# Perturbation of m-isometries by nilpotent operators.

1. Introduction

The notion of m-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 map T : E [right arrow] E (m [greater than or equal to] 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 [member of] E,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1)

We say that T is a strict (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 [right arrow] X, where X is a Banach space with norm [parallel]*[parallel], is an (m, q)-isometry if and only if, for all x [member of] X,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (2)

In the setting of Hilbert spaces, the case q = 2 can be expressed in a special way. Agler [1] gives the following definition: a linear bounded operator T : H [right arrow] H acting on a Hilbert space H is an (m, 2)-isometry if

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (3)

(m, 2)-isometries on Hilbert spaces will be called for short misometries.

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

In Section 3 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 (2n + 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 (2n - 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 [member of] B(H), [T.sup.*] denotes its adjoint. Moreover, m [greater than or equal to] 1 is an integer and q > 0 a real number.

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

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

Let G be a commutative group and denote its operation by +. Given a sequence a = [([a.sub.n]).sub.n [greater than or equal to] 0] in G, the difference sequence Da = [(Da).sub.n [greater than or equal to] 0] is defined by [(Da).sub.n] := [a.sub.n+1] - [a.sub.n]. The powers of D are defined recursively by [D.sup.0]a := a, [D.sup.k+1]a = D([D.sup.k]a). It is easy to show that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (4)

for all k [greater than or equal to] 0 and n [greater than or equal to] 0 integers.

A sequence a in a group G is called an arithmetic progression of order h = 0, 1, 2 ..., if [D.sup.h+1] a = 0. Equivalently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (5)

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.sub.n], for every n = 0, 1, 2 ...; that is, there are [[gamma].sub.h], [[gamma].sub.h-1], ..., [[gamma].sub.1], [[gamma].sub.0] [member of] G, which depend only on a, such that, for every n = 0, 1, 2, ...,

[a.sub.n] = p(n) = [h.summation over (i=0)] [[gamma].sub.i][n.sup.i]. (6)

We say that the sequence a 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 p of (6) has degree h.

Moreover, a sequence a in a group G is an arithmetic progression of order h if and only if, for all n [greater than or equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]; (7)

that is,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (8)

Now we give a basic result about m-isometries.

Theorem 1. Let H be a Hilbert space. An operator T [member of] B(H) is a strict m-isometry if and only if there are [A.sub.m-1] [not equal to] 0, [A.sub.m-2], ..., [A.sub.1], [A.sub.0] in B(H), which depend only on T, such that, for every n = 0, 1, 2 ...,

[T.sup.*n][T.sup.n] = [m-1.summation over (i=0)] [A.sub.i][n.sup.i]; (9)

that is, the sequence [([T.sup.*n][T.sup.n]).sub.n [greater than or equal to] 0] is an arithmetic progression of strict order m-1 in B(H).

Proof. If T [member of] B(H) is a strict m-isometry, then it satisfies (3). Hence, for each integer i [greater than or equal to] 0,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (10)

but

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (11)

By (5), the operator sequence [([T.sup.*n][T.sup.n]).sub.n [greater than or equal to] 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.sup.*n][T.sup.n]; that is, there are operators [A.sub.m-1] = 0, [A.sub.m-2], ..., [A.sub.1], [A.sub.0] in B(H), such that, for every n = 0, 1, 2 ...,

[T.sup.*n][T.sup.n] = [A.sub.m-1][n.sup.m-1] + [A.sub.m-2][n.sup.m-2] + ... + [A.sub.1]n + [A.sub.0]. (12)

Conversely, if [([T.sup.*n][T.sup.n]).sub.n [greater than or equal to] 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]). Let E be a metric space and let T : E [right arrow] 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 operator Q [member of] B(H) is nilpotent of order n (n [greater than or equal to] 1 integer), or n-nilpotent, if [Q.sup.n] = 0 and [Q.sup.n-1] [not equal to] 0.

In any finite dimensional Hilbert space H, strict misometries can be characterized in a very simple way: a linear operator T [member of] 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) ([12, page 134] and [10, Theorem 2.7]).

It was proved in [10, Theorem 2.2] that if A [member of] B(H) is an isometry and Q [member of] B(H) is an n-nilpotent operator such that TQ = QT, then T + Q is a strict (2n - 1)-isometry. Now we obtain a partial generalization of this result: if T [member of] B(H) is an m-isometry and Q [member of] B(H) is an n-nilpotent operator commuting with T, then T + Q is a (2n + m - 2)-isometry. However, T + Q is not necessarily a strict (2n + m - 2)isometry. For example, if T is an isometry and Q any nnilpotent operator (n > 1) such that TQ = QT, then T = T + Q+ (-Q) is not a strict (4n - 3)-isometry.

Theorem 3. Let H be a Hilbert space. Let T [member of] B(H) be an misometry and Q [member of] B(H) an n-nilpotent operator (n [greater than or equal to] 1 integer) such that TQ = QT. Then T + Q is (2n + m - 2)-isometry.

Proof. Fix an integer k [greater than or equal to] 0 and denote h := min {k, n - 1}. Then we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (13)

From (9) we obtain, for certain [A.sub.m-1], ..., [A.sub.0][member of] B (H),

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (14)

Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (15)

Note that [MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] are real polynomials in k of degree less than or equal to h [less than or equal to] n-1, and [(k - j).sup.r] and [(k - i).sup.r] have degree r [less than or equal to] [m.sup.-1]. Hence [q.sub.r,i,j] and [p.sub.r,i,j] are real polynomials of degree less than or equal to m-1+2(n-1) = 2n+m-3. Consequently we can write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (16)

which is a polynomial in k, of degree less than or equal to 2n+ m - 3 with coefficients in B(H). By Theorem 1, the operator T + Q is an (2n + m - 2)-isometry. ?

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

Theorem 4. Let H be a Hilbert space. Let T [member of] B(H) be an isometry and let Q [member of] B(H) be an n-nilpotent operator (n [greater than or equal to] 1 integer) such that TQ = QT. Then T + Q is a strict (2n - 1)isometry.

Proof. By Theorem 3 we obtain that T + Q is a (2n - 1)isometry; that is, [([(T + Q).sup.*k][(T + Q).sup.k]).sub.k [greater than or equal to] 0] is an arithmetic progression of order less than or equal to 2n-2. Nowwe prove that it is an arithmetic progression of strict order 2n - 2, or equivalently the polynomial (9) has degree 2n-2. Note that as T is an isometry we have [T.sup.*k][T.sup.k] = I, for every positive integer k.

As in the proof of Theorem 3, for any integer k [greater than or equal to] 0, we have that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (17)

where h := min {k, n - 1}.

The coefficient of [k.sup.2n-2] in the polynomial [(T + Q).sup.*k][(T + Q).sup.k] is

[(1/(n-1)!).sup.2] [Q.sup.*n-1][Q.sup.n-1], (18)

which is null if and only if [Q.sup.*n-1][Q.sup.n-1] = o, that is, if and only if [Q.sup.n-1] = 0. Therefore, if Q is nilpotent of order n, then [(T + Q).sup.*k][(T + Q).sup.k] can be written as a polynomial in k, of degree 2n-2 and coefficients in B(H). Consequently T + Q is a strict (2n - 1)-isometry.

Now we obtain the following corollary of Theorem 4.

Corollary 5. Let H be a Hilbert space. Let Q [member of] B(H) be an nnilpotent operator (n [greater than or equal to] 1 integer). Then I+Q is a strict (2n-1)- isometry.

Recall that an operator T [member of] B(H) is N-supercyclic (N [greater than or equal to] 1 integer) if there exists a subspace F c H of dimension N such that its orbit {[T.sup.n]x : n [greater than or equal to] 0, x [member of] F} is dense in H. Moreover, T is called supercyclic 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 Nsupercyclic, for any N [greater than or equal to] 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. Let H be an infinite dimensional Hilbert space. If T [member of] 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

Theorem 4 is not true for finite-dimensional Banach spaces even for m = 1.

Denote [l.sup.d.sub.p] := ([C.sup.d], [[parallel]*[parallel].sub.p]).

Example 1. Let Q: [C.sup.2] [right arrow] [C.sup.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.sup.2.sub.p] for any 1 [less than or equal to] p < [infinity] and p [not equal to] 2;

(2) I + Q is not a (3, p)-isometry on [l.sup.2.sub.[infinity]] for any p > 0;

(3) I + Q is a strict (2k + 1,2k)-isometry on ([C.sup.2], [[parallel]*[parallel].sub.2k]) for any k = 1, 2, 3, ....

Proof. For (x, y) [member of] [C.sup.2] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (19)

Write

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (20)

(1) We consider two cases: 1 < p < [infinity] and p = 1.

(a) Case 1 < p < [infinity]. For x = 0, y = 1, and q = p, we have

A(0,1;p,p) = [3.sup.P] + 1 - 3 * [2.sup.p] - 3 + 6 - 1 = [3.sup.P] - 3 * [2.sup.P] + 3. (21)

So A(0,1;p,p) = 0 if and only if [3.sup.P-1] + 1 = [2.sup.P], which is true only when p = 2 or p = 1 since the function f(t) = [3.sup.t-1] + 1 - [2.sup.t] is null only for t = 1 and t = 2.

Consequently I + Q is not a (3, p)-isometry on [l.sup.2.sub.p] if p [not equal to] 2 and 1 < p < [infinity].

(b) Case p = 1. In order to prove that I + Q is not a (3,1)-isometry on [l.sup.2.sub.1], we take the vector (1, -1) and obtain that

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (22)

(2) For (x, y) [member of] [C.sup.2] we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (23)

In particular, for x := 1 and y := -1,

A(1,-1; [infinity], p) = [2.sup.P] - 1 [not equal to] 0. (24)

Therefore I + Q is not a (3, _p)-isometry on lp for any p > 0.

(3) First we prove by induction on k that I + Q is a (2k + 1,2k)-isometry on [l.sup.2.sub.2k] for any k = 1, 2, 3 ..., Note that, for (x, y) [member of] [C.sup.2],

[(I + Q).sup.s] (x, y) = (x + sy, y), (s = 0, 1, 2 ...). (25)

By Corollary 5, the operator I + Q is a strict (3,2)-isometry on [l.sup.2.sub.2] Hence I + Q is a strict (2k + 1,2fc)-isometry on [l.sup.2.sub.2] for all k = 1, 2, 3 ... [9, Corollary 4.6]. Thus for (x, y) [member of] [C.sup.2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (26)

Suppose that I+Q is a (2i-1,2i-2)-isometry on [l.sup.2.sub.2i-2] for every i = 2, 3, ..., k. Hence I + Q is also a (2k + 1,2i - 2)-isometry on [l.sup.2.sub.2i-2]. Then, for (x, y) [member of] [C.sup.2],

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (27)

Therefore,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (28)

Taking into account equality (28) we can write (26) in the following way:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (29)

Therefore I + Q is a (2k + 1,2k)-isometry on [l.sup.2.sub.2k].

Now we prove that I+Q is a strict (2k +1, 2fc)-isometry on [l.sup.2.sub.2k]. Suppose on the contrary that I + Q is a (2k, 2k)-isometry on [l.sup.2.sub.2k]. Then,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (30)

for all (x, y) [member of] [C.sup.2]. So

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (31)

for all (x, y) [member of] [C.sup.2]. In particular, for y = 1 and x = 0, 1, 2, ..., we have

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (32)

So [([s.sup.2k]).sup.[infinity].sub.s=0] is an arithmetic progression of order 2k - 2, which is a contradiction with (6). ?

Remark 7. Notice that, in any Hilbert space of dimension n, there are strict m-isometries only for any m [less than or equal to] 2n-1. However, as the above example shows, there are strict (2k + 1,2k)-isometries for any integer k in a Banach space of dimension 2.

The following example gives an operator of the form I + 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. Let X 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 [right arrow] X by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (33)

Then Q [member of] 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 that I+Q is not an isometry since the function f [member of] X given by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (34)

satisfies [parallel]f[parallel] = 1 and [parallel](I + Q)f[parallel] = 2.

For m = 2,3,4, ... consider the function fm e X defined by

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (35)

Note that [f.sub.m](3/4) = 1/(1 - m) = [min.sub.0[less than or equal to] t [less than or equal to] 1] [f.sub.m](t) (Figure 1).

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (36)

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (37)

since k(1/ (m- 1)) [less than or equal to] 1. But as m(1/ (m - 1)) >1 we obtain

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (38)

Consequently,

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (39)

Therefore I + Q is not an (m, q)-isometry for any m = 1, 2, 3 ... and any q > 0.

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.

http://dx.doi.org/10.1155/2014/745479

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 Innovation, Spain, Project no. MTM2011-26538. The third author was supported by Grant no. 14-07880S of GA CR and RVO:67985840.

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Teresa Bermudez, (1) Antonio Martinon, (1) Vladimir Muller, (2) and Juan Agustin Noda (1)

(1) Departamento de Analisis Matematico, Universidad de La Laguna, La Laguna, 38271 Tenerife, Spain

(2) Mathematical Institute, Czech Academy of Sciences, 115 67 Prague, Czech Republic

Correspondence should be addressed to Teresa Bermudez; tbermude@ull.es

Received 8 April 2014; Accepted 18 July 2014; Published 9 September 2014