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SPECTRAL PROPERTIES OF k-QUASI-2-ISOMETRIC OPERATORS
SPECTRAL PROPERTIES OF k-QUASI-2-ISOMETRIC OPERATORS
The Pure and Applied Mathematics. 2015. Aug, 22(3): 275-283
Copyright © 2015, Korean Society of Mathematical Education
  • Received : July 04, 2015
  • Published : August 31, 2015
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About the Authors
Junli Shen
College of Computer and Information Technology, Henan Normal University, Xinxiang 453007, ChinaEmail address:shenjunli08@126.com
Fei Zuo
Henan Engineering Laboratory for Big Data Statistical Analysis and Optimal Control, School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, ChinaEmail address:zuofei2008@sina.com

Abstract
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 -quasi-2-isometric operator if T k ( T ∗2 T 2 − 2 T T + I ) Tk = 0, which is a generalization of 2-isometric operator. In this paper, we consider basic structural properties of k -quasi-2-isometric operators. Moreover, we give some examples of k -quasi-2-isometric operators. Finally, we prove that generalized Weyl’s theorem holds for polynomially k -quasi-2-isometric operators.
Keywords
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 ( Tn ) is closed and the induced operator T [n] : R ( Tn ) ∋ x Tx R ( Tn ) is Fredholm, i.e., R ( T [n] ) = R ( T n+1 ) is closed, α ( T [n] ) < ∞ and β ( T [n] ) = dim R ( Tn )/ R ( T [n] ) < ∞. Similarly, a B -Fredholm operator T is called B -Weyl if i ( T [n] ) = 0. The B -Weyl spectrum σBW ( T ) is defined by σBW ( 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 Sturm-Lioville results for a subclass of the Toeplitz operators. These results were suggested by the study of operators T which satisfies the equation,
  • T∗2T2− 2T∗T+I= 0.
Such T are natural generalizations of isometric operators ( T T = I ) and are called 2-isometric operators. It is known that an isometric operator is a 2-isometric operator. 2-isometric operators have been studied by many authors and they have many interesting properties (see [2 , 3 , 7 , 8 , 9 , 14] ).
In order to extend 2-isometric operators we introduce k -quasi-2-isometric 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
  • T∗k(T∗2T2− 2T∗T+I)Tk= 0.
It is clear that each 2-isometric operator is a k -quasi-2-isometric operator and each k -quasi-2-isometric operator is a ( k + 1)-quasi-2-isometric operator.
In this paper we give a necessary and sufficient condition for T to be a k -quasi-2-isometric operator. Moreover, we study characterizations of weighted shift operators which are k -quasi-2-isometric operators. Finally, we prove polynomially k -quasi-2-isometric 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 -quasi-2-isometric operators.
Theorem 2.1. If Tk does not have a dense range, then the following statements are equivalent:
(1) T is a k-quasi-2-isometric operator;
(2)
PPT Slide
Lager Image
on
PPT Slide
Lager Image
, where T 1 is a 2- isometric operator and
PPT Slide
Lager Image
Furthermore , σ ( T ) = σ ( T 1 ) ∪ {0}.
Proof . (1) ⇒ (2) Consider the matrix representation of T with respect to the decomposition
PPT Slide
Lager Image
:
PPT Slide
Lager Image
Let P be the projection onto
PPT Slide
Lager Image
. Since T is a k -quasi-2-isometric operator, we have
PPT Slide
Lager Image
Therefore
PPT Slide
Lager Image
On the other hand, for any x = ( x 1 , x 2 ) ∈ H , we have
PPT Slide
Lager Image
which implies
PPT Slide
Lager Image
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
PPT Slide
Lager Image
on
PPT Slide
Lager Image
, where
PPT Slide
Lager Image
and
PPT Slide
Lager Image
. Since
PPT Slide
Lager Image
we have
PPT Slide
Lager Image
where
PPT Slide
Lager Image
. It follows that T k ( T ∗2 T 2 − 2 T T + I ) Tk = 0 on
PPT Slide
Lager Image
. Thus T is a k -quasi-2-isometric operator.               ☐
Corollary 2.2. If
PPT Slide
Lager Image
is a k-quasi- 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
PPT Slide
Lager Image
, it follows that
  •               ☐
Corollary 2.3. If T is a k-quasi -2- isometric operator and R ( Tk ) 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 k-quasi- 2- isometric operator, then so is Tn for every natural number n.
Proof. We decompose T as
  • on.
Then by Theorem 2.1,
PPT Slide
Lager Image
. Hence T 1 is a 2-isometric operator, by [14 , Theorem 2.1],
PPT Slide
Lager Image
is a 2-isometric operator. Since
Tn is a k -quasi-2-isometric operator for every natural number n by Theorem 2.1.               ☐
Lemma 2.5. T is a k-quasi- 2 -isometric operator if and only if
for every x H .
Theorem 2.6. Let T be a k-quasi -2- isometric operator and M be an invariant subspace for T. Then the restriction T | M is also a k-quasi -2- isometric operator.
Proof. For x M , we have
Thus T | M is a k -quasi-2-isometric 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 = αne n+1 for all n ≥ 0, where
PPT Slide
Lager Image
is the canonical orthogonal basis for l 2 and | αn | ≠ 0 for each n ≥ 0. Then the following statement holds: Wα is a k -quasi-2-isometric operator if and only if
where
Proof . By calculation,
PPT Slide
Lager Image
, by definition, Wα is a k -quasi-2-isometric operator if and only if
PPT Slide
Lager Image
              ☐
Remark 2.8. Let Wα be the unilateral weighted shift with weight sequence ( αn ) n ≥0 and | αn | ≠ 0 for each n ≥ 0. From Example 2.7 we obtain the following characterizations:
1. Wα is a k -quasi-2-isometric operator if and only if
for n k .
2. {| αn |} is a decreasing sequence of real numbers converging to 1 for n k .
3.
PPT Slide
Lager Image
for n k + 1.
4. Let 2 = | αk |, 1 = | α k+1 | = | α k+2 | = | α k+3 | = ⋯. Then Wα is a ( k + 1)-quasi-2-isometric operator but not a k -quasi-2-isometric operator.
In the sequel, we focus on polynomially k -quasi-2-isometric operators.
We say that T is a polynomially k -quasi-2-isometric operator if there exists a nonconstant complex polynomial p such that p ( T ) is a k -quasi-2-isometric operator. It is clear that a k -quasi-2-isometric operator is a polynomially k -quasi-2-isometric operator. The following example provides an operator which is a polynomially k -quasi-2-isometric operator but not a k -quasi-2-isometric operator.
Example 2.9. Let
PPT Slide
Lager Image
Then T is a polynomially k-quasi- 2- isometric operator but not a k-quasi -2- isometric operator.
Proof. Since
we have
Then
Therefore T is not a k -quasi-2-isometric 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 -quasi-2-isometric 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 k-quasi -2 -isometric operator. Then T is polaroid.
Proof . We first show that a k -quasi-2-isometric operator is polaroid. We consider the following two cases: Case I: If the range of Tk is dense, then T is a 2-isometric operator, T is polaroid. Since an invertible 2-isometric operator is a unitary operator by [2 , Proposition 1.23], and if T is a non-invertible 2-isometric operator, then iso σ ( T ) is empty.
Case II: If the range of Tk is not dense, by Theorem 2.1, we have
Let λ ∈ iso σ ( T ). Suppose that T 1 is a non-invertible 2-isometric 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 2-isometric 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 -quasi-2-isometric operator is polaroid. If T is a polynomially k -quasi-2-isometric operator, then p ( T ) is a k -quasi-2-isometric 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 k-quasi- 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 k-quasi -2 -isometric operator. Then T has SVEP.
Proof . We first suppose that T is a k -quasi-2-isometric operator. We consider the following two cases:
Case I: If the range of Tk is dense, then T is a 2-isometric operator, T has SVEP by [8 , Theorem 2].
Case II: If the range of Tk is not dense, by Theorem 2.1, we have
Suppose ( T z ) f ( z ) = 0, f ( z ) = f 1 ( z ) ⊕ f 2 ( z ) on
PPT Slide
Lager Image
. 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 2-isometric 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 -quasi-2-isometric operator. Then p ( T ) is a k -quasi-2-isometric 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. σBW ( T ) = σ D ( T ), where σD ( 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 2-isometric operator. Then we have the following result:
Theorem 2.13. If T is a polynomially k-quasi -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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