Perturbation of -Isometries by Nilpotent Operators
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.
The notion of -isometric operators on Hilbert spaces was introduced by Agler . See also [2–5]. Recently Sid Ahmed  has defined -isometries on Banach spaces, Bayart  introduced -isometries on Banach spaces, and -isometries on metric spaces were considered in . Moreover, Hoffman et al.  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  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 , 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 ,
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.
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 .
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
(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 .
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.
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.
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