Essentially 𝜆-Hankel Operators

Abstract

The notion of essentially 𝜆-Hankel operators is introduced on the space 𝐻². In addition to the discussion of some algebraic and topological properties of the set essHank_𝜆 , the set of all essentially 𝜆-Hankel operators on 𝐻², it is shown that an essentially Toeplitz Rhaly operator with determining sequence ⟨𝑎_𝑛⟩ is in essHank_𝜆(𝜆≠0) if and only if lim_𝑛→∞(𝑛+1)|𝑎_𝑛|=0.

1. Introduction

The notion of Toeplitz operators was introduced by Toeplitz [1] in the year 1911. Hankel operators are the formal companions of Toeplitz operators. It is well known that Toeplitz and Hankel operators are characterized as solutions to the operator equations 𝑆^∗𝑇𝑆=𝑇 and 𝑆^∗𝐻=𝐻𝑆, respectively, where 𝑆 denotes the unilateral forward shift on 𝐻². The solutions of the operator equation 𝑆^∗𝑋𝑆=𝜆𝑋 (for an arbitrary complex number 𝜆) were described by Sun in the year 1984 [2]. In the year 2002, Avendaño [3] introduced the notion of 𝜆-Hankel operators as those operators 𝑋 which satisfy the operator equation 𝑆^∗𝑋−𝑋𝑆=𝜆𝑋. In a different direction, Avendaño [4] studied the notion of Hankel operators in reference to the Calkin algebra ℬ(𝐻²)/𝒦(𝐻²) and introduced the notion of essentially Hankel operators on 𝐻². Motivated by these developments, in this paper, we introduce and study the notion of essentially 𝜆-Hankel operators on the space 𝐻².

For a fixed complex number 𝜆, we denote the set of all essentially 𝜆-Hankel operators on 𝐻² by essHank_𝜆. The set essHank_𝜆 is shown to be a norm-closed vector subspace of ℬ(𝐻²) containing no essentially invertible operator. It is shown that for a general 𝜆, essHank_𝜆 is neither an algebra of operators on 𝐻² nor a self-adjoint set. Although the set essHank_𝜆 contains no nonzero Toeplitz operators, it turns out to be invariant under multiplication by Toeplitz operators. It turns out that essToep∩essHank_𝜆 is an algebra without identity, where essToep denotes the class of all essentially Toeplitz operators on 𝐻². In particular, for purely imaginary 𝜆, essToep∩essHank_𝜆 is a 𝐶^∗-algebra. We also show that if 𝜆≠0, essToep∩essHank_𝜆 contains no noncompact Rhaly operators.

We begin with the following

Definition 1 (see [5]). A bounded linear operator 𝑇 on 𝐻² is said to be an essentially Toeplitz operator if it satisfies 𝑆^∗𝑇𝑆−𝑇=𝐾, (1) for some compact operator 𝐾 on 𝐻². The set of all essentially Toeplitz operators is denoted by essToep.

Definition 2 (see [4]). A bounded linear operator 𝑋 on 𝐻² is said to be an essentially Hankel operator if it satisfies 𝑆^∗𝑋−𝑋𝑆=𝐾, (2) for some compact operator 𝐾 on 𝐻². The set of all essentially Hankel operators on 𝐻² is denoted by essHank. For more details, one can refer to [4].

Definition 3 (see [3]). 𝑋 is said to be a 𝜆-Hankel operator if it satisfies 𝑆^∗𝑋−𝑋𝑆=𝜆𝑋.
Clearly, a 0-Hankel operator is just a Hankel operator.

Definition 4 (see [6]). Given a sequence 𝑎=⟨𝑎_𝑛⟩ of scalars, the Rhaly matrix (terraced matrix) 𝑅_𝑎 [6] is defined as 𝑅_𝑎=𝑎₁𝑎000⋯₂𝑎₂𝑎00⋯₃𝑎₃𝑎₃.0⋯⋮⋮⋮⋮⋯ (3) This means that Rhaly matrices are lower triangular matrices with constant row segments. It is known that [6] if {𝑛𝑎_𝑛} is bounded, then the Rhaly matrix 𝑅_𝑎 represents a bounded linear operator on the space ℓ² (identified with 𝐻²) and ‖𝑅_𝑎‖≤2sup_𝑛∈ℕ|𝑛𝑎_𝑛|.

We now introduce the notion of essentially 𝜆-Hankel operators on the space 𝐻² as follows

Definition 5. For a fixed complex number 𝜆, a bounded linear operator 𝑋 on 𝐻² is said to be an essentially 𝜆-Hankel operator if it satisfies the operator equation 𝑆^∗−𝜆𝐼𝑋−𝑋𝑆=𝐾, (4) for some compact operator 𝐾 on 𝐻², 𝑆 denoting the unilateral forward shift on 𝐻².
We denote the set of all essentially 𝜆-Hankel operators on 𝐻² by essHank_𝜆. Some basic facts and observations about essHank_𝜆 which follow from the definition itself are as follows: (i)essHank_𝜆∩𝒦(𝐻²)=𝒦(𝐻²), where 𝒦(𝐻²) denotes the ideal of all compact operators on 𝐻²,(ii)since the zero operator on 𝐻² is compact, every 𝜆-Hankel operator is in essHank_𝜆,(iii) if 𝜆=0, then essHank_𝜆=essHank₀=essHank, where essHank is the set of all essentially Hankel operators on 𝐻² introduced by Avendaño [4],(iv)if 𝐻 is a Hankel operator, then 𝑆^∗𝐻=𝐻𝑆. Therefore, every Hankel operator is in essHank₀.(v)we see that the operator 𝑆^∗−𝑆−𝜆𝐼 cannot be compact on 𝐻², for any complex number 𝜆. Therefore 𝐼∉essHank_𝜆, for any 𝜆, where 𝐼 denotes the identity operator on 𝐻²,(vi)for 𝑋₁,𝑋₂∈essHank_𝜆, 𝑋₁𝑋₂∈essHank_𝜆 if and only if 𝑋₁𝑆^∗−𝑆−𝜆𝐼𝑋₂∈𝒦𝐻², (5)(vii)from the definition itself, it is clear that if 𝑋 is a 𝜆-Hankel operator and 𝐾 is a compact operator on 𝐻², then 𝑋+𝐾 is in essHank_𝜆. That is, compact perturbations of 𝜆-Hankel operators are in essHank_𝜆. It was shown by Avendaño that the reverse implication is not true for the case 𝜆=0. Avendaño [4] proved that the Cesaro operator (i.e., the Rhaly operator corresponding to the sequence ⟨1/𝑛⟩) whose matrix with respect to the standard orthonormal basis is given by 11000…212100…313130……………… (6) is in essHank₀ but is not expressible in 0-Hankel plus compact form,(viii)essHank_𝜆 is a norm-closed vector subspace of ℬ(𝐻²), the space of all bounded linear operators on 𝐻².

Proof. For 𝛼,𝛽 in ℂ and 𝑋₁,𝑋₂∈essHank_𝜆, we have 𝑆^∗−𝜆𝐼𝛼𝑋₁+𝛽𝑋₂−𝛼𝑋₁+𝛽𝑋₂𝑆=𝛼𝑆^∗−𝜆𝐼𝑋₁−𝑋₁𝑆+𝛽𝑆^∗−𝜆𝐼𝑋₂−𝑋₂𝑆=𝛼𝐾₁+𝛽𝐾₂, (7) where 𝐾₁,𝐾₂∈𝒦(𝐻²). Therefore, 𝛼𝑋₁+𝛽𝑋₂∈essHank_𝜆. Also, if ⟨𝑋_𝑛⟩→𝑋 in ℬ(𝐻²), where each 𝑋_𝑛 is in essHank_𝜆, then ⟨(𝑆^∗−𝜆𝐼)𝑋_𝑛−𝑋_𝑛𝑆⟩→(𝑆^∗−𝜆𝐼)𝑋−𝑋𝑆 in ℬ(𝐻²). As each 𝑋_𝑛∈essHank_𝜆 and 𝒦(𝐻²) is a uniformly closed subspace of ℬ(𝐻²), it follows that (𝑆^∗−𝜆𝐼)𝑋−𝑋𝑆∈𝒦(𝐻²). Thus, 𝑋∈essHank_𝜆. Hence, the conclusionis clear.

We see that if 𝑋∈essHank₀, then 𝑆^∗𝑋−𝑋𝑆=𝐾,where𝐾∈𝒦𝐻². (8) On taking adjoints on both the sides, we get 𝑆^∗𝑋^∗−𝑋^∗𝑆=−𝐾^∗,where𝐾^∗∈𝒦𝐻². (9) Therefore, 𝑋^∗∈essHank₀.

Thus, essHank₀ is a self-adjoint set. We show that for a general complex number 𝜆, this is not the case. That is, essHank_𝜆 is not a self-adjoint set in general. For this we begin with the following.

Theorem 6. If 𝜆 and 𝜇 are distinct complex numbers, then essHank_𝜆∩essHank_𝜇=𝒦𝐻². (10)

Proof. Let 𝑋∈essHank_𝜆∩essHank_𝜇. Then, 𝑆^∗−𝜆𝐼𝑋−𝑋𝑆=𝐾₁,𝑆^∗−𝜇𝐼𝑋−𝑋𝑆=𝐾₂, (11) where 𝐾₁,𝐾₂∈𝒦(𝐻²). Subtracting the previous two equations we have 𝜆−𝜇𝑋∈𝒦𝐻², (12) where 𝜆−𝜇≠0. This implies that 𝑋∈𝒦(𝐻²). Therefore, essHank_𝜆∩essHank_𝜇⊆𝒦(𝐻²). Reverse inclusion is obvious by the definition.

Theorem 7. If 𝑋∈essHank_𝜆, then 𝑋^∗∈essHank_𝜇, where 𝜇=−𝜆.

Proof. Let 𝑋∈essHank_𝜆. Then, 𝑆^∗−𝜆𝐼𝑋−𝑋𝑆=𝐾, (13) for some compact operator 𝐾 on 𝐻². Taking adjoints on both sides of (13), we obtain 𝑋^∗𝑆−𝜆𝐼−𝑆^∗𝑋^∗=𝐾^∗, (14) where 𝐾^∗∈𝒦(𝐻²). Thus, [𝑆^∗−(−𝜆)𝐼]𝑋^∗−𝑋^∗𝑆∈𝒦(𝐻²). This means that 𝑋^∗∈essHank_𝜇, where 𝜇=−𝜆.

Corollary 8. A necessary condition for a noncompact operator 𝑋 in essHank_𝜆, 𝜆≠0, to be self-adjoint is that 𝜆 is purely imaginary.

Proof. Let 𝑋∈essHank_𝜆, where 𝑋 is a noncompact operator on 𝐻². Then, 𝑋^∗∈essHank_𝜇, where 𝜇=−𝜆. Now, if 𝑋=𝑋^∗, then 𝑋,𝑋^∗∈essHank_𝜆∩essHank_𝜇. As 𝑋 is non-compact, we have 𝜆=𝜇. That is, 𝜆+𝜆=0. That is, 𝜆 is purely imaginary.

Theorem 9. Let 𝑋∈essHank_𝜆. Then, 0∈𝜎_𝑒(𝑋), where 𝜎_𝑒(𝑋) denotes the essential spectrum of the operator 𝑋.

Proof. Let 𝑋∈essHank_𝜆. Then, 𝑆^∗−𝜆𝐼𝑋−𝑋𝑆=𝐾, (15) where 𝐾 is a compact operator on 𝐻².
Case (i). 𝜆=0: In this case, 𝑋 satisfies 𝑆^∗𝑋−𝑋𝑆=𝐾, (16) where 𝐾∈𝒦(𝐻²). Clearly, 𝑋 cannot be Fredholm, for if 𝑋 is a Fredholm operator of index 𝑛, then 𝑆^∗𝑋 and 𝑋𝑆+𝐾 are Fredholm operators with indices 𝑛+1 and 𝑛−1, respectively, leading to 1=−1, which is absurd. Therefore, 0∈𝜎_𝑒(𝑋).
Case (ii). 𝜆≠0: In this case, if 𝑋 is essentially invertible, then (𝑆^∗−𝜆𝐼)−𝑋𝑆𝑋⁻¹ is a compact operator on 𝐻². This leads to the essential similarly of 𝑆^∗−𝜆𝐼 and 𝑆, which is a contradiction as 𝜎_𝑒(𝑆)=𝜎_𝑒(𝑆^∗)=𝕋, where 𝕋 denotes the unit circle in the complex plane. So, 0∈𝜎_𝑒(𝑋) in this case also.

In the next theorem, we show that there is no nonzero Toeplitz operator in essHank_𝜆. For this we need the following lemmas.

Lemma 10. A nonzero Toeplitz operator cannot be a 𝜆-Hankel operator.

Proof. Let 𝑇 be a nonzero Toeplitz operator. Then, 𝑆^∗𝑇𝑆=𝑇. (17) If possible, suppose that 𝑇 is 𝜆-Hankel also. Then, 𝑆^∗−𝜆𝐼𝑇=𝑇𝑆. (18) From (17) and (18), it follows that (𝜆𝑇+𝑇𝑆)𝑆=𝑇. (19) That is, 𝑇𝑒_𝑛+2=𝑇𝑒_𝑛−𝜆𝑇𝑒_𝑛+1, (20) for all 𝑛≥0. This means that 𝑇 is finite dimensional and hence a compact operator on 𝐻². But nonzero Toeplitz operators are never compact. Thus, we have a contradiction and the conclusion follows.

Lemma 11. If 𝑇 is a nonzero Toeplitz operator, then so is 𝑆^∗𝑇−𝑇𝑆, 𝑆 denoting the unilateral forward shift on 𝐻².

Using the previous two lemmas, we now prove that the set essHank_𝜆 contains no nonzero Toeplitz operator.

Theorem 12. essHank_𝜆∩𝒥={0}, where 𝒥 denotes the set of all Toeplitz operators on 𝐻².

Proof. Let 𝑇∈essHank_𝜆∩𝒥. Then, 𝑆^∗𝑆𝑇𝑆=𝑇,_∗−𝜆𝐼𝑇−𝑇𝑆=𝐾, (21) for some compact operator 𝐾 on 𝐻². Since 𝑇 is a Toeplitz operator, 𝑆^∗𝑇−𝑇𝑆 is also a Toeplitz operator on 𝐻². It follows that (𝑆^∗−𝜆𝐼)𝑇−𝑇𝑆 is a Toeplitz operator on 𝐻². As a nonzero Toeplitz operator cannot be compact, we must have (𝑆^∗−𝜆𝐼)𝑇−𝑇𝑆=0. That is, 𝑇 is a 𝜆-Hankel operator on 𝐻². Now, using Lemma 10, we get that 𝑇=0. Hence, essHank_𝜆∩𝒥={0}. (22)

In the next theorem, we show that the set essHank_𝜆 is invariant under multiplication by Toeplitz operators.

Theorem 13. If 𝑋∈essHank_𝜆 and 𝑇_𝜙 is any Toeplitz operator on 𝐻², then 𝑋𝑇_𝜙 and 𝑇_𝜙𝑋 both are in essHank_𝜆.

Proof. Let 𝑇_𝜙 be a Toeplitz operator on the space 𝐻². Then, we have 𝑆^∗𝑇_𝜙𝑆=𝑇_𝜙. (23) Since 𝑆 is essentially unitary, we have the commutator of 𝑇_𝜙 with 𝑆, and that with 𝑆^∗ is compact on 𝐻². Now, let 𝑋∈essHank_𝜆. Then, (i)consider 𝑋𝑇_𝜙𝑆=(𝑋𝑆)𝑇_𝜙mod𝒦𝐻²=𝑆^∗−𝜆𝐼𝑋𝑇_𝜙mod𝒦𝐻². (24) Therefore, 𝑋𝑇_𝜙𝑆−𝑆^∗−𝜆𝐼𝑋𝑇_𝜙∈𝒦𝐻². (25)(ii)Consider 𝑆^∗−𝜆𝐼𝑇_𝜙𝑋=𝑆^∗𝑇_𝜙𝑋−𝜆𝑇_𝜙𝑋=𝑇_𝜙𝑆^∗𝑋−𝜆𝑇_𝜙𝑋mod𝒦𝐻²=𝑇_𝜙(𝜆𝑋+𝑋𝑆)−𝜆𝑇_𝜙𝑋mod𝒦𝐻²=𝑇_𝜙𝑋𝑆mod𝒦𝐻². (26) Therefore, 𝑇_𝜙𝑋𝑆−𝑆^∗−𝜆𝐼𝑇_𝜙𝑋∈𝒦𝐻². (27)Hence, the conclusion is clear.

More generally, we have the following.

Theorem 14. If 𝑋∈essHank_𝜆 and 𝑇∈essToep, then 𝑋𝑇, 𝑇𝑋∈essHank_𝜆.

Proof. Let 𝑋∈essHank_𝜆 and 𝑇∈essToep. Then, 𝑆^∗−𝜆𝐼𝑋−𝑋𝑆∈𝒦𝐻²,𝑆𝑇−𝑇𝑆∈𝒦𝐻² (28) Now, 𝑆^∗)−()𝑆=𝑆−𝜆𝐼(𝑋𝑇𝑋𝑇_∗−𝜆𝐼(𝑋𝑇)−𝑋(𝑆𝑇)mod𝒦𝐻²=𝑆^∗−𝜆𝐼𝑋−𝑋𝑆𝑇mod𝒦𝐻²∈𝒦𝐻². (29) Therefore, 𝑋𝑇∈essHank_𝜆.
Also, 𝑆^∗−𝜆𝐼(𝑇𝑋)−(𝑇𝑋)𝑆=𝑇𝑆^∗𝑋−𝜆𝑇𝑋−𝑇𝑋𝑆mod𝒦𝐻²=𝑇𝑆^∗−𝜆𝐼𝑋−𝑋𝑆mod𝒦𝐻²∈𝒦𝐻². (30) Therefore, 𝑇𝑋∈essHank_𝜆.
We mention here that the previous result was proved for the special case 𝜆=0 by Avendaño [7]. It is easy to see that essToep is a 𝐶^∗-algebra. This fact together with the previous theorem gives us that essToep∩essHank_𝜆 is an algebra of operators on 𝐻² as shown in the following.

Theorem 15. essToep∩essHank_𝜆 is an algebra of operators on 𝐻² with no identity.

Proof. Since essToep is a 𝐶^∗-algebra and essHank_𝜆 is a vector subspace of ℬ(𝐻²), it follows that essToep∩essHank_𝜆 is a vector subspace of ℬ(𝐻²). Also, if 𝑋₁,𝑋₂∈essToep∩essHank_𝜆, then using Theorem 14, we get that 𝑋₁𝑋₂∈essToep∩essHank_𝜆. Thus, essToep∩essHank_𝜆 is an algebra of operators on 𝐻². As 𝐼∉essHank_𝜆, the theorem is proved.

Remark 16. We have seen that if 𝜆 is purely imaginary, then essHank_𝜆 is a self-adjoint set. Therefore, for purely imaginary 𝜆, essToep∩essHank_𝜆 turns to be a 𝐶^∗-algebra.

Theorem 17. If 𝑋∈essHank_𝜆, then 𝑆Ker𝑋⊆Ker(𝑋+𝐾), (31) for some compact operator 𝐾 on 𝐻².

Proof. Let 𝑋∈essHank_𝜆. Then, 𝑆^∗−𝜆𝐼𝑋−𝑋𝑆=𝐾, (32) for some compact operator 𝐾 on 𝐻².
If 𝑓∈Ker𝑋, then 𝑋𝑓=0. From (32) it follows that (𝑋𝑆+𝐾)𝑓=0. That is, (𝑋+𝐾𝑆^∗)𝑆𝑓=0. This implies that (𝑋+𝐾^′)𝑆𝑓=0, where 𝐾^′=𝐾𝑆^∗∈𝒦(𝐻²). Thus, 𝑆𝑓∈Ker(𝑋+𝐾^′), where 𝐾^′∈𝒦(𝐻²). Hence, the result is clear.

Corollary 18. If 𝑋∈essHank_𝜆, then 𝑆Range𝑋^⟂⊆Range(𝑋+𝐾)^⟂, (33) for some compact operator 𝐾 on 𝐻².

Proof. Let 𝑋∈essHank_𝜆. Then, 𝑋^∗∈essHank_𝜇, where 𝜇=−𝜆. Applying Theorem 17 to 𝑋^∗, we get 𝑆Ker𝑋^∗⊆Ker𝑋^∗+𝐾, (34) for some compact operator 𝐾 on 𝐻². This means that 𝑆Ker𝑋^∗⊆Ker𝑋+𝐾^∗∗. (35) Therefore, 𝑆Range𝑋^⟂⊆Range(𝑋+𝐾^∗)^⟂, (36) where 𝐾^∗∈𝒦(𝐻²). So, the conclusion follows.

Remark 19. It is known that [3, 7] kernel of a 𝜆-Hankel operator is an invariant subspace of 𝑆, and closure of the range of a 𝜆-Hankel operator is an invariant subspace of 𝑆^∗. We mention here that these facts about 𝜆-Hankel operators follow as deductions to Theorem 17 and Corollary 18. For if 𝑋 is 𝜆-Hankel then 𝐾=0 in Theorem 17 and Corollary 18 leading to 𝑆(Ker𝑋)⊆Ker𝑋 and 𝑆^∗(Range𝑋)⊆Range𝑋, respectively.

It is known that a Rhaly operator 𝑅_𝑎 is essentially Toeplitz if and only if it is essentially Hankel. That is, 𝑅_𝑎∈essToep if and only if 𝑅_𝑎∈essHank₀. We show that this is not the case if 𝜆≠0. In fact, an essentially Toeplitz Rhaly operator is in essHank_𝜆(𝜆≠0) if and only if it is compact. Precisely, we have the following.

Theorem 20. Let 𝑅_𝑎∈essToep be a Rhaly operator with determining sequence 𝑎=⟨𝑎_𝑛⟩∈ℓ². Then 𝑅_𝑎∈essHank_𝜆(𝜆≠0) if and only if lim_𝑛→∞(𝑛+1)|𝑎_𝑛|=0.

Proof. Let 𝑅_𝑎∈essToep. Then, 𝑅_𝑎∈essHank₀ [4]. Now if 𝑅_𝑎∈essHank_𝜆(𝜆≠0), then we have 𝑅_𝑎∈essHank₀∩essHank_𝜆. As 𝜆≠0, we have 𝑅_𝑎∈𝒦(𝐻²), by Theorem 6. It is known that [8] a Rhaly operator 𝑅_𝑎 with determining sequence 𝑎=⟨𝑎_𝑛⟩ is compact if and only if lim_𝑛→∞(𝑛+1)|𝑎_𝑛|=0. The desired result follows.

Acknowledgment

The support of UGC Research Grant no.8-1(2)/2010(MRP/NRCB) for carrying out the research work is gratefully acknowledged.