#### Abstract

We prove that if is an -isometry on a Hilbert space and an -nilpotent operator commuting with , then is a -isometry. Moreover, we show that a similar result for -isometries on Banach spaces is not true.

#### 1. Introduction

The notion of -isometric operators on Hilbert spaces was introduced by Agler [1]. See also [25]. Recently Sid Ahmed [6] has defined -isometries on Banach spaces, Bayart [7] introduced -isometries on Banach spaces, and -isometries on metric spaces were considered in [8]. Moreover, Hoffman et al. [9] have studied the role of the second parameter . Recall the main definitions.

A map ( integer and real), defined on a metric space with distance , is called an -isometry if, for all , We say that is a strict -isometry if either or is an -isometry with but is not an -isometry. Note that -isometries are isometries.

The above notion of an -isometry can be adapted to Banach spaces in the following way: a bounded linear operator , where is a Banach space with norm , is an -isometry if and only if, for all ,

In the setting of Hilbert spaces, the case can be expressed in a special way. Agler [1] gives the following definition: a linear bounded operator acting on a Hilbert space is an -isometry if -isometries on Hilbert spaces will be called for short -isometries.

The paper is organized as follows. In the next section we collect some results about applications of arithmetic progressions to -isometric operators.

In Section 3 we prove that, in the setting of Hilbert spaces, if is an -isometry, is an -nilpotent operator, and they commute, and then is a -isometry. This is a partial generalization of the following result obtained in [10, Theorem 2.2]: if is an isometry and is a nilpotent operator of order commuting with , then is a strict -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 -isometries, for any positive integer and any positive real number .

Notation. Throughout this paper denotes a Hilbert space and the algebra of all linear bounded operators on . Given , denotes its adjoint. Moreover, is an integer and a real number.

#### 2. Preliminaries: Arithmetic Progressions and -Isometries

In this section we give some basic properties of -isometries. We need some preliminaries about arithmetic progressions and their applications to -isometries. In [11], some results about this topic are recollected.

Let be a commutative group and denote its operation by +. Given a sequence in , the difference sequence is defined by . The powers of are defined recursively by , . It is easy to show that for all and integers.

A sequence in a group is called an arithmetic progression of order , if . Equivalently, for . It is well known that the sequence in is an arithmetic progression of order if and only if there exists a polynomial in , with coefficients in and of degree less than or equal to , such that , for every ; that is, there are , which depend only on , such that, for every , We say that the sequence is an arithmetic progression of strict order , if or if it is of order but is not of order ; that is, the polynomial of (6) has degree .

Moreover, a sequence in a group is an arithmetic progression of order if and only if, for all , that is,

Now we give a basic result about -isometries.

Theorem 1. Let be a Hilbert space. An operator is a strict -isometry if and only if there are in , which depend only on , such that, for every , that is, the sequence is an arithmetic progression of strict order in .

Proof. If is a strict -isometry, then it satisfies (3). Hence, for each integer , but By (5), the operator sequence is an arithmetic progression of strict order . Therefore, from (6) we obtain that there is a polynomial of degree in , with coefficients in satisfying ; that is, there are operators in , such that, for every ,
Conversely, if is an arithmetic progression of strict order , then (10) and (11) hold. Taking we obtain (3), so is a strict -isometry.

Now we recall an elementary property of -isometries on metric spaces which will be used in the next sections.

Proposition 2 (see [8, Proposition 3.11]). Let be a metric space and let be an -isometry. If is an invertible strict -isometry, then is odd.

#### 3. -Isometry Plus -Nilpotent

Recall that an operator is nilpotent of order ( integer), or -nilpotent, if and .

In any finite dimensional Hilbert space , strict -isometries can be characterized in a very simple way: a linear operator is a strict -isometry if and only if is odd and , where and are commuting operators on and is unitary and a nilpotent operator of order ([12, page 134] and [10, Theorem 2.7]).

It was proved in [10, Theorem 2.2] that if is an isometry and is an -nilpotent operator such that , then is a strict -isometry. Now we obtain a partial generalization of this result: if is an -isometry and is an -nilpotent operator commuting with , then is a -isometry. However, is not necessarily a strict -isometry. For example, if is an isometry and any -nilpotent operator () such that , then is not a strict -isometry.

Theorem 3. Let be a Hilbert space. Let be an -isometry and an -nilpotent operator ( integer) such that . Then is -isometry.

Proof. Fix an integer and denote . Then we have From (9) we obtain, for certain , Write Note that and are real polynomials in of degree less than or equal to , and and have degree . Hence and are real polynomials of degree less than or equal to . Consequently we can write which is a polynomial in , of degree less than or equal to with coefficients in . By Theorem 1, the operator is an -isometry.

For isometries it is possible to say more [10, Theorem 2.2].

Theorem 4. Let be a Hilbert space. Let be an isometry and let be an -nilpotent operator ( integer) such that . Then is a strict -isometry.

Proof. By Theorem 3 we obtain that is a -isometry; that is, is an arithmetic progression of order less than or equal to . Now we prove that it is an arithmetic progression of strict order , or equivalently the polynomial (9) has degree . Note that as is an isometry we have , for every positive integer .
As in the proof of Theorem 3, for any integer , we have that where .
The coefficient of in the polynomial is which is null if and only if , that is, if and only if . Therefore, if is nilpotent of order , then can be written as a polynomial in , of degree and coefficients in . Consequently is a strict -isometry.

Now we obtain the following corollary of Theorem 4.

Corollary 5. Let be a Hilbert space. Let be an -nilpotent operator ( integer). Then is a strict -isometry.

Recall that an operator is -supercyclic ( integer) if there exists a subspace of dimension such that its orbit is dense in . Moreover, is called supercyclic if it is -supercyclic. See [13, 14].

Bayart [7, Theorem 3.3] proved that on an infinite dimensional Banach space an -isometry is never -supercyclic, for any . 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 -isometries plus commuting nilpotent operators.

Corollary 6. Let be an infinite dimensional Hilbert space. If is an -isometry that commutes with a nilpotent operator , then is never -supercyclic for any .

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

Theorem 4 is not true for finite-dimensional Banach spaces even for .

Denote .

Example 1. Let be defined by ; hence is a -nilpotent operator. The following assertions hold: (1) is not a -isometry on for any and ;(2) is not a -isometry on for any ;(3) is a strict -isometry on for any .

Proof. For we have Write
(1) We consider two cases: and .(a)Case . For , , and , we have So if and only if , which is true only when or since the function is null only for and .Consequently is not a -isometry on if and .(b)Case . In order to prove that is not a -isometry on , we take the vector and obtain that
(2) For we have In particular, for and , Therefore is not a -isometry on for any .
(3) First we prove by induction on that is a -isometry on for any . Note that, for , By Corollary 5, the operator is a strict -isometry on . Hence is a strict -isometry on for all [9, Corollary 4.6]. Thus for , Suppose that is a -isometry on for every . Hence is also a -isometry on . Then, for , Therefore, Taking into account equality (28) we can write (26) in the following way: Therefore is a -isometry on .
Now we prove that is a strict -isometry on . Suppose on the contrary that is a -isometry on . Then, for all . So for all . In particular, for and , we have So is an arithmetic progression of order , which is a contradiction with (6).

Remark 7. Notice that, in any Hilbert space of dimension , there are strict -isometries only for any . However, as the above example shows, there are strict -isometries for any integer in a Banach space of dimension 2.

The following example gives an operator of the form with a nilpotent operator such that is not an -isometry for any integer and any .

Example 2. Let be the Banach space of all real continuous functions on such that endowed with the supremun norm. Define by Then is -nilpotent operator. Moreover, is not an -isometry for any and any .

Proof. It is clear that is not an isometry since the function given by satisfies and .
For consider the function defined by
Note that (Figure 1).

Fix . For , we have

If , then since . But as we obtain Consequently, Therefore is not an -isometry for any and any .

#### Disclosure

After submitting this paper for publication we received from Le and Gu et al. the papers [16, 17], in which they obtained (independently) Theorem 3. Their arguments are different from ours, using the Hereditary Functional Calculus.

#### Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

The first author is partially supported by Grant of Ministerio de Ciencia e Innovación, Spain, Project no. MTM2011-26538. The third author was supported by Grant no. 14-07880S of GA ČR and RVO:67985840.