Gopal Dattand Ritu Aggarwal

1Department of Mathematics,PGDAV College,University of Delhi,Delhi–110065, India.

2Department of Mathematics,University of Delhi,Delhi–110007,India.

A Note on the Operator Equation Generalizing the Notion of Slant Hankel Operators

Gopal Datt1,?and Ritu Aggarwal2

1Department of Mathematics,PGDAV College,University of Delhi,Delhi–110065, India.

2Department of Mathematics,University of Delhi,Delhi–110007,India.

.The operator equation λMzX=XMzk,for k ≥ 2,λ ∈ C,is completely solved. Further,some algebraic and spectral properties of the solutions of the equation are discussed.

Hankel operators,slant Hankel operators,generalized slant Toeplitz operators,generalized slant Toeplitz operators,spectrum of an operator.

AMS SubjectClassif i cations:47B35

1 Introduction

Let Z,C and T denote the set of integers,set of complex numbers and the unit circle, respectively.Let L2(T)(simply written as L2)denote the classical Hilbert space with standard orthonormal basis{en:n ∈ Z},where en(z)=znfor each z ∈ T.The symbol H2denotes the space generated by{en:n ≥ 0}.The symbol L∞is used to denote the space of all essentially bounded measurable functions on T and H∞=L∞∩ H2.The theory of Hankel operators,which is a beautiful area of mathematical analysis,admits of vast applications.In 1861,Hankel[12]began the study of f i nite matrices whose entries depend only on the sum of the coordinates and such objects are called Hankel matrices.In 1881, Kronecker[14]obtained f i rst theorem about inf i nite Hankel matrices that characterizes Hankel matrices of f i nite rank.

The development of the theory of Hankel operators led to different generalizations of the original concept,like,slant Hankel operators,λ-Hankel operators and(λ,μ)-Hankel operators(see[2,5]and[8]).A lot of progress has taken place in the study of Hankel operators on Bergman spaces on the disk,Dirichlet type spaces,Bergman and Hardy spaces on the unit ball in Cnor on symmetric domains,etc.[16].

Hankel operators on the space L2are characterized by the operator equation MzX= XMz,whereas on the Hardy space H2these are characterized by the operator equation U?X=XU,where U is the forward unilateral shift operator on the Hardy space H2. We refer to[10,11,16]and the references therein for the basic study of Hankel operators on these spaces.Motivated by the approach initiated by Barr′ia and Halmos[2], various equations,like,MzX=XMz2(solutions of which are named as slant Hankel operators[2]),U?X ? XU= λX, λ ∈ C(solutions of which are named as λ-Hankel operators[5])etc.are attained by mathematicians.In this row,generalized slant Hankel operators[3]have also been obtained which are nothing but the solution of the operator equation MzX=XMzk,for k≥2 and are named as kth-order slant Hankel operators.Work of Avenda ?no[5]dragged our attention to the operator equation λ MzX=XMzk,for k ≥ 2 and λ ∈ C.Clearly,for λ =1,this equation characterizes the kth-order slant Hankel operators and if further k=2 then it is nothing but the equation characterizing slant Hankel operators.

From the work of Nehari[15],it is known that each Hankel operator is induced by an essentially bounded measurable symbol φ ∈ L∞and is denoted as Hφ.Not much is known about spectral properties of Hankel operators in terms of the inducing symbol. Power[17]described the essential spectrum of Hφfor piecewise continuous functions φ ∈ L∞.In this paper,we completely solve the operator equation λ MzX=XMzk,for k ≥ 2 and λ ∈ C.We describe some of the spectral properties of the solutions of the equation λ MzX=XMzk,for k ≥ 2 and λ ∈ C.We achieve the containment of a closed disc in the spectrum of each non-zero operator satisfying the equation λ MzX=XMzk,for k ≥ 2 and λ∈C.

2 Operator equation:λMzX=XMzkfor k≥ 2,λ ∈ C

In last two decades various operator equations generalizing the notion of Hankel operators have been discussed,for the details and importance of which we suggest the references[2,3]and[4].The purpose here is to call attention to the operator equation λ MzX=XMzk,for an integer k ≥ 2 and λ ∈ C.Throughout the paper,k is assumed to be an integer greater than or equal to 2.We begin with the following result.

Theorem 2.1.The only solution of the operator equation λ MzX=XMzk,|λ |/=1 is the zero operator.

Proof.Suppose that X satisf i es λ MzX=XMzk.First,consider the case|λ|＜ 1.Def i ne a map τ :B(L2)→ B(L2)as τ(X)= λMzXMzk.Then ‖τ‖ ≤ |λ|＜ 1 and(I? τ)is invertible. Now(I? τ)X=0,which implies that X=0.

Now consider the case|λ|＞ 1.This time we def i ne the mapping τ as τ(X)=MzXMzk. Now ‖τ ‖≤ 1 so(λ I? τ )is invertible and this provides that X=0.This completes the proof.

Now in view of thelast result,we are left to solve the operatorequation λMzX=XMzkfor|λ|=1.We consider the operators Wkand J(the f l ip operator)on L2def i ned as

and Jen=e?nfor each n ∈ Z.Then the facts WkJ=JWk,Mφ(z)Wk=WkMφ(zk)and Mφ(z)J= JMφ(z),where φ ∈ L∞,are well known about these operators.It is interesting to know that each kth-order slant Hankel operator on L2is of the form WkJMφfor some φ ∈ L∞(see[3]). Using these facts,we claim the following.

Theorem 2.2.Let λ ∈ C be such that|λ|=1.The operator equation λMzX=XMzkadmits of non-zero solutions and each non-zero solution is of the form X=DλWkJMφfor some φ ∈ L∞, where Dλis the composition operator on L2induced by zi.e.,Dλf(z)=ffor all f∈ L2. Proof.Suppose X is an operatorof the form DλWkJMφfor some φ∈ L∞,where symbol Dλis used in the sense it is def i ned in the statement.Now it can be seen that λ MzDλ=DλMzand Mz(WkJMφ)=(WkJMφ)Mzk,which provides that λ MzX=XMzk.

Conversely,suppose that X is an operator satisfying λMzX=XMzk.Pre-multiplying by Dλ,wegetthatX=XMzk.Therefore,X isa kth-orderslant Hankeloperator on L2and hence we get the result.

Now onward,we are focussed to study the behavior of the solutions of the equation λ MzX=XMzk,|λ|=1.In fact,here onward,the term solution is always used in reference to the solution of this equation only.The notion of Toeplitz operators,Hankel operators, slant Toeplitzoperatorsand slant Hankel operatorsare characterized in terms of matrices (see[2,7,13,17])and in the same direction we would like to have a look at the matrix characterization to the solutions of the equation λ MzX=XMzk,|λ|=1.For φ = ∑n∈Zanenin L∞and λ ∈ C with|λ|=1,the solution X=DλWkJMφsatisf i es

for each i,j∈ Z.So the matrix representation of the solution X is

We now have the following characterization to the solutions in terms of matrices.

Theorem 2.3.A necessary and suff i cient condition for an operator X on L2to be a solution of the equation λMzX=XMzk,|λ|=1 is that its matrix[aij]with respect to the standard orthonormal basis{en:n ∈ Z}satisf i es

for every i,j∈ Z.

Theorem 2.2 can be restated in the following form.

Theorem 2.4.Let λ ∈ C be such that|λ|=1.An operator X on L2is a solution of the operator equation λ MzX=XMzkif and only if it is of the form X=DλWkJMφfor some φ ∈ L∞.

As each solution is induced by an element of L∞and also depend on the choice of λ, we’ll denote the solution X of the form X=DλWkJMφby Xφ,λ.It is clear that for φ ∈ L∞,‖Xφ,λ‖ = ‖DλWkJMφ‖ ≤ ‖φ‖∞.We see the following.

Proof.Proof follows as

where ψ and ξλare elements of L∞given by

Thus,we complete the proof.

Without any extra efforts,along the lines of techniques used in[4],it is easy to check the following observation about the set Sλof all the solutions of the equation λMzX= XMzk,|λ|=1.

Theorem 2.6.We have the following:

2.Sλis weakly and hence strongly closed.

3.Sλis a norm closed subspace of B(L2),the algebra of all bounded operators on L2.

4.Sλis not self adjoint.

Proof.We just give proof for(4).It follows using Theorem 2.3 and the fact that for φ =∑n∈Zanen∈ L∞,the adjoint of a non-zero Xφ,λis=MφJ(rèn)?W?kDλand hence

for each i,j∈ Z.

In order to see whether Sλform an algebra or not,we f i rst claim the following about a solution Xφ,λ,φ ∈ L∞.

Lemma 2.1.DλWkJXφ,λis a solution if and only if Xφ,λis zero.

Proof.Let φ= ∑n∈Zanen∈ L∞and let DλWkJXφ,λbe a solution.Thenonapplying Theorem 2.3,we have

for each i,j∈ Z.This gives

which implies that ak(ki)?j= λ?kak2(i?1)?(j+k)for each i,j∈ Z.Hence,if i=0 then we get= λ?ka?k2?j?kfor each j∈ Z.This gives that for integer t,at= λ?kna?n(k2+k)+t→ 0 as n → 0.As a matter of fact φ =0 and Xφ,λ=0.Converse is obvious.

For φ,ψ ∈ L∞,the product of two solutions Xφ,λand Xψ,λsatisf i es

If we multiply the solution Xφ,λwith a Laurent(multiplication)operator A(=Mψ), where φ,ψ ∈ L∞then Xφ,λA=DλWkJMφMψ=DλWkJMφψand AXφ,λ=MψDλWkJMφ= DλMψ(λz)WkJMφ=DλJ=Mψ()WkMφ=DλJWkMψ(λzk)φ(z).These observations land at the conclusion that a solution Xφ,λcommutes with a Laurent operator Mψif and only if ψφ = ψφ.It is interesting to attain the following about these products.

Theorem 2.7.The product of a solution and a Laurent operator is always a solution.

Proof.For any φ ∈ L∞,a straight forward computation shows that for any Laurent operator M,λMz(Xφ,λA)=A=(Xφ,λMzk)A=(Xφ,λA)Mzk.Similarly,λMz(Xφ,λA)= (λMzA)Xφ,λ=A(λMzXφ,λ)=A(Xφ,λMzk)=(Xφ,λA)Mzk.Hence the result.

It is shown in[3]that the only compact kth-order slant Hankel operator is the zero operator.When we use this with the fact that the composition operator Dλis unitary,we get the following.

Theorem 2.8.The solution Xφ,λis compact if and only if it is zero operator.

Now we discuss the isometric behavior of the solutions and f i nd a result similar to the result obtained for generalized λ-slant Toeplitz operators in[9].If φ ∈ L∞is unimodular(i.e.,|φ|=1)then simple computation shows that Xφ,λis co-isometry(i.e.,is isometry).However,by the same technique as used in[9],we get the following.

Theorem 2.9.For φ ∈ L∞,Xφ,λis co-isometry if and only if

for a.e. θ∈ [0,2π].

There is a dearth of hyponormal solutions and we f i nd that the only solution Xφ,λwhich is hyponormal is the zero operator.

Theorem 2.10.For φ ∈ L∞and|λ|=1,the solution Xφ,λis hyponormal if and only if Xφ,λ=0. Proof.Suppose φ = ∑n∈Zanen∈ L∞and Xφ,λis hyponormal.Then for all f∈ L2,

This,in particular,for f=e0gives

which implies that a?kn?m=0 for m=1,2,···,k? 1 and for all n ∈ Z.Now on substituting f=e1in the inequality,we f i nd

which yields that a?kn=0 for all n ∈ Z.Thus φ =0 and so Xφ,λ=0.This completes the proof as converse is trivial.

3 Spectral behavior of solutions of the equation λMzX=XMzk

In this section,our aim is to investigate information about the spectral behavior of solutions of the equation λ MzX=XMzk,|λ|=1.We also prove that the spectrum of the solution contains a closed disc for an invertible symbol in L∞,which is a well known result in case of kth-order slant Toeplitz operators[1].For an operator A on a Hilbert space, the symbols σ(A)and σp(A)are used to denote the spectrum and the point spectrum of A respectively.The result here are just stated as can be obtained without any extra efforts by adopting the methods used to obtain the same for kth-order slant Toeplitz operators in[1].

For φ ∈ L∞,we write

Theorem 3.1.If φ is invertible in L∞,then σp(X?,λ)= σp(Xφ(zk),λ).

Theorem 3.2.For φ ∈ L∞,σ(X?,λ)= σ(Xφ(zk),λ).

Our next result shows the containment of a closed disc in the spectrum of a solution of the operator equation λMzX=XMzk.

Theorem 3.3.For any invertible φ in L∞,σ(X?,λ)contains a closed disc,where X is a solution of the equation λ MzX=XMzk.

Proof.Let μ be any non-zero complex number.As φ is invertible in L∞so is φ?1.Now suppose thatis onto.Then for each f ∈ L2,we have

where Pkis the projection on the closed span of{ekn:n ∈ Z}in L2.Now,pick 0/=g0in (I? Pk)(L2).Beingis onto,we fi nd a f∈ L2such that

Since g0∈ (I? Pk)(L2),we haveMφ?1J?(μ?1? JMφWk)f=0.This,on using the facts that μ /=0,Wkis co-isometry(i.e.,=I)and Mφ?1and J are invertible,yields that(μ?1Dλ? JMφWk)f=0.This shows that

It implies that μ?1∈ σp(Xφ(zk),λ).Nowis onto(in fact invertible)for each μ ∈ ρthe resolvent ofso on applying Theorem 3.1,we get that

where ? = ∑n∈Z〈φ,en〉λne?n.As spectrum of any operator is compact it follows that σ(X?,λ)contains a disc of eigenvalues of X?,λ.

Remark 3.1.Radius of closed disc contained in σ(X?,λ)iswhere r(A)denotes the spectral radius of the operator A.For,

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Received 8 June 2015;Accepted(in revised version)23 September 2016

?Corresponding author.Email addresses:gopal.d.sati@gmail.com(G.Datt),rituaggawaldu@rediffmail. com(R.Aggarwal)