Perturbation of m-Isometries by Nilpotent Operators

and Applied Analysis 3 Proof. Fix an integer k ≥ 0 and denote h := min{k, n − 1}. Then we have (T + Q) ∗k (T + Q) k

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  = 2 can be expressed in a special way.Agler [1] gives the following definition: a linear bounded operator  :  →  acting on a Hilbert space  is an (, 2)-isometry if (, 2)-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 (2 +  − 2)-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 (2 − 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 (, )-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,  ≥ 1 is an integer and  > 0 a real number.

Preliminaries: Arithmetic Progressions and (𝑚,𝑞)-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  = (  ) ≥0 in , the difference sequence  = () ≥0 is defined by ()  :=  +1 −   .The powers of  are defined recursively by  0  := ,  +1  = (  ).It is easy to show that for all  ≥ 0 and  ≥ 0 integers.A sequence  in a group  is called an arithmetic progression of order ℎ = 0, 1, 2 . .., if  ℎ+1  = 0. Equivalently, for  = 0, 1, 2, . ... It is well known that the sequence  in  is an arithmetic progression of order ℎ if and only if there exists a polynomial () in , with coefficients in  and of degree less than or equal to ℎ, such that () =   , for every  = 0, 1, 2 . ..; that is, there are  ℎ ,  ℎ−1 , . . .,  1 ,  0 ∈ , which depend only on , such that, for every  = 0, 1, 2, . .., We say that the sequence  is an arithmetic 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  of ( 6) has degree ℎ.Moreover, a sequence  in a group  is an arithmetic progression of order ℎ if and only if, for all  ≥ 0, that is, Now we give a basic result about -isometries.

𝑚-Isometry Plus 𝑛-Nilpotent
Recall that an operator  ∈ () is nilpotent of order  ( ≥ 1 integer), or -nilpotent, if   = 0 and  −1 ̸ = 0.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 ( + 1) /2 ([12, page 134] and [10, Theorem 2.7]).
For isometries it is possible to say more [10, Theorem 2.2].
Proof.By Theorem 3 we obtain that  +  is a (2 − 1)isometry; that is, (( + ) *  ( + )  ) ≥0 is an arithmetic progression of order less than or equal to 2−2.Now we prove that it is an arithmetic progression of strict order 2 − 2, or equivalently the polynomial (9) has degree 2−2.Note that as  is an isometry we have  *    = , for every positive integer .
As in the proof of Theorem 3, for any integer  ≥ 0, we have that where ℎ := min{,  − 1}.
The coefficient of  2−2 in the polynomial which is null if and only if  * −1  −1 = 0, that is, if and only if  −1 = 0. Therefore, if  is nilpotent of order , then ( + ) *  ( + )  can be written as a polynomial in , of degree 2 − 2 and coefficients in ().Consequently  +  is a strict (2 − 1)-isometry.
Now we obtain the following corollary of Theorem 4.
Bayart [7,Theorem 3.3] proved that on an infinite dimensional Banach space an (, )-isometry is never supercyclic, for any  ≥ 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 -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 .

Some Examples in the Setting of Banach Spaces
Theorem 4 is not true for finite-dimensional Banach spaces even for  = 1.Denote ℓ   := (C  , ‖⋅‖  ).