Let
T
be a bounded linear operator on a complex Hilbert space
H
. For a positive integer
k
, an operator
T
is said to be a
k
quasi2isometric operator if
T
^{∗k}
(
T
^{∗2}
T
^{2}
− 2
T
^{∗}
T
+
I
)
T^{k}
= 0, which is a generalization of 2isometric operator. In this paper, we consider basic structural properties of
k
quasi2isometric operators. Moreover, we give some examples of
k
quasi2isometric operators. Finally, we prove that generalized Weyl’s theorem holds for polynomially
k
quasi2isometric operators.
1. INTRODUCTION
Let
H
be an infinite dimensional separable Hilbert space. We denote by
B
(
H
) the algebra of all bounded linear operators on
H
, write
N
(
T
) and
R
(
T
) for the null space and range space of
T
, and also, write
σ
(
T
),
σ
_{a}
(
T
) and iso
σ
(
T
) for the spectrum, the approximate point spectrum and the isolated point spectrum of
T
, respectively.
An operator
T
is called Fredholm if
R
(
T
) is closed,
α
(
T
) = dim
N
(
T
) < ∞ and
β
(
T
) = dim
H/R
(
T
) < ∞. Moreover if
i
(
T
) =
α
(
T
) −
β
(
T
) = 0, then
T
is called Weyl. The Weyl spectrum
w
(
T
) of
T
is defined by
w
(
T
) := {
λ
∈ ℂ :
T
−
λ
is not Weyl}. Following
[12]
, we say that Weyl’s theorem holds for
T
if
σ
(
T
)\
w
(
T
) =
π
_{00}
(
T
), where
π
_{00}
(
T
) := {
λ
∈ iso
σ
(
T
) : 0 < dim
N
(
T
−
λ
) < ∞}.
More generally, Berkani investigated
B
Fredholm theory (see
[4
,
5
,
6]
). An operator
T
is called
B
Fredholm if there exists
n
∈ ℕ such that
R
(
T^{n}
) is closed and the induced operator
T
_{[n]}
:
R
(
T^{n}
) ∋
x
→
Tx
∈
R
(
T^{n}
) is Fredholm, i.e.,
R
(
T
_{[n]}
) =
R
(
T
^{n+1}
) is closed,
α
(
T
_{[n]}
) < ∞ and
β
(
T
_{[n]}
) = dim
R
(
T^{n}
)/
R
(
T
_{[n]}
) < ∞. Similarly, a
B
Fredholm operator
T
is called
B
Weyl if
i
(
T
_{[n]}
) = 0. The
B
Weyl spectrum
σ_{BW}
(
T
) is defined by
σ_{BW}
(
T
) = {
λ
∈ ℂ :
T
−
λ
is not
B
Weyl}. We say that generalized Weyl’s theorem holds for
T
if
σ
(
T
) \
σ
_{BW}
(
T
) =
E
(
T
), where
E
(
T
) denotes the set of all isolated points of the spectrum which are eigenvalues (no restriction on multiplicity). Note that, if generalized Weyl’s theorem holds for
T
, then so does Weyl’s theorem
[5]
.
In
[1]
Agler obtained certain disconjugacy and SturmLioville results for a subclass of the Toeplitz operators. These results were suggested by the study of operators
T
which satisfies the equation,
Such
T
are natural generalizations of isometric operators (
T
^{∗}
T
=
I
) and are called 2isometric operators. It is known that an isometric operator is a 2isometric operator. 2isometric operators have been studied by many authors and they have many interesting properties (see
[2
,
3
,
7
,
8
,
9
,
14]
).
In order to extend 2isometric operators we introduce
k
quasi2isometric operators defined as follows:
Definition 1.1.
For a positive integer
k
, an operator
T
is said to be a
k

quasi
2
isometric operator
if
It is clear that each 2isometric operator is a
k
quasi2isometric operator and each
k
quasi2isometric operator is a (
k
+ 1)quasi2isometric operator.
In this paper we give a necessary and sufficient condition for
T
to be a
k
quasi2isometric operator. Moreover, we study characterizations of weighted shift operators which are
k
quasi2isometric operators. Finally, we prove polynomially
k
quasi2isometric operators satisfy generalized Weyl’s theorem.
2. MAIN RESULTS
We begin with the following theorem which is the essence of this paper; it is a structure theorem for
k
quasi2isometric operators.
Theorem 2.1.
If
T^{k}
does not have a dense range, then the following statements are equivalent:
(1)
T is a kquasi2isometric operator;
(2)
on
, where
T
_{1}
is a
2
isometric operator and
Furthermore
,
σ
(
T
) =
σ
(
T
_{1}
) ∪ {0}.
Proof
. (1) ⇒ (2) Consider the matrix representation of
T
with respect to the decomposition
:
Let
P
be the projection onto
. Since
T
is a
k
quasi2isometric operator, we have
Therefore
On the other hand, for any
x
= (
x
_{1}
,
x
_{2}
) ∈
H
, we have
which implies
Since
σ
(
T
)∪
M
=
σ
(
T
_{1}
)∪
σ
(
T
_{3}
), where
M
is the union of the holes in
σ
(
T
) which happen to be subset of
σ
(
T
_{1}
) ∩
σ
(
T
_{3}
) by Corollary 7 of
[11]
, and
σ
(
T
_{1}
) ∩
σ
(
T
_{3}
) has no interior point and
T
_{3}
is nilpotent, we have
σ
(
T
) =
σ
(
T
_{1}
) ∪ {0}.
(2) ⇒ (1) Suppose that
on
, where
and
. Since
we have
where
. It follows that
T
^{∗k}
(
T
^{∗2}
T
^{2}
− 2
T
^{∗}
T
+
I
)
T^{k}
= 0 on
. Thus
T
is a
k
quasi2isometric operator. ☐
Corollary 2.2.
If
is a kquasi
2
isometric operator and
T
_{1}
is invertible, then T is similar to a direct sum of a
2
isometric operator and a nilpotent operator.
Proof
. Since
T
_{1}
is invertible, we have
σ
(
T
_{1}
) ∩
σ
(
T
_{3}
) =
φ
. Then there exists an operator
S
such that
T
_{1}
S
−
S
T
_{3}
=
T
_{2}
[15]
. Since
, it follows that
Corollary 2.3.
If T is a kquasi
2
isometric operator and
R
(
T^{k}
)
is dense
,
then T is a
2
isometric operator.
Proof.
This is a result of Theorem 2.1. ☐
Corollary 2.4.
If T is a kquasi
2
isometric operator, then so is T^{n} for every natural number n.
Proof.
We decompose
T
as
Then by Theorem 2.1,
. Hence
T
_{1}
is a 2isometric operator, by
[14
, Theorem 2.1],
is a 2isometric operator. Since
T^{n}
is a
k
quasi2isometric operator for every natural number
n
by Theorem 2.1. ☐
Lemma 2.5.
T
is a kquasi
2
isometric operator if and only if
for every x
∈
H
.
Theorem 2.6.
Let T be a kquasi
2
isometric operator and M be an invariant subspace for T. Then the restriction T

_{M} is also a kquasi
2
isometric operator.
Proof.
For
x
∈
M
, we have
Thus
T

_{M}
is a
k
quasi2isometric operator. ☐
Example 2.7.
Given a bounded sequence
α
:
α
_{0}
,
α
_{1}
,
α
_{2}
, . . . (called weights), the unilateral weighted shift
W
_{α}
associated with
α
is the operator on
l
_{2}
defined by
W
_{α}
e
_{n}
=
α_{n}e
_{n+1}
for all
n
≥ 0, where
is the canonical orthogonal basis for
l
_{2}
and 
α_{n}
 ≠ 0 for each
n
≥ 0. Then the following statement holds:
W_{α}
is a
k
quasi2isometric operator if and only if
where
Proof
. By calculation,
, by definition,
W_{α}
is a
k
quasi2isometric operator if and only if
☐
Remark 2.8.
Let
W_{α}
be the unilateral weighted shift with weight sequence (
α_{n}
)
_{n}
_{≥0}
and 
α_{n}
 ≠ 0 for each
n
≥ 0. From Example 2.7 we obtain the following characterizations:
1.
W_{α}
is a
k
quasi2isometric operator if and only if
for
n
≥
k
.
2. {
α_{n}
} is a decreasing sequence of real numbers converging to 1 for
n
≥
k
.
3.
for
n
≥
k
+ 1.
4. Let 2 = 
α_{k}
, 1 = 
α
_{k+1}
 = 
α
_{k+2}
 = 
α
_{k+3}
 = ⋯. Then
W_{α}
is a (
k
+ 1)quasi2isometric operator but not a
k
quasi2isometric operator.
In the sequel, we focus on polynomially
k
quasi2isometric operators.
We say that
T
is a polynomially
k
quasi2isometric operator if there exists a nonconstant complex polynomial
p
such that
p
(
T
) is a
k
quasi2isometric operator. It is clear that a
k
quasi2isometric operator is a polynomially
k
quasi2isometric operator. The following example provides an operator which is a polynomially
k
quasi2isometric operator but not a
k
quasi2isometric operator.
Example 2.9.
Let
Then T is a polynomially kquasi
2
isometric operator but not a kquasi
2
isometric operator.
Proof.
Since
we have
Then
Therefore
T
is not a
k
quasi2isometric operator.
On the other hand, consider the complex polynomial
h
(
z
) = (
z
− 1)
^{2}
+ 1. Then
h
(
T
) =
I
, and hence
T
is a polynomially
k
quasi2isometric operator. ☐
Recall that an operator
T
is said to be isoloid if every isolated point of
σ
(
T
) is an eigenvalue of
T
and polaroid if every isolated point of
σ
(
T
) is a pole of the resolvent of
T
. In general, if
T
is polaroid, then it is isoloid. However, the converse is not true.
Theorem 2.10.
Let T be a polynomially kquasi
2
isometric operator. Then T is polaroid.
Proof
. We first show that a
k
quasi2isometric operator is polaroid. We consider the following two cases: Case I: If the range of
T^{k}
is dense, then
T
is a 2isometric operator,
T
is polaroid. Since an invertible 2isometric operator is a unitary operator by
[2
, Proposition 1.23], and if
T
is a noninvertible 2isometric operator, then iso
σ
(
T
) is empty.
Case II: If the range of
T^{k}
is not dense, by Theorem 2.1, we have
Let
λ
∈ iso
σ
(
T
). Suppose that
T
_{1}
is a noninvertible 2isometric operator. Then
σ
(
T
) =
D
, where
D
is the closed unit disk. Since
σ
(
T
) =
σ
(
T
_{1}
) ∪ {0}, we have iso
σ
(
T
) is empty; thus
T
_{1}
is a invertible 2isometric operator and
λ
∈ iso
σ
(
T
_{1}
) or
λ
= 0,
T
_{1}
is a unitary operator,
T
_{3}
is nilpotent. It is easy to prove that
T
−
λ
has finite ascent and descent, i.e.,
λ
is a pole of the resolvent of
T
, therefore
T
is polaroid.
Next we show that a polynomially
k
quasi2isometric operator is polaroid. If
T
is a polynomially
k
quasi2isometric operator, then
p
(
T
) is a
k
quasi2isometric operator for some nonconstant polynomial
p
. Hence it follows from the first part of the proof that
p
(
T
) is polaroid. Now apply
[10
, Lemma 3.3] to conclude that
p
(
T
) polaroid implies
T
polaroid. ☐
Corollary 2.11.
Let T be a polynomially kquasi
2
isometric operator. Then T is isoloid.
An operator
T
is said to has the single valued extension property (abbreviated SVEP) if, for every open subset
G
of ℂ, any analytic function
f
:
G
→
H
such that (
T
−
z
)
f
(
z
) ≡ 0 on
G
, we have
f
(
z
) ≡ 0 on
G
.
Theorem 2.12.
Let T be a polynomially kquasi
2
isometric operator. Then T has SVEP.
Proof
. We first suppose that
T
is a
k
quasi2isometric operator. We consider the following two cases:
Case I: If the range of
T^{k}
is dense, then
T
is a 2isometric operator,
T
has SVEP by
[8
, Theorem 2].
Case II: If the range of
T^{k}
is not dense, by Theorem 2.1, we have
Suppose (
T
−
z
)
f
(
z
) = 0,
f
(
z
) =
f
_{1}
(
z
) ⊕
f
_{2}
(
z
) on
. Then we can write
And
T
_{3}
is nilpotent,
T
_{3}
has SVEP, hence
f
_{2}
(
z
) = 0, (
T
_{1}
−
z
)
f
_{1}
(
z
) = 0. Since
T
_{1}
is a 2isometric operator,
T
_{1}
has SVEP by
[8
, Theorem 2], then
f
_{1}
(
z
) = 0. Consequently,
T
has SVEP.
Now suppose that
T
is a polynomially
k
quasi2isometric operator. Then
p
(
T
) is a
k
quasi2isometric operator for some nonconstant complex polynomial
p
, and hence
p
(
T
) has SVEP. Therefore,
T
has SVEP by
[13
, Theorem 3.3.9]. ☐
Since the SVEP for
T
entails that generalized Browder’s theorem holds for
T
, i.e.
σ_{BW}
(
T
) =
σ
_{D}
(
T
), where
σ_{D}
(
T
) denotes the Drazin spectrum, a sufficient condition for an operator
T
satisfying generalized Browder’s theorem to satisfy generalized Weyl’s theorem is that
T
is polaroid. In
[14]
, Patel showed that Weyl’s theorem holds for 2isometric operator. Then we have the following result:
Theorem 2.13.
If T is a polynomially kquasi
2
isometric operator, then generalized Weyl’s theorem holds for T, so does Weyl’s theorem.
Proof.
It is obvious from Theorem 2.10, Theorem 2.12 and the statements of the above. ☐
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