Some examples of $m$-isometries

TL;DR

This paper characterizes the spectra of strict m-isometries on finite-dimensional Hilbert spaces, classifies m-isometries on R^2, and explores volume preservation and construction methods for m-isometries.

Contribution

It provides a complete description of spectra for strict m-isometries, classifies all m-isometries on R^2, and introduces a construction method for higher-order m-isometries.

Findings

Spectra of strict m-isometries on finite-dimensional spaces are characterized.

On R^2, m-isometries are limited to 3-isometries and isometries of a specific form.

m-isometries preserve volume on real Hilbert spaces.

Abstract

We obtain the admissible sets on the unit circle to be the spectrum of a strict $m$-isometry on an $n$-finite dimensional Hilbert space. This property gives a better picture of the correct spectrum of an $m$-isometry. We determine that the only $m$-isometries on $\mathbb{R}^2$ are $3$-isometries and isometries giving by $\pm I+Q$, where $Q$ is a nilpotent operator. Moreover, on real Hilbert space, we obtain that $m$-isometries preserve volumes. Also we present a way to construct a strict $(m+1)$-isometry with an $m$-isometry given, using ideas of Aleman and Suciu \cite[Proposition 5.2]{AS} on infinite dimensional Hilbert space.

Equations

$$⟨Tx,Ty⟩=⟨T^{\ast }Tx,y⟩,$$

$$(yx-1{)}^m(T):=\sum _{k=0}^m(-1{)}^{m-k}\left(\genfrac{}{}{0ex}{}{m}{k}\right)T^{\ast k}{T}^k=0;$$

$$\sum _{k=0}^m(-1{)}^{m-k}(\genfrac{}{}{0ex}{}{m}{k})\Vert T^{k}x{\Vert }^2=0,$$

$$I_{m}(H):=\{T\in L(H)\;\;:\;\;T\mbox{ is an $m$-isometry}\}\;.$$

$$T\cong T_{|H_1}\oplus \cdots \oplus T_{|H_s},$$

$$R_{\theta}:=\left(\begin{array}[]{cc}\cos\theta&-\sin\theta\\ \sin\theta&\cos\theta\end{array}\right)\;\mbox{ and }S_{\theta}:=\left(\begin{array}[]{cc}\cos\theta&\sin\theta\\ \sin\theta&-\cos\theta\end{array}\right)\;,$$

$$M:=\left(\begin{array}[]{cc}0&1\\ 0&0\end{array}\right)\;,\;\;\;\;\;N:=\left(\begin{array}[]{cc}0&0\\ 1&0\end{array}\right)\;Q_{k}:=\left(\begin{array}[]{cc}1&k\\ -\frac{1}{k}&-1\end{array}\right)\;,$$

$$R_{\theta }M=M{R}_{\theta }⟺R_{\theta }N=N{R}_{\theta }⟺R_{\theta }{Q}_k=Q_k{R}_{\theta }⟺\sin \theta =0⟺\theta =0\text{mbox}or\theta =\pi .$$

$$S_{\theta }M=M{S}_{\theta }⟺S_{\theta }N=N{S}_{\theta }⟺S_{\theta }{Q}_k=Q_k{S}_{\theta }⟺\sin \theta =\cos \theta =0,$$

$$\{{\lambda }_1{x}_1+{\lambda }_2{x}_2+\dots +{\lambda }_k{x}_k:0\le {\lambda }_i\le 1,\text{mbox}fori=1,2,\dots ,k\}.$$

$$A(x_1,x_2,\dots ,x_n):$$

$$Q_j(x_1,x_2,\dots ,x_n):$$

$$A{Q}_j(x_1,x_2,\dots ,x_n)=Q_{j}A(x_1,x_2,\dots ,x_n)=(0,x_3,x_4,\dots ,x_{j+1},0,\dots ,0),$$

$$\Vert T^{3}x{\Vert }^2-3\Vert T^{2}x{\Vert }^2+3\Vert Tx{\Vert }^2-\Vert x{\Vert }^2\ne 0,$$

$$T(x_1,\cdots ,x_n):=({\lambda }_1{x}_1,\cdots ,{\lambda }_n{x}_n)$$

$$S:=\left(\begin{array}[]{cccc}\lambda_{1}&0&\dots&0\\ 0&\lambda_{2}&\ddots&\vdots\\ \vdots&\ddots&\ddots&0\\ 0&\dots&0&\lambda_{n}\end{array}\right)$$

$$\sigma (T)=\{{\lambda }_1,...,{\lambda }_r\},\text{mbox}withr\ge n-k+2.$$

$$T=({\lambda }_1{I}_{n_1}\oplus ...\oplus {\lambda }_r{I}_{n_r})+(Q_1\oplus ...\oplus Q_r),$$

$${\beta }_{\ell }(S,x):=\frac{1}{\ell !}\sum _{j=0}^{\ell }(-1{)}^{\ell -j}(\genfrac{}{}{0ex}{}{\ell }{j})\Vert S^{j}x{\Vert }^2.$$

$$p(z)=\sum _{n\ge 0}{p}_n{z}^n$$

$$Lp(z):=\sum _{n\ge 1}{p}_n{z}^{n-1}=\frac{p(z)-p_0}{z}.$$

$$\Vert p{\Vert }_2^2:=\sum _{n\ge 0}|p_n{|}^2.$$

$$\Vert |p\Vert {|}_k^2:=\Vert p{\Vert }_2^2+\sum _{n\ge 0}\Vert (L^{nk}p)(T)x_0{\Vert }^2,$$

$$\sum _{k=0}^m(-1{)}^k(\genfrac{}{}{0ex}{}{n}{k})=(-1{)}^m\left(\genfrac{}{}{0ex}{}{n-1}{m}\right),$$

$$\Vert |M_z^{kj}p\Vert {|}_k^2=\Vert |p\Vert {|}_k^2+\sum _{i=1}^j\Vert T^{ki}p(T)x_0{\Vert }^2,$$

$${\beta }_{\ell +1}(M_z^k,p)=\frac{\ell !}{(\ell +1)!}{\beta }_{\ell }(T^k,T^{k}p(T)x_0).$$

$$\Vert |M_z^{kj}p\Vert {|}_k^2=\Vert |p\Vert {|}_k^2+\sum _{i=1}^j\Vert T^{ki}p(T)x_0{\Vert }^2,$$

$$\Vert |M_z^{k}p\Vert {|}_k^2=\Vert |p\Vert {|}_k^2+\Vert T^{k}p(T)x_0{\Vert }^2,$$

$$\Vert |M_z^{k}p\Vert {|}_k^2=\Vert |z^{k}p\Vert {|}_k^2$$

$$\Vert |M_z^{k(j+1)}p\Vert {|}_k^2$$

$${\beta }_{\ell +1}(M_z^k,p)$$

Full text

1. Introduction

Let $H$ be a Hilbert space. Denote by $L(H)$ the algebra of bounded linear operators on $H$. For $T\in L(H)$ we consider the adjoint operator $T^{*}\in L(H)$, which is the unique map that satisfies

[TABLE]

for every $x,y\in H$. Given $T\in L(H)$, denote by $Ker(T)$ and $R(T)$, the kernel and range of $T$, respectively. For a positive integer $m$, an $m$-isometry is an operator $T\in L(H)$ which satisfies the condition

[TABLE]

equivalently

[TABLE]

for every $x\in H$. A strict $m$-isometry is an $m$-isometry which is not an $(m-1)$-isometry. This class of operators was introduced by Agler in [2] and was studied by Agler and Stankus in [4, 5, 6].

Let $n$ be a positive integer. Recall that $Q\in L(H)$ is $n$-nilpotent if $Q^{n}=0$ and $Q^{n-1}\neq 0$.

A notion related with $m$-isometries is the following. An operator $T\in L(H)$ is isometric $n$-Jordan if there exist an isometry $A\in L(H)$ and an $n$-nilpotent $Q\in L(H)$ such that $T=A+Q$ with $AQ=QA$.

Theorem 1.1**.**

[13, Theorem 2.2]** Any isometric $n$-Jordan operator is a strict $(2n-1)$-isometry.

Actually, a much stronger result is true. Indeed in [15, Theorem 3], it is obtained a generalization of Theorem 1.1 for $m$-isometries: if $T$ is an $m$-isometry, $Q$ is an $n$-nilpotent operator and they commute, then $T+Q$ is a $(2n+m-2)$-isometry. See also [25, 28]. Moreover, the study of isometric $n$-Jordan operators concerning with $m$-isometries on Banach spaces context has been studied in [15].

Another way of generalization was obtained in [13, Proposition 2.6] for sub-isometry $n$-Jordan operator. Recall that $T$ is a sub-isometry $n$-Jordan operator if $T$ is the restriction of an isometry $n$-Jordan operator $J$ to an invariant subspace of $J$.

Notice that Theorem 1.1 gives an easy way to construct examples of $m$-isometries, for an odd $m$. It is sufficient to choose the identity operator as the isometry and any $n$-nilpotent operator with $n=\frac{m+1}{2}$.

At a first glance, we could think that all the $m$-isometries come from isometric $n$-Jordan. However, this is not true, since there are strict $m$-isometries for even $m$, see [8, Proposition 9]. What can we say about $m$-isometries with odd $m$? Recently, Yarmahmoodi and Hedayatian have proven that the only isometric $n$-Jordan weighted shift operators are isometries [30, Theorem 1]. So, there are $m$-isometries that are not isometric $n$-Jordan, since Athavale in [8] gave examples of strict $m$-isometries with the weighted shift operator for all integers $m$.

Whenever, if $H$ is finite dimensional is possible to say more.

Some authors have given examples of $m$-isometries. For example with the unilateral or bilateral weighted shift [1, 12, 14, 18] and with the composition operator [14, 16, 23]. Another way to construct examples of $m$-isometries is developing different tools like tensor product [19], functional calculus [24], on Hilbert-Schmidt class [17] and with $C_{0}$-semigroups [10, 21, 29].

The purpose of this paper is to make a clear picture of $m$-isometries on finite dimensional Hilbert space. In Section 2, we begin with the study of $m$-isometries on $\mathbb{R}^{2}$ and on $\mathbb{R}^{n}$, with $n\geq 3$. We give all the 3-isometries on $\mathbb{R}^{2}$. Also, we obtain the expression of $m$-isometries and study how this class of operators change volumes on $\mathbb{R}^{n}$. Moreover, we study the case of complex Hilbert space, where we prove the admissible sets on the unit circle to be the spectrum of an $m$-isometry. In Section 3, we reproduce similar ideas of Aleman and Suciu [7, Proposition 5.2] to define a 3-isometry using a given 2-isometry. In fact, we obtain a way to construct a strict $(m+1)$-isometry using a weaker condition than a strict $m$-isometry.

In particular, we will answer the following problems.

Problem 1.2**.**

Let $T\in L(H)$ with $H$ an $n$-finite dimensional Hilbert space and $m$ an odd integer. Are all strict $m$-isometries of the form $\lambda I+Q$, where $Q$ is a nilpotent operator and $\lambda$ is a complex number with modulus 1?

Problem 1.3**.**

Let $T\in L(\mathbb{R}^{n})$. How does an $m$-isometry $T$ change volumes?

Problem 1.4**.**

Let $H$ be any $n$-finite dimensional Hilbert space and let $T$ be an $m$-isometry with odd $m$. What can we say about the spectrum?

2. $m$-isometries on finite dimensional Hilbert space

Recall some important properties of the spectrum of an $m$-isometry.

Denote $\overline{\mathbb{D}}$ and $\partial\mathbb{D}$ the closed unit disk and the unit circle, respectively.

Lemma 2.1**.**

Let $m$ be a positive integer, $H$ be a Hilbert space and $T\in L(H)$ be an $m$-isometry. Then

1. $(1)$

[4, Lemma $1.21$]* $\sigma(T)=\overline{\mathbb{D}}$ or $\sigma(T)\subseteq\partial\mathbb{D}$.* 2. $(2)$

[3, Lemma $19$]* The eigenvectors of $T$ corresponding to distinct eigenvalues are orthogonal.*

Remark 2.2**.**

1. (1)

Notice that any $m$-isometry on a finite dimensional space is bijective. 2. (2)

It is well known that if $Q$ is $k$-nilpotent on an $n$-dimensional vector space, then $k\leq n$.

Denote

[TABLE]

The following theorem gives a nice picture of $m$-isometries on finite dimensional spaces.

Theorem 2.3**.**

([13, Theorem 2.7], [3, page 134]) Let $H$ be an $n$-finite dimensional Hilbert space and $T\in L(H)$. Then

1. (1)

$T$* is a strict $m$-isometry if and only if $T$ is an isometric $k$-Jordan operator, where $m=2k-1$ with $k\leq n$.* 2. (2)

$I_{1}(H)=I_{2}(H)\subsetneq I_{3}(H)=I_{4}(H)\subsetneq\ldots\subsetneq I_{2n-1}(H)=I_{j}(H)$* for all $j\geq 2n-1$.*

Proof.

We include the proofs for completeness.

(1) Assume that $T$ is a strict $m$-isometry on $H$. Then the spectrum of $T$, $\sigma(T)=\{\lambda_{1},\lambda_{2},\ldots,\lambda_{s}\}$, where $\lambda_{i}$ are eigenvalues of modulus 1, since the spectrum of $T$ must be in the unit circle and $m$ is odd [4, Lemma 1.21 & Proposition 1.23]. By part (2) of Lemma 2.1, the spectral subspaces of $T$, $H_{i}:=Ker(T-\lambda_{i})^{n_{i}}$ are mutually orthogonal and

[TABLE]

where $n_{1},\ldots,n_{s}$ are positive integers such that $Ker(T-\lambda_{i})^{n_{i}}=Ker(T-\lambda_{i})^{N}$ for all $N\geq n_{i}$. Moreover, for all $j\in\{1,\ldots,s\}$, we have that $\sigma(T_{|H_{j}})=\{\lambda_{j}\}$ and $T_{|H_{j}}$ is of the form $\lambda_{j}+Q_{j}$ for some nilpotent operator $Q_{j}$. So, $T=A+Q$ for some isometry, in fact unitary diagonal operator $A$ and some nilpotent operator $Q$ such that $AQ=QA$.

The converse is consequence of Theorem 1.1.

(2) Let us prove that $I_{2\ell-1}(H)=I_{2\ell}(H)$ for all $\ell\in\mathbb{N}$. Recall that if $T$ is $(2\ell)$-isometry, then $T$ is bijective and so $T$ is $(2\ell-1)$-isometry [4, Proposstion 1.23]. Moreover, the highest degree of nilpotent operator on $n$-dimensional Hilbert space is $n$. The result is a consequence of Theorem 1.1. ∎

2.1. $m$-isometries on real Hilbert spaces

Next, we study the $m$-isometries on $\mathbb{R}^{n}$.

Based on the above results, we obtain all $m$-isometries on $\mathbb{R}^{2}$.

Theorem 2.4**.**

If $T\in L(\mathbb{R}^{2})$ is a strict $m$-isometry, then $m=1$ or $m=3$ and $T=A+Q$, where $A$ is an isometry and $Q$ is a nilpotent operator of order 2 that commutes.

Recall that isometries on $\mathbb{R}^{2}$ are given by

[TABLE]

where

1. (1)

$R_{\theta}$ is a rotation (about [math]) and its determinant, $\det(R_{\theta})$ is $1$ and 2. (2)

$S_{\theta}$ is a symmetry respect to the straight line of equation $x_{2}=\tan(\theta/2)x_{1}$ and $\det(S_{\theta})=-1$.

And the non-zero nilpotent operators on $\mathbb{R}^{2}$ are $\lambda M,\;\lambda N$ and $\lambda Q_{k}$ where

[TABLE]

with $k\neq 0$ and $\lambda\in\mathbb{C}\setminus\{0\}$.

We are interested in studying isometries that commute with nilpotent operators on $\mathbb{R}^{2}$.

Lemma 2.5**.**

The unique isometries on $L(\mathbb{R}^{2})$ that commute with a non-zero nilpotent operator are the trivial cases, that is, $\pm I$.

Proof.

Simple calculations prove that

[TABLE]

That is, the unique isometries of type $R_{\theta}$ which commute with some non-zero nilpotent (hence with all the nilpotent) are $R_{0}=I$ and $R_{\pi}=-I$.

Analogously, we have that

[TABLE]

which it is impossible. Hence there are not isometries $S_{\theta}$ which commute with some non-zero nilpotent operator. ∎

Taking into account Theorem 2.3 we give the unique strict 3-isometries on $\mathbb{R}^{2}$. Indeed, we answer Problem 1.2 for $n=2$ in the following result.

Theorem 2.6**.**

The strict $3$-isometries on $\mathbb{R}^{2}$ are of the form $\pm I+Q$, where $Q$ is a non-zero nilpotent operator given in (2.3).

Proof.

It is immediate by Theorem 2.4 and Lemma 2.5.

∎

Let $T\in L(\mathbb{R}^{n})$ with $n\geq 3$ and let us consider the following $n$ conditions:

1. $(M_{k})$

$\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;S_{k}(Tx_{1},Tx_{2},\ldots,Tx_{k})=S_{k}(x_{1},x_{2},\ldots,x_{k})$

for all $x_{1},x_{2},\ldots,x_{k}\in\mathbb{R}^{n}$ and $k=1,2,\ldots,n$, where $S_{k}(x_{1},x_{2},\ldots,x_{k})$ denotes the $k$-dimensional measure of the set

[TABLE]

Lemma 2.7**.**

Let $T\in L(\mathbb{R}^{n})$. Then

1. (1)

[26, Teorema II]** $T$ satisfies the conditions $(M_{1}),\;(M_{2}),\;\cdots\;,(M_{n-1})$ if and only if $T$ is an isometry. 2. (2)

[20]** The condition $(M_{n})$ is equivalent to $det(T)=\pm 1$.

An easy application of Theorem 1.1 gives that, for example in $\mathbb{R}^{3}$, we have strict 3-isometries giving by $\pm I+Q$, where $Q$ is a 2-nilpotent operator and strict 5-isometries giving by $\pm I+Q$, where $Q$ is a 3-nilpotent operator.

The next result gives answer to Problems 1.2 and 1.3 for $n\geq 3$, where $n$ is the dimension of the Hilbert space.

Theorem 2.8**.**

Let $n\geq 3$. Then the following properties follow:

1. (1)

There are non-trivial strict $m$-isometries on $L(\mathbb{R}^{n})$ for any odd $m$ less than $2n-1$, that is, there exists an isometry different from $\pm I$ such that commutes with a non-zero $k$-nilpotent operator with $k\in\{1,2,\cdots,n-1\}$. 2. (2)

The $m$-isometries preserve volumes.

Proof.

(1) Define

[TABLE]

Then $A$ is an isometry and $Q_{j}$ is a $j$-nilpotent operator such that

[TABLE]

for all $(x_{1},x_{2},\ldots,x_{n})\in\mathbb{R}^{n}$. By Theorem 1.1, we get that $A+Q_{j}$ is a non trivial strict $(2j-1)$-isometry for $j=1,\ldots,n-1$.

(2) By Lemma 2.7, it will be enough to prove that $det(A+Q)=\pm 1$ for all isometries $A$ that commute with a nilpotent operator $Q$. Since $AQ=QA$, then $\sigma(A+Q)=\sigma(A)$ by [31, Proposition 1.1]. According to the spectrum of an isometry on a finite dimensional space, we have that the spectrum of $A$ is a closed subset of the unit circle. By [9, page 150], the determinant of $T$ is the product of the eigenvalues of $T$, counting multiplicity. Hence $det(T)=\pm 1$.

∎

The converse of part (2) of Theorem 2.8 is not true, as prove the following example.

Example 2.9**.**

Let T:=\left(\begin{array}[]{ccc}1&0&0\\ 0&2&1\\ 0&1&1\end{array}\right). Then $det(T)=1$ and $T$ is not a 3-isometry, since

[TABLE]

for $x:=(1,1,0)$.

2.2. On complex Hilbert space

We recall the following results about the spectrum of $m$-isometries.

Lemma 2.10**.**

[13, Theorem 4.4]* Let $H$ be an infinite dimensional Hilbert space.*

1. (1)

If $K$ is any compact subset of $\partial\mathbb{D}$, then there exists a strict $m$-isometry for any odd number $m$ such that $\sigma(T)=K$. 2. (2)

If $K$ is the closed unit disk, then there exits a strict $m$-isometry for any integer number $m$.

The main aim of this section is to solve Problem 1.4.

Let $T\in L(\mathbb{C}^{n})$ be an $m$-isometry. It is clear that $\sigma(T)\subseteq\partial\mathbb{D}$ by part (1) of Lemma 2.1 and $\sigma(T)$ has at most $n$ different eigenvalues. Indeed if $K:=\{\lambda_{1},\cdots,\lambda_{n}\}$ with $\lambda_{i}$ different complex numbers on the unit circle, then it is possible to define an isometry $T$ such that $\sigma(T)=K$. In particular, the following operator

[TABLE]

is an isometry on $\mathbb{C}^{n}$ with $\sigma(T)=\{\lambda_{1},\cdots,\lambda_{n}\}$.

In the following theorem we prove that any $m$-isometry with $m\geq 3$ on $\mathbb{C}^{n}$ can not have $n$ different eigenvalues.

Theorem 2.11**.**

Any strict $(2k-1)$-isometry on $\mathbb{C}^{n}$ with $2\leq k\leq n$ has at most $n-1$ distinct eigenvalues.

Proof.

Assume that $T\in L(\mathbb{C}^{n})$ is a strict $(2k-1)$-isometry with $\sigma(T)=\{\lambda_{1},...,\lambda_{n}\}$ where $\lambda_{1},...,\lambda_{n}$ are different eigenvalues of $T$. Then $T$ could be written as $T=PSP^{-1}$, for some $P\in L(\mathbb{C}^{n})$ where

[TABLE]

and $|\lambda_{i}|=1$ for $i\in\{1,\cdots,n\}$, by part (1) of Lemma 2.1. Since $T$ is a strict $(2k-1)$-isometry, by part (2) of Lemma 2.1, the operator $P$ is a unitary operator. This means that $T$ is unitarily equivalent to $S$, therefore $T$ is a unitary, which is a contradiction. ∎

Theorem 2.12**.**

The strict $(2k-1)$-isometries on $\mathbb{C}^{n}$, with $2\leq k\leq n$ are of the form $(\lambda_{1}I_{n_{1}}\oplus...\oplus\lambda_{\ell}I_{n_{\ell}})+Q$, with $\ell\in\{1,...,n-k+1\}$, where $Q$ is a $k$-nilpotent, $|\lambda_{j}|=1$ for all $j\in\{1,...,\ell\}$ and $n_{1}+...+n_{\ell}=n$.

Proof.

Suppose that $T$ is a strict $(2k-1)$-isometry. By Theorem 2.3, we have that $T=U+Q$, where $U$ is a unitary operator and $Q$ is a $k$-nilpotent operator such that $UQ=QU$.

Assume, by contradiction, that $T$ has at least $n-k+2$ distinct eigenvalues. That means

[TABLE]

Then $\mathbb{C}^{n}=H_{\lambda_{1}}\oplus...\oplus H_{\lambda_{r}}$ , where $H_{\lambda_{i}}:=Ker(T-\lambda_{i}I)^{n_{i}}$ and ${n_{i}}$ is the order of multiplicity of the eigenvalue $\lambda_{i}$. Denote $T_{|H_{i}}$ the restriction operator of $T$ to $H_{i}$, for $1\leq i\leq r$. Then $T_{|H_{i}}=\lambda_{i}I_{n_{i}}+Q_{i}$, where $Q_{i}$ is a $h_{i}$-nilpotent with $1\leq h_{i}\leq n_{i}$. By part (2) of Lemma 2.1, we conclude that $T$ could be written as

[TABLE]

where $Q_{1}\oplus...\oplus Q_{r}$ is a $k_{0}$-nilpotent, with $k_{0}:=\max_{i=1,...,r}\{h_{i}\}$ and $k_{0}<k$. Then we get a contradiction. ∎

Corollary 2.13**.**

If $T\in L(\mathbb{C}^{n})$ is a strict $(2k-1)$-isometry, with $2\leq k\leq n$, then $\sigma(T)\subseteq\{\lambda_{1},...,\lambda_{n-k+1}\}\subseteq\partial\mathbb{D}$.

Corollary 2.14**.**

Any $(2n-1)$-isometry on $\mathbb{C}^{n}$ is of the form $\lambda I+Q$, where $Q$ is an $n$-nilpotent operator and $\lambda\in\partial\mathbb{D}$. In particular the spectrum is a single point on the unit circle.

3. Construction of an $(m+1)$-isometry from an $m$-isometry

In this section we present a method to construct a Hilbert space $H_{k}$ and an $(m+1)$-isometry on $H_{k}$ from an $m$-isometry $T^{k}$ on a Hilbert space for some integer $k$. Our result is based on the construction given by Aleman and Suciu in [7, Proposition 5.2] for $m=2$ and $k=1$.

Henceforth $H$ will denote an infinite dimensional Hilbert space.

Given $S\in L(H)$, $x\in H$ and an integer $\ell\geq 1$, it is defined

[TABLE]

Note that $S$ is an $m$-isometry if and only if $\beta_{m}(S,x)=0$ for all vector $x\in H$.

Consider $\mathbb{C}[z]$ the space of all complex polynomials. Given $p\in\mathbb{C}[z]$, we write

[TABLE]

and define $Lp\in\mathbb{C}[z]$ in the following way:

[TABLE]

We have that $\mathbb{C}[z]$ is an inner product space with the norm $\|.\|_{2}$ given by

[TABLE]

Also if we consider a new norm on $\mathbb{C}[z]$ defined by

[TABLE]

it is obtained that $\mathbb{C}[z]$ is an inner product space with $\||.\||_{k}$. Denote $H_{k}$ its completion with the new norm.

The following combinatorial result will be useful.

Lemma 3.1**.**

[22, Eq. $0.151$ $(4)$]* If $m$ is any positive integer, then*

[TABLE]

for any integer $n\geq m+1$.

Recall that the class of $m$-isometries is stable under powers. However, the converse is not true. See [11, 27].

Theorem 3.2**.**

Let $T\in L(H)$ such that $T^{k}$ is a strict $m$-isometry on $R(T^{k})$, for some $k$, and $x_{0}\in H\setminus\{0\}$ such that $\beta_{m-1}(T^{k},T^{k}x_{0})\neq 0$.

1. (1)

For every $p\in\mathbb{C}[z]$ and $j\in\mathbb{N}$,

[TABLE]

where $M_{z}$ denotes the multiplication operator defined by $M_{z}p:=zp$. 2. (2)

For every $p\in\mathbb{C}[z]$ and $\ell\geq 1$,

[TABLE] 3. (3)

The extension of $M_{z}^{k}$ to $H_{k}$ is an $(m+1)$-isometry.

Proof.

(1) Let $p$ be any polynomial and $j\in\mathbb{N}$. Then will prove that

[TABLE]

by induction. For $j=1$ we need to prove that

[TABLE]

for any polynomial $p$.

Let $p(z):=\sum_{n\geq 0}p_{n}z^{n}$. Then

[TABLE]

Then (3.6) holds.

Suppose that (3.5) is true for $j$. Let us prove it for $j+1$. Then

[TABLE]

So we prove (3.5).

(2) For $\ell\in\mathbb{N}$, we have

[TABLE]

where $p$ is any polynomial.

Using Lemma 3.1, in the last sum, we have that

[TABLE]

So,

[TABLE]

So, (3.4) is proved.

(3) It is enough to prove that $\beta_{m+1}(M_{z}^{k},p)=0$ for any $p\in\mathbb{C}[z]$. This is a consequence of (3.4), since $T^{k}$ is an $m$-isometry on $R(T^{k})$. ∎

Corollary 3.3**.**

[7, Proposition 5.2]** Let $T$ be a $2$-isometry on a Hilbert space $H$. Fix $x_{0}\in H\setminus\{0\}$ and let $H_{1}$ be the completion of the space of analytic polynomials with respect to the norm

[TABLE]

Then the multiplication operator by the independent variable $M_{z}p:=zp$ extends to a $3$-isometry on $H_{1}$.

Acknowledgements: The first author is partially supported by grant of Ministerio de Ciencia e Innovación, Spain, project no. MTM2016-75963-P. The third author was supported in part by Departamento de An lisis Matem tico of Universidad de La Laguna and Le Laboratoire de Recherche Math matiques et Applications LR17ES11.