1
,
γ
0
∈
G, which depend only on a, such that, for every n
=
0,1
,
2
,
…,
(6)
a
n
=
p
n
=
∑
i
=
0
h
γ
i
n
i
.
We say that the sequence a is anarithmetic progression of strict order
h
=
0,1
,
2
…, if h
=
0 or if it is of order h
>
0 but is not of order h
-
1; that is, the polynomial p of (6) has degree h.Moreover, a sequencea in a group G is an arithmetic progression of order h if and only if, for all n
≥
0,
(7)
a
n
=
∑
k
=
0
h
-
1
h
-
k
n
n
-
1
⋯
n
-
k
︷
⋯
n
-
h
k
!
h
-
k
!
a
k
;
that is,
(8)
a
n
=
∑
k
=
0
h
-
1
h
-
k
n
k
n
-
k
-
1
h
-
k
a
k
.Now we give a basic result aboutm-isometries.Theorem 1.
LetH be a Hilbert space. An operator T
∈
B
(
H
) is a strict m-isometry if and only if there are A
m
-
1
≠
0
,
A
m
-
2
,
…
,
A
1
,
A
0 in B
(
H
), which depend only on T, such that, for every n
=
0,1
,
2
…,
(9)
T
*
n
T
n
=
∑
i
=
0
m
-
1
A
i
n
i
;
that is, the sequence T
*
n
T
n
n
≥
0 is an arithmetic progression of strict order m
-
1 in B
(
H
).Proof.
IfT
∈
B
(
H
) is a strict m-isometry, then it satisfies (3). Hence, for each integer i
≥
0,
(10)
∑
k
=
0
m
-
1
m
-
k
m
k
T
*
i
T
*
k
T
k
T
i
=
∑
k
=
0
m
-
1
m
-
k
m
k
T
*
k
+
i
T
k
+
i
=
0
,
but
(11)
∑
k
=
0
m
-
1
-
1
m
-
1
-
k
m
-
1
k
T
*
k
T
k
≠
0
.
By (5), the operator sequence (
T
*
n
T
n
)
n
≥
0 is an arithmetic progression of strict order m
-
1. Therefore, from (6) we obtain that there is a polynomial p
(
n
) of degree m
-
1 in n, with coefficients in B
(
H
) satisfying p
(
n
)
=
T
*
n
T
n; that is, there are operators A
m
-
1
≠
0
,
A
m
-
2
,
…
,
A
1
,
A
0 in B
(
H
), such that, for every n
=
0,1
,
2
…,
(12)
T
*
n
T
n
=
A
m
-
1
n
m
-
1
+
A
m
-
2
n
m
-
2
+
⋯
+
A
1
n
+
A
0
.
Conversely, if(
T
*
n
T
n
)
n
≥
0 is an arithmetic progression of strict order m
-
1, then (10) and (11) hold. Taking i
=
0 we obtain (3), so T is a strict m-isometry.Now we recall an elementary property of(
m
,
q
)-isometries on metric spaces which will be used in the next sections.Proposition 2 (see [8, Proposition 3.11]).
LetE be a metric space and let T
:
E
→
E be an (
m
,
q
)-isometry. If T is an invertible strict (
m
,
q
)-isometry, then m is odd.


## 3.m-Isometry Plus n-Nilpotent

Recall that an operatorQ
∈
B
(
H
) isnilpotent of order
n (n
≥
1 integer), or n
-nilpotent, if Q
n
=
0 and Q
n
-
1
≠
0.In any finite dimensional Hilbert spaceH, strict m-isometries can be characterized in a very simple way: a linear operator T
∈
B
(
H
) is a strict m-isometry if and only if m is odd and T
=
A
+
Q, where A and Q are commuting operators on H and A is unitary and Q a nilpotent operator of order m
+
1
/
2 ([12, page 134] and [10, Theorem 2.7]).It was proved in [10, Theorem 2.2] that if A
∈
B
(
H
) is an isometry and Q
∈
B
(
H
) is an n-nilpotent operator such that T
Q
=
Q
T, then T
+
Q is a strict (
2
n
-
1
)-isometry. Now we obtain a partial generalization of this result: if T
∈
B
(
H
) is an m-isometry and Q
∈
B
(
H
) is an n-nilpotent operator commuting with T, then T
+
Q is a (
2
n
+
m
-
2
)-isometry. However, T
+
Q is not necessarily a strict (
2
n
+
m
-
2
)-isometry. For example, if T is an isometry and Q any n-nilpotent operator (n
>
1) such that T
Q
=
Q
T, then T
=
T
+
Q
+
(
-
Q
) is not a strict (
4
n
-
3
)-isometry.Theorem 3.
LetH be a Hilbert space. Let T
∈
B
(
H
) be an m-isometry and Q
∈
B
(
H
) an n-nilpotent operator (n
≥
1 integer) such that T
Q
=
Q
T. Then T
+
Q is (
2
n
+
m
-
2
)-isometry.Proof.
Fix an integerk
≥
0 and denote h
≔
min
⁡
{
k
,
n
-
1
}. Then we have
(13)
T
+
Q
*
k
T
+
Q
k
=
∑
i
=
0
h
k
i
Q
*
i
T
*
k
-
i
∑
j
=
0
h
k
j
T
k
-
j
Q
j
=
∑
i
,
j
=
0
h
k
i
k
j
Q
*
i
T
*
k
-
i
T
k
-
j
Q
j
=
∑
0
≤
i
<
j
≤
h
k
i
k
j
Q
*
i
T
*
j
-
i
T
*
k
-
j
T
k
-
j
Q
j
WW
+
∑
0
≤
j
≤
i
≤
h
k
i
k
j
Q
*
i
T
*
k
-
i
T
k
-
i
T
i
-
j
Q
j
.
From (9) we obtain, for certain A
m
-
1
,
…
,
A
0
∈
B
(
H
),
(14)
T
+
Q
*
k
T
+
Q
k
=
∑
0
≤
i
<
j
≤
h
k
i
k
j
Q
*
i
T
*
j
-
i
∑
r
=
0
m
-
1
A
r
k
-
j
r
Q
j
+
∑
0
≤
j
≤
i
≤
h
k
i
k
j
Q
*
i
∑
r
=
0
m
-
1
A
r
k
-
i
r
T
i
-
j
Q
j
.
Write
(15)
B
r
,
i
,
j
≔
Q
*
i
T
*
j
-
i
A
r
Q
j
∈
B
H
,
C
r
,
i
,
j
≔
Q
*
i
A
r
T
i
-
j
Q
j
∈
B
H
,
q
r
,
i
,
j
≔
k
i
k
j
k
-
j
r
,
p
r
,
i
,
j
≔
k
i
k
j
k
-
i
r
.
Note that k
i and k
j are real polynomials in k of degree less than or equal to h
≤
n
-
1, and (
k
-
j
)
r and (
k
-
i
)
r have degree r
≤
m
-
1. Hence q
r
,
i
,
j and p
r
,
i
,
j are real polynomials of degree less than or equal to m
-
1
+
2
(
n
-
1
)
=
2
n
+
m
-
3. Consequently we can write
(16)
T
+
Q
*
k
T
+
Q
k
=
∑
r
=
0
m
-
1
∑
0
≤
i
<
j
≤
h
B
r
,
i
,
j
q
r
,
i
,
j
+
∑
r
=
0
m
-
1
∑
0
≤
j
≤
i
≤
h
C
r
,
i
,
j
p
r
,
i
,
j
,
which is a polynomial in k, of degree less than or equal to 2
n
+
m
-
3 with coefficients in B
(
H
). By Theorem 1, the operator T
+
Q is an (
2
n
+
m
-
2
)-isometry.For isometries it is possible to say more [10, Theorem 2.2].Theorem 4.
LetH be a Hilbert space. Let T
∈
B
(
H
) be an isometry and let Q
∈
B
(
H
) be an n-nilpotent operator (n
≥
1 integer) such that T
Q
=
Q
T. Then T
+
Q is a strict (
2
n
-
1
)-isometry.Proof.
By Theorem3 we obtain that T
+
Q is a (
2
n
-
1
)-isometry; that is, (
(
T
+
Q
)
*
k
(
T
+
Q
)
k
)
k
≥
0 is an arithmetic progression of order less than or equal to 2
n
-
2. Now we prove that it is an arithmetic progression of strict order 2
n
-
2, or equivalently the polynomial (9) has degree 2
n
-
2. Note that as T is an isometry we have T
*
k
T
k
=
I, for every positive integer k.
As in the proof of Theorem3, for any integer k
≥
0, we have that
(17)
T
+
Q
*
k
T
+
Q
k
=
∑
i
,
j
=
0
h
k
i
k
j
Q
*
i
T
*
k
-
i
T
k
-
j
Q
j
=
∑
0
≤
i
<
j
≤
h
k
i
k
j
Q
*
i
T
*
j
-
i
Q
j
+
∑
0
≤
j
≤
i
≤
h
k
i
k
j
Q
*
i
T
i
-
j
Q
j
,
where h
≔
min
⁡
{
k
,
n
-
1
}.
The coefficient ofk
2
n
-
2 in the polynomial (
T
+
Q
)
*
k
(
T
+
Q
)
k is
(18)
1
n
-
1
!
2
Q
*
n
-
1
Q
n
-
1
,
which is null if and only if Q
*
n
-
1
Q
n
-
1
=
0, that is, if and only if Q
n
-
1
=
0. Therefore, if Q is nilpotent of order n, then (
T
+
Q
)
*
k
(
T
+
Q
)
k can be written as a polynomial in k, of degree 2
n
-
2 and coefficients in B
(
H
). Consequently T
+
Q is a strict (
2
n
-
1
)-isometry.Now we obtain the following corollary of Theorem4.Corollary 5.
LetH be a Hilbert space. Let Q
∈
B
(
H
) be an n-nilpotent operator (n
≥
1 integer). Then I
+
Q is a strict (
2
n
-
1
)-isometry.Recall that an operatorT
∈
B
(
H
) is N
-supercyclic (N
≥
1 integer) if there exists a subspace F
⊂
H of dimension N such that its orbit {
T
n
x
:
n
≥
0
,
x
∈
F
} is dense in H. Moreover, T is calledsupercyclic if it is 1-supercyclic. See [13, 14].Bayart [7, Theorem 3.3] proved that on an infinite dimensional Banach space an (
m
,
q
)-isometry is never N-supercyclic, for any N
≥
1. In the setting of Banach spaces, Yarmahmoodi et al. [15, Theorem 2.2] showed that any sum of an isometry and a commuting nilpotent operator is never supercyclic. For Hilbert space operators we extend the result [15, Theorem 2.2] to m-isometries plus commuting nilpotent operators.Corollary 6.
LetH be an infinite dimensional Hilbert space. If T
∈
B
(
H
) is an m-isometry that commutes with a nilpotent operator Q, then T
+
Q is never N-supercyclic for any N.


## 4. Some Examples in the Setting of Banach Spaces

Theorem4 is not true for finite-dimensional Banach spaces even for m
=
1.Denotel
p
d
≔
(
C
d
,
·
p
).Example 1.
LetQ
:
C
2
→
C
2 be defined by Q
(
x
,
y
)
≔
(
y
,
0
); hence Q is a 2-nilpotent operator. The following assertions hold: (1)
I
+
Q is not a (
3
,
p
)-isometry on l
p
2 for any 1
≤
p
<
∞ and p
≠
2;
(2)
I
+
Q is not a (
3
,
p
)-isometry on l
∞
2 for any p
>
0;
(3)
I
+
Q is a strict (
2
k
+
1,2
k
)-isometry on (
C
2
,
·
2
k
) for any k
=
1,2
,
3
,
….Proof.
For(
x
,
y
)
∈
C
2 we have
(19)
I
+
Q
x
,
y
=
x
+
y
,
y
,
I
+
Q
2
x
,
y
=
x
+
2
y
,
y
,
I
+
Q
3
x
,
y
=
x
+
3
y
,
y
.
Write
(20)
A
x
,
y
;
p
,
q
≔
I
+
Q
3
x
,
y
p
q
-
3
I
+
Q
2
x
,
y
p
q
+
3
I
+
Q
x
,
y
p
q
-
x
,
y
p
q
.
(1) We consider two cases:1
<
p
<
∞ and p
=
1.(a)
Case1
<
p
<
∞. For x
=
0, y
=
1, and q
=
p, we have
(21)
A
0,1
;
p
,
p
=
3
p
+
1
-
3
·
2
p
-
3
+
6
-
1
=
3
p
-
3
·
2
p
+
3
.
SoA
(
0,1
;
p
,
p
)
=
0 if and only if 3
p
-
1
+
1
=
2
p, which is true only when p
=
2 or p
=
1 since the function f
(
t
)
=
3
t
-
1
+
1
-
2
t is null only for t
=
1 and t
=
2.
ConsequentlyI
+
Q is not a (
3
,
p
)-isometry on l
p
2 if p
≠
2 and 1
<
p
<
∞.
(b)
Casep
=
1. In order to prove that I
+
Q is not a (
3,1
)-isometry on l
1
2, we take the vector (
1
,
-
1
) and obtain that
(22)
A
1
,
-
1
;
1,1
=
I
+
Q
3
1
,
-
1
1
-
3
I
+
Q
2
1
,
-
1
1
+
3
I
+
Q
1
,
-
1
1
-
1
,
-
1
1
≠
0
.
(2) For(
x
,
y
)
∈
C
2 we have
(23)
A
x
,
y
;
∞
,
p
≔
I
+
Q
3
x
,
y
∞
p
-
3
I
+
Q
2
x
,
y
∞
p
+
3
I
+
Q
x
,
y
∞
p
-
x
,
y
∞
p
=

max
⁡
x
+
3
y
,
y
p

-

3
max
⁡
x
+
2
y
,
y
p
+
3
max
⁡
x
+
y
,
y
p
-
max
⁡
x
,
y
p
.
In particular, for x
:
=
1 and y
:
=
-
1,
(24)
A
1
,
-
1
;
∞
,
p
=
2
p
-
1
≠
0
.
Therefore I
+
Q is not a (
3
,
p
)-isometry on l
∞
2 for any p
>
0.
(3) First we prove by induction onk that I
+
Q is a (
2
k
+
1,2
k
)-isometry on l
2
k
2 for any k
=
1
,
2
,
3
…. Note that, for (
x
,
y
)
∈
C
2,
(25)
I
+
Q
s
x
,
y
=
x
+
s
y
,
y
,
s
=
0,1
,
2
…
.
By Corollary 5, the operator I
+
Q is a strict (
3,2
)-isometry on l
2
2. Hence I
+
Q is a strict (
2
k
+
1,2
k
)-isometry on l
2
2 for all k
=
1
,
2
,
3
… [9, Corollary 4.6]. Thus for (
x
,
y
)
∈
C
2,
(26)
∑
s
=
0
2
k
+
1
-
1
2
k
+
1
-
s
2
k
+
1
s
x
+
s
y
2
+
y
2
k
=
0
.
Suppose that I
+
Q is a (
2
i
-
1,2
i
-
2
)-isometry on l
2
i
-
2
2 for every i
=
2,3
,
…
,
k. Hence I
+
Q is also a (
2
k
+
1,2
i
-
2
)-isometry on l
2
i
-
2
2. Then, for (
x
,
y
)
∈
C
2,
(27)
∑
s
=
0
2
k
+
1
-
1
2
k
+
1
-
s
2
k
+
1
s
x
+
s
y
2
i
-
2
+
y
2
i
-
2
=
0
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
m
i
m
m
2
≤
i
≤
k
.
Therefore,
(28)
∑
s
=
0
2
k
+
1
-
1
2
k
+
1
-
s
2
k
+
1
s
x
+
s
y
2
i
-
2
=
0
,
m
m
m
m
m
m
m
m
m
m
m
m
m
m
2
≤
i
≤
k
.
Taking into account equality (28) we can write (26) in the following way:
(29)
0
=
∑
s
=
0
2
k
+
1
-
1
2
k
+
1
-
s
2
k
+
1
s
∑
i
=
0
k
k
i
x
+
s
y
2
i
y
2
k
-
i
=
∑
i
=
0
k
-
1
k
i
y
2
k
-
i
∑
s
=
0
2
k
+
1
-
1
2
k
+
1
-
s
2
k
+
1
s
x
+
s
y
2
i
+
∑
s
=
0
2
k
+
1
-
1
2
k
+
1
-
s
2
k
+
1
s
x
+
s
y
2
k
=
∑
s
=
0
2
k
+
1
-
1
2
k
+
1
-
s
2
k
+
1
s
x
+
s
y
2
k
+
y
2
k
.
Therefore I
+
Q is a (
2
k
+
1,2
k
)-isometry on l
2
k
2.
Now we prove thatI
+
Q is a strict (
2
k
+
1,2
k
)-isometry on l
2
k
2. Suppose on the contrary that I
+
Q is a (
2
k
,
2
k
)-isometry on l
2
k
2. Then,
(30)
∑
s
=
0
2
k
-
1
-
1
2
k
-
1
-
s
2
k
-
1
s
x
+
s
y
2
k
+
y
2
k
=
0
for all (
x
,
y
)
∈
C
2. So
(31)
∑
s
=
0
2
k
-
1
-
1
2
k
-
1
-
s
2
k
-
1
s
x
+
s
y
2
k
=
0
for all (
x
,
y
)
∈
C
2. In particular, for y
=
1 and x
=
0,1
,
2
,
…, we have
(32)
∑
s
=
0
2
k
-
1
-
1
2
k
-
1
-
s
2
k
-
1
s
x
+
s
2
k
=
0
.
So (
s
2
k
)
s
=
0
∞ is an arithmetic progression of order 2
k
-
2, which is a contradiction with (6).Remark 7.
Notice that, in any Hilbert space of dimensionn, there are strict m-isometries only for any m
≤
2
n
-
1. However, as the above example shows, there are strict (
2
k
+
1,2
k
)-isometries for any integer k in a Banach space of dimension 2.The following example gives an operator of the formI
+
Q with Q a nilpotent operator such that I
+
Q is not an (
m
,
q
)-isometry for any integer m and any q
>
0.Example 2.
LetX be the Banach space of all real continuous functions f on [
0,1
] such that f
(
1
)
=
0 endowed with the supremun norm. Define Q
:
X
→
X by
(33)
Q
f
t
≔
f
t
+
1
2
,
if
0
≤
t
≤
1
2
,
0
,
if
1
2
<
t
≤
1
.
Then Q
∈
B
(
X
) is 2-nilpotent operator. Moreover, I
+
Q is not an (
m
,
q
)-isometry for any m
=
1,2
,
3
,
… and any q
>
0.Proof.
It is clear thatI
+
Q is not an isometry since the function f
∈
X given by
(34)
f
t
≔
1
,
if
0
≤
t
≤
1
2
,
-
2
t
+
2
,
if
1
2
<
t
≤
1
satisfies f
=
1 and (
I
+
Q
)
f
=
2.
Form
=
2,3
,
4
,
… consider the function f
m
∈
X defined by
(35)
f
m
t
≔
-
4
t
+
1
,
if
0
≤
t
≤
1
4
,
0
,
if
1
4
<
t
≤
1
2
,
-
4
m
-
1
t
+
2
m
-
1
,
if
1
2
<
t
≤
3
4
,
4
m
-
1
t
-
4
m
-
1
,
if
3
4
<
t
≤
1
.
Note thatf
m
(
3
/
4
)
=
1
/
1
-
m
=
min
⁡
0
≤
t
≤
1
f
m
(
t
) (Figure 1).Figure 1
Graphics of functionsf
3, f
5, and f
7.Fixq
>
0. For k
=
0,1
,
2
,
…, we have
(36)
I
+
Q
k
f
m
q
=
I
+
k
Q
f
m
q
=
sup
⁡
0
≤
t
≤
1
f
m
t
+
k
Q
f
m
t
q
.If0
≤
k
≤
m
-
1, then
(37)
I
+
Q
k
f
m
q
=
f
m
0
+
k
f
m
1
2
q
=
1
,
since k
1
/
m
-
1
≤
1. But as m
1
/
m
-
1
>
1 we obtain
(38)
I
+
Q
m
f
m
q
=
f
m
1
4
+
m
f
m
3
4
q
=
m
m
-
1
q
>
1
.
Consequently,
(39)
∑
k
=
0
m
-
1
m
-
k
m
k
I
+
Q
k
f
m
q
=
∑
k
=
0
m
-
1
-
1
m
-
l
m
k
+
I
+
Q
m
f
m
q
=
-
1
+
m
m
-
1
q
≠
0
.
Therefore I
+
Q is not an (
m
,
q
)-isometry for any m
=
1,2
,
3
… and any q
>
0.



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*Source: 745479-2014-09-09.xml*