SPECTRAL PROPERTIES OF k-QUASI-2-ISOMETRIC OPERATORS

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^∗2T² − 2T^∗T + I)T^k = 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.

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), σa(T) and isoσ(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) = π00(T), where π00(T) := {λ ∈ isoσ(T) : 0 < dimN(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(Tn+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) \ σBW (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) on , where T1 is a 2-isometric operator and Furthermore, σ(T) = σ(T1) ∪ {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-quasi-2-isometric operator, we have

Therefore

On the other hand, for any x = (x1, x2) ∈ H, we have

which implies

Since σ(T)∪M = σ(T1)∪σ(T3), where M is the union of the holes in σ(T) which happen to be subset of σ(T1) ∩ σ(T3) by Corollary 7 of [11], and σ(T1) ∩ σ(T3) has no interior point and T3 is nilpotent, we have σ(T) = σ(T1) ∪ {0}.

(2) ⇒ (1) Suppose that on , where and . Since

we have

where . It follows that T∗k(T∗2T2 − 2T∗T + I)Tk = 0 on . Thus T is a k-quasi-2-isometric operator.               ☐

Corollary 2.2. If is a k-quasi-2-isometric operator and T1 is invertible, then T is similar to a direct sum of a 2-isometric operator and a nilpotent operator.

Proof. Since T1 is invertible, we have σ(T1) ∩ σ(T3) = φ. Then there exists an operator S such that T1S − ST3 = T2 [15]. Since , 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, . Hence T1 is a 2-isometric operator, by [14, Theorem 2.1], 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 : α0, α1, α2, . . . (called weights), the unilateral weighted shift Wα associated with α is the operator on l2 defined by Wαen = αnen+1 for all n ≥ 0, where is the canonical orthogonal basis for l2 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, , by definition, Wα is a k-quasi-2-isometric operator if and only if               ☐

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. for n ≥ k + 1.

4. Let 2 = |αk|, 1 = |αk+1| = |αk+2| = |αk+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 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 isoσ(T) is empty.

Case II: If the range of Tk is not dense, by Theorem 2.1, we have

Let λ ∈ isoσ(T). Suppose that T1 is a non-invertible 2-isometric operator. Then σ(T) = D, where D is the closed unit disk. Since σ(T) = σ(T1) ∪ {0}, we have isoσ(T) is empty; thus T1 is a invertible 2-isometric operator and λ ∈ isoσ(T1) or λ = 0, T1 is a unitary operator, T3 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) = f1(z) ⊕ f2(z) on . Then we can write

And T3 is nilpotent, T3 has SVEP, hence f2(z) = 0, (T1 − z)f1(z) = 0. Since T1 is a 2-isometric operator, T1 has SVEP by [8, Theorem 2], then f1(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) = σD(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.               ☐