# Essentiallyλ-Hankel Operators

Authors: S. C. Arora; Jyoti Bhola
Journal: Journal of Mathematics
 (2013)

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## Abstract

The notion of essentiallyλ-Hankel operators
is introduced on the space H2. In addition to the discussion of some
algebraic and topological properties of the set essHankλ , the set of all
essentially λ-Hankel operators on H2, it is shown that an essentially
Toeplitz Rhaly operator with determining sequence 〈an〉 is in essHankλλ≠0 if and only if
limn→∞n+1an=0.


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## Body

## 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 numberλ, we denote the set of all essentially λ-Hankel operators on H2 by essHankλ. The set essHankλ 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, essHankλ is neither an algebra of operators on H2 nor a self-adjoint set. Although the set essHankλ contains no nonzero Toeplitz operators, it turns out to be invariant under multiplication by Toeplitz operators. It turns out that essToep∩essHankλ is an algebra without identity, where essToep denotes the class of all essentially Toeplitz operators on H2. In particular, for purely imaginary λ, essToep∩essHankλ is a C*-algebra. We also show that if λ≠0, essToep∩essHankλ contains no noncompact Rhaly operators.We begin with the followingDefinition 1 (see [5]).
A bounded linear operatorT on H2 is said to be an essentially Toeplitz operator if it satisfies
(1)S*TS-T=K,
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 operatorX on H2 is said to be an essentially Hankel operator if it satisfies
(2)S*X-XS=K,
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 sequencea=〈an〉 of scalars, the Rhaly matrix (terraced matrix) Ra [6] is defined as
(3)Ra=[a1000⋯a2a200⋯a3a3a30⋯⋮⋮⋮⋮⋯].
This means that Rhaly matrices are lower triangular matrices with constant row segments. It is known that [6] if {nan} is bounded, then the Rhaly matrix Ra represents a bounded linear operator on the space ℓ2 (identified with H2) and ∥Ra∥≤2supn∈ℕ|nan|.We now introduce the notion of essentiallyλ-Hankel operators on the space H2 as followsDefinition 5.
For a fixed complex numberλ, a bounded linear operator X on H2 is said to be an essentially λ-Hankel operator if it satisfies the operator equation
(4)(S*-λI)X-XS=K,
for some compact operator K on H2, S denoting the unilateral forward shift on H2.
We denote the set of all essentiallyλ-Hankel operators on H2 by essHankλ. Some basic facts and observations about essHankλ which follow from the definition itself are as follows: (i)
essHankλ∩𝒦(H2)=𝒦(H2), where 𝒦(H2) denotes the ideal of all compact operators on H2,(ii)
since the zero operator onH2 is compact, every λ-Hankel operator is in essHankλ,(iii)
ifλ=0, then essHankλ=essHank0=essHank, where essHank is the set of all essentially Hankel operators on H2 introduced by Avendaño [4],(iv)
ifH is a Hankel operator, then S*H=HS. Therefore, every Hankel operator is in essHank0.(v)
we see that the operatorS*-S-λI cannot be compact on H2, for any complex number λ. Therefore I∉essHankλ, for any λ, where I denotes the identity operator on H2,(vi)
forX1,X2∈essHankλ, X1X2∈essHankλ if and only if
(5)X1(S*-S-λI)X2∈𝒦(H2),(vii)
from the definition itself, it is clear that ifX is a λ-Hankel operator and K is a compact operator on H2, then X+K is in essHankλ. That is, compact perturbations of λ-Hankel operators are in essHankλ. 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
(6)[1000…121200…1313130………………]
is in essHank0 but is not expressible in 0-Hankel plus compact form,(viii)
essHankλ is a norm-closed vector subspace of ℬ(H2), the space of all bounded linear operators on H2.Proof.
Forα,β in ℂ and X1,X2∈essHankλ, we have
(7)(S*-λI)(αX1+βX2)-(αX1+βX2)S=α[(S*-λI)X1-X1S]+β[(S*-λI)X2-X2S]=αK1+βK2,
where K1,K2∈𝒦(H2). Therefore, αX1+βX2∈essHankλ. Also, if 〈Xn〉→X in ℬ(H2), where each Xn is in essHankλ, then 〈(S*-λI)Xn-XnS〉→(S*-λI)X-XS in ℬ(H2). As each Xn∈essHankλ and 𝒦(H2) is a uniformly closed subspace of ℬ(H2), it follows that (S*-λI)X-XS∈𝒦(H2). Thus, X∈essHankλ. Hence, the conclusionis clear.We see that ifX∈essHank0, then
(8)S*X-XS=K,whereK∈𝒦(H2).
On taking adjoints on both the sides, we get
(9)S*X*-X*S=-K*,whereK*∈𝒦(H2).
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, essHankλ is not a self-adjoint set in general. For this we begin with the following.Theorem 6.
Ifλ and μ are distinct complex numbers, then
(10)essHankλ∩essHankμ=𝒦(H2).Proof.
LetX∈essHankλ∩essHankμ. Then,
(11)(S*-λI)X-XS=K1,(S*-μI)X-XS=K2,
where K1,K2∈𝒦(H2). Subtracting the previous two equations we have
(12)(λ-μ)X∈𝒦(H2),
where λ-μ≠0. This implies that X∈𝒦(H2). Therefore, essHankλ∩essHankμ⊆𝒦(H2). Reverse inclusion is obvious by the definition.Theorem 7.
IfX∈essHankλ, then X*∈essHankμ, where μ=-λ-.Proof.
LetX∈essHankλ. Then,
(13)(S*-λI)X-XS=K,
for some compact operator K on H2. Taking adjoints on both sides of (13), we obtain
(14)X*(S-λ-I)-S*X*=K*,
where K*∈𝒦(H2). Thus, [S*-(-λ-)I]X*-X*S∈𝒦(H2). This means that X*∈essHankμ, where μ=-λ-.Corollary 8.
A necessary condition for a noncompact operatorX in essHankλ, λ≠0, to be self-adjoint is that λ is purely imaginary.Proof.
LetX∈essHankλ, where X is a noncompact operator on H2. Then, X*∈essHankμ, where μ=-λ-. Now, if X=X*, then X,X*∈essHankλ∩essHankμ. As X is non-compact, we have λ=μ. That is, λ+λ-=0. That is, λ is purely imaginary.Theorem 9.
LetX∈essHankλ. Then, 0∈σe(X), where σe(X) denotes the essential spectrum of the operator X.Proof.
LetX∈essHankλ. Then,
(15)(S*-λI)X-XS=K,
where K is a compact operator on H2.
Case (i). λ=0: In this case, X satisfies
(16)S*X-XS=K,
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∈σe(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 σe(S)=σe(S*)=𝕋, where 𝕋 denotes the unit circle in the complex plane. So, 0∈σe(X) in this case also.In the next theorem, we show that there is no nonzero Toeplitz operator inessHankλ. For this we need the following lemmas.Lemma 10.
A nonzero Toeplitz operator cannot be aλ-Hankel operator.Proof.
LetT be a nonzero Toeplitz operator. Then,
(17)S*TS=T.
If possible, suppose that T is λ-Hankel also. Then,
(18)(S*-λI)T=TS.
From (17) and (18), it follows that
(19)(λT+TS)S=T.
That is,
(20)Ten+2=Ten-λTen+1,
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.
IfT 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 setessHankλ contains no nonzero Toeplitz operator.Theorem 12.
essHank
λ
∩
𝒥
=
{
0
}, where 𝒥 denotes the set of all Toeplitz operators on H2.Proof.
LetT∈essHankλ∩𝒥. Then,
(21)S*TS=T,(S*-λI)T-TS=K,
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,
(22)essHankλ∩𝒥={0}.In the next theorem, we show that the setessHankλ is invariant under multiplication by Toeplitz operators.Theorem 13.
IfX∈essHankλ and Tϕ is any Toeplitz operator on H2, then XTϕ and TϕX both are in essHankλ.Proof.
LetTϕ be a Toeplitz operator on the space H2. Then, we have
(23)S*TϕS=Tϕ.
Since S is essentially unitary, we have the commutator of Tϕ with S, and that with S* is compact on H2. Now, let X∈essHankλ. Then, (i)
consider(24)(XTϕ)S=(XS)Tϕ(mod𝒦(H2))=[(S*-λI)X]Tϕ(mod𝒦(H2)).
Therefore,
(25)(XTϕ)S-(S*-λI)(XTϕ)∈𝒦(H2).(ii)
Consider(26)(S*-λI)(TϕX)=S*TϕX-λTϕX=TϕS*X-λTϕX(mod𝒦(H2))=Tϕ(λX+XS)-λTϕX(mod𝒦(H2))=(TϕX)S(mod𝒦(H2)).
Therefore,
(27)(TϕX)S-(S*-λI)(TϕX)∈𝒦(H2).Hence, the conclusion is clear.More generally, we have the following.Theorem 14.
IfX∈essHankλ and T∈essToep, then XT, TX∈essHankλ.Proof.
LetX∈essHankλ and T∈essToep. Then,
(28)(S*-λI)X-XS∈𝒦(H2),ST-TS∈𝒦(H2)
Now,
(29)(S*-λI)(XT)-(XT)S=(S*-λI)(XT)-X(ST)(mod𝒦(H2))=[(S*-λI)X-XS]T(mod𝒦(H2))∈𝒦(H2).
Therefore, XT∈essHankλ.
Also,(30)(S*-λI)(TX)-(TX)S=TS*X-λTX-TXS(mod𝒦(H2))=T[(S*-λI)X-XS](mod𝒦(H2))∈𝒦(H2).
Therefore, TX∈essHankλ.
We mention here that the previous result was proved for the special caseλ=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∩essHankλ 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.
SinceessToep is a C*-algebra and essHankλ is a vector subspace of ℬ(H2), it follows that essToep∩essHankλ is a vector subspace of ℬ(H2). Also, if X1,X2∈essToep∩essHankλ, then using Theorem 14, we get that X1X2∈essToep∩essHankλ. Thus, essToep∩essHankλ is an algebra of operators on H2. As I∉essHankλ, the theorem is proved.Remark 16.
We have seen that ifλ is purely imaginary, then essHankλ is a self-adjoint set. Therefore, for purely imaginary λ, essToep∩essHankλ turns to be a C*-algebra.Theorem 17.
IfX∈essHankλ, then
(31)S(KerX)⊆Ker(X+K),
for some compact operator K on H2.Proof.
LetX∈essHankλ. Then,
(32)(S*-λI)X-XS=K,
for some compact operator K on H2.
Iff∈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.
IfX∈essHankλ, then
(33)S(RangeX¯)⊥⊆[Range(X+K)¯]⊥,
for some compact operator K on H2.Proof.
LetX∈essHankλ. Then, X*∈essHankμ, where μ=-λ-. Applying Theorem 17 to X*, we get
(34)S(KerX*)⊆Ker(X*+K),
for some compact operator K on H2. This means that
(35)S(KerX*)⊆Ker[(X+K*)*].
Therefore,
(36)S(RangeX¯)⊥⊆[Range(X+K*)¯]⊥,
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 operatorRa is essentially Toeplitz if and only if it is essentially Hankel. That is, Ra∈essToep if and only if Ra∈essHank0. We show that this is not the case if λ≠0. In fact, an essentially Toeplitz Rhaly operator is in essHankλ(λ≠0) if and only if it is compact. Precisely, we have the following.Theorem 20.
LetRa∈essToep be a Rhaly operator with determining sequence a=〈an〉∈ℓ2. Then Ra∈essHankλ(λ≠0) if and only if  limn→∞(n+1)|an|=0.Proof.
LetRa∈essToep. Then, Ra∈essHank0 [4]. Now if Ra∈essHankλ(λ≠0), then we have Ra∈essHank0∩essHankλ. As λ≠0, we have Ra∈𝒦(H2), by Theorem 6. It is known that [8] a Rhaly operator Ra with determining sequence a=〈an〉 is compact if and only if limn→∞(n+1)|an|=0. The desired result follows.



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*Source: 865396-2013-01-09.xml*