JOPER Journal of Operators 2314-5072 2314-5064 Hindawi Publishing Corporation 10.1155/2015/172754 172754 Research Article On Some Algebraic and Operator-Theoretic Properties of λ-Toeplitz Operators http://orcid.org/0000-0002-7287-0406 Nikpour Mehdi Fan Dashan Department of Science and Mathematics The American University of Afghanistan Darulaman Road, Kabul Afghanistan 2015 612015 2015 04 09 2014 16 12 2014 17 12 2014 612015 2015 Copyright © 2015 Mehdi Nikpour. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Based on a spectral problem raised by Barría and Halmos, a new class of Hardy-Hilbert space operators, containing the classical Toeplitz operators, is introduced, and some of their Toeplitz-like algebraic and operator-theoretic properties are studied and explored.

1. Introduction

All of the work I am about to describe takes place in the Hardy-Hilbert space of the unit circle U, denoted by H2(U)(=H2), and consists of all square-integrable (with respect to the normalized arc-length measure dθ/2π) functions, on U, whose negative Fourier coefficients all vanish; that is, (1)H2=fL2U02πddf^(n)=12π02πf(eıθ)e-ınθdθ=0,nZ+. For more details and basic properties of Hardy spaces, the reader is referred to [1, Chapters 1 and 2] or [2, Chapter 17].

Two of the most intensely studied classes of bounded operators on H2 are Toeplitz and Hankel operators. Originally, an infinite matrix is called Toeplitz (resp., Hankel) if its entries depend just on the difference (resp., the sum) of their indices. Hence, Toeplitz matrices are the ones with constant diagonals, and Hankel matrices are those with constant skew-diagonals. They both play a decisive role in a very wide circle of problems in operator theory, C*-algebras, moment problems, interpolation by holomorphic or meromorphic functions, inverse spectral problems, orthogonal polynomials, prediction theory, Wiener-Hopf equations, boundary problems of function theory, the extension theory of symmetric operators, singular integral equations, models of statistical physics, and many others. Also, there exists a vast literature on the theory of Toeplitz and Hankel operators; see, for example, .

Maybe a naïve reason also for their importance is the fact that Toeplitz and Hankel operators are compressions of (bounded) multiplication operators and their flipped, respectively, to H2. Indeed, any essentially bounded function ϕ on U induces, in a natural way, three bounded operators, one on L2(U) and the two others on H2, as follows.

The Multiplication operator Mϕ is given by Mϕf=ϕf, for fL2(U).

The Toeplitz operator Tϕ is defined, in terms of the orthogonal projection P from L2(U) onto H2(U), as the compression of Mϕ to H2; Tϕf=PMϕf, for fH2(U).

The Hankel operator Hϕ is defined as the compression of the “flipped” Mϕ onto H2(U); Hϕf=PJMϕf, for fH2(U), where J is the unitary self-adjoint operator on L2(U) (the so-called flip operator) defined by (Jf)(ζ):=ζ¯f(ζ¯), for ζU, mapping H2 onto (H2) (the orthogonal complement of H2) and (H2) onto H2.

In each case, ϕ is called the symbol of the operator.

These classes of operators can also be considered as solutions to some linear operator-equations involving the Toeplitz operator Teıθ, known as the unilateral forward shift, and its Hilbert-adjoint Te-ıθ, usually called the unilateral backward shift. Indeed, it is well known that an operator H is Hankel if and only if Te-ıθH=HTeıθ (Hankel equation) and that an operator T is Toeplitz if and only if Te-ıθTTeıθ=T (Toeplitz equation).

Generalizations of such operator-equations have been studied and explored for some time. For instance, in  the operator-equation S*XT=X, for arbitrary contractions S and T acting on different Hilbert spaces, has been studied. Pták in  studied the solutions to the operator-equation S*X=XT, where S and T are contractions.

Here we study an operator-equation, on B(H2), which is a slight modification to the Toeplitz equation; namely, Te-ıθXTeıθ=λX, for an arbitrary complex number λ. This operator-equation appeared in  and it was asked what its operator-solutions could be, what algebraic and operator-theoretic properties those solutions had, and how these operator-solutions relate to the case λ=1 (Toeplitz operators). Fortunately, this problem is a spectral one; that is, its solutions are the eigen-operators of a bounded operator-valued linear transformation on B(H2), which have been found and characterized by Sun in .

In this paper we study and develop some algebraic and operator-theoretic properties of λ-Toeplitz operators as the bounded operator-solutions to the operator-equation Te-ıθXTeıθ=λX, for an arbitrary complex number λ. In most cases, it is shown that λ-Toeplitz operators behave the same as the classical Toeplitz operators, on H2. We also introduce the classes of analytic and coanalytic λ-Toeplitz operators (Definition 9), which generalize the most commonly considered classes of Toeplitz operators, and apply them to study the multiplicative properties of λ-Toeplitz operators. In Theorem 14, we show that a product of two λ-Toeplitz operators is again one precisely when each operator is either analytic or coanalytic, which generalizes [13, Theorem 8]. We then give an example (Example 16) to show that, unlike the Toeplitz case, two analytic (resp., coanalytic) λ-Toeplitz operators need not commute, (which violates [13, Theorem 9] in the λ-Toeplitz operators’ context). Though, we can still obtain some necessary and sufficient conditions for pairs of (co-)analytic λ-Toeplitz operators to commute (Theorem 17). We also obtain an interesting result (Corollary 18) on the problem of invertibility for λ-Toeplitz operators and its connection with our notions of analyticity and coanalyticity, which generalizes [13, Corollary 2]. Finally, we study the relation between analytic/coanalytic λ-Toeplitz operators and the classical Hankel operators (Theorem 19). This work justifies Barría and Halmos’ suggestion, in , that the notion of λ-Toeplitzness may be worthy of study.

Finally, it should be mentioned that this work has its roots in  and been inspired by [12, 14].

We close this section by setting up the notations required for what is to follow.

1.1. Notations

B ( H 2 ) is the C*-algebra of all bounded linear operators on H2.

K stands for the two-sided ideal of all compact operators in B(H2).

For TB(H2),

σ ( T ) denotes the spectrum of T;

σ p ( T ) denotes the set of eigenvalues for T.

The standard tensor notation will be used for operators of rank one: for f and g vectors in H2, the operator fg is defined by (fg)h=h,gH2f.

The standard orthonormal basis (en)n=0+ for H2, where ens are functions in H2 defined as enζ=ζn, for ζU.

For fH2 and n=0,1,2,, f^(n) stands for the nth-Fourier coefficient of f; that is, f^(n)=f,en.

H ( U ) consists of all boundary functions of bounded holomorphic functions on U.

2. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M100"><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:math></inline-formula>-Toeplitzness

Here we give two approaches for defining the concept of “λ-Toeplitzness”: matricial and operator-theoretic approaches.

Definition 1.

One calls a singly infinite matrix a λ-Toeplitz matrix if, on each diagonal (parallel to the main diagonal), the entries are in continued proportion; that is, the matrix (amn)m,n=0 is a λ-Toeplitz matrix if there exists a λC such that am+1,n+1=λam,n, for m,n=0,1,2,.

For a fixed complex number λ, a typical example of a singly infinite λ-Toeplitz matrix is (2)a0a-1a-2a-3a1λa0λa-1λa-2a2λa1λ2a0λ2a-1a3λa2λ2a1λ3a0. In , it is shown that Toeplitz operators, on H2, are the solutions of the operator-equation Te-ıθXTeıθ=X, on B(H2). Equivalently, this means that they are the eigen-operators of the following operator-valued linear transformation (let us call it Toeplitz mapping on B(H2)): (3)Γ:B(H2)B(H2)XTe-ıθXTeıθ, corresponding to the eigenvalue 1. This suggests a general context in which Toeplitz operators can be embedded.

Definition 2.

One calls an operator, in B(H2), a λ-Toeplitz operator if it is an eigen-operator of the Toeplitz mapping Γ corresponding to one of its eigenvalues.

More precisely, for λσp(Γ), the set Tλ=ker(λI-Γ) consists of λ-Toeplitz operators corresponding to λ, or, equivalently, (4)Tλ=XBH2ΓX=Te-ıθXTeıθ=λX. It can be easily checked that the Toeplitz mapping is a contraction; moreover, since Te-ıθTeıθ=I, we should have Γ=1. Thus, there is no λ-Toeplitz operator for λ>1; that is, σ(Γ)U¯. But every diagonal operator with diagonal (λn)n=0, for λU¯, is a solution for [λI-Γ](X)=0. Thus, U¯σp(Γ)(σ(Γ)). Therefore, σp(Γ)=σ(Γ)=U¯; that is, the only eigen-operators for Γ are the ones corresponding to the eigenvalues living in U¯.

Observation 1.

Bounded operator-solutions to Te-ıθXTeıθ=λX, on B(H2), exist if and only if λU¯. For more details on the form of a λ-Toeplitz operator, see .

Theorem 3 (Sun [<xref ref-type="bibr" rid="B18">12</xref>]).

Let λC. The operator-equation Te-ıθXTeıθ=λX has bounded solutions if and only if λ1. One then has the following.

If λ=1, all solutions are of the form Dλ¯T, where T is a Toeplitz operator and Dλ is the diagonal unitary operator defined as Dλen=λnen for all n.

If λ<1, all solutions are compact operators of the form (5)n=0λnTeınθfen+enTeınθg

for some f and gH2.

For convenience, let us divide λ-Toeplitz operators into two main classes: unimodular λ-Toeplitz operators and nonunimodular λ-Toeplitz operators, which are the ones corresponding to the eigenvalues of the Toeplitz mapping Γ on the unit circle and the unit disk, respectively.

Remark 4.

Some immediate consequences of Sun’s Theorem are that the nonunimodular λ-Toeplitz operators are compact (so are not invertible) and unimodular λ-Toeplitz operators are not. Indeed, in the latter case, the only compact unimodular λ-Toeplitz operator is the zero operator.

Remark 5.

By Definition 2, if XTλ, for some λU¯, then Te-ıθXTeıθ=λX. Hence, the entries of its matrix representation (am,n)m,n=0, with respect to the monomial basis for H2, satisfy (6)am+1,n+1=Xen+1,em+1H2=XTeıθen,TeıθemH2=Te-ıθXTeıθen,emH2=λXen,emH2=λam,n,form,n=0,1,2,. This yields (7)am,nm,n=0=a0,0a0,1a0,2a0,3a1,0λa0,0λa0,1λa0,2a2,0λa1,0λ2a0,0λ2a0,1a3,0λa2,0λ2a1,0λ3a0,0, where (8)Xe0=n=0an,0eınθ,X*e0=n=0a¯0,neınθ.

3. Basic Properties of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M168"><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:math></inline-formula>-Toeplitz Operators

Recall that T1 consists of all (classical) Toeplitz operators, and it turns out, as we will see later, that other λ-Toeplitz operators behave like them. Also, notice that, for each λU¯,  Tλ forms a complex vector subspace of B(H2).

Here we look at the following straightforward properties of the λ-Toeplitz operators. First, from Definition 2, we observe that, for each λU¯, Tλ is topologically well behaved. Indeed, for XTλ, since Γ(X)=Te-ıθXTeıθ is weakly continuous in its middle factor, Tλ is weakly closed, and, therefore, a fortiori, it is strongly and uniformly closed.

The next result, inspired by [14, Theorem 4.5], states that self-adjointness only exists among real λ-Toeplitz operators; that is, λ-Toeplitz operators correspond to real eigenvalues for Γ.

Proposition 6.

For λU¯ and XTλ, one has the following.

X*Tλ¯.

If X0 and X=X*, then λR.

Proof.

(i) Since XTλ, we have Te-ıθXTeıθ=λX, from which, by taking adjoints, we get Te-ıθX*Teıθ=λ¯X*. So X*Tλ¯.

(ii) If X is a nonzero self-adjoint element of Tλ, then XTλTλ¯. But this means that (9)λX=Te-ıθXTeıθ=Te-ıθX*Teıθ=λ¯X*=λ¯X, which implies λ=λ¯, or, equivalently, λR.

Remark 7.

As a consequence of Sun’s Theorem, every nonunimodular λ-Toeplitz operator is compact. This, in turn, states that they are not only noninvertible, but also nonessentially invertible. But the situation is different for unimodular ones. Indeed, the only compact unimodular one is the zero operator: for λU, letting XTλ, n and n+k be nonnegative integers, we have (10)Xen,en+kH2(U)=λnXe0,ekH2(U). Now, if X is a compact operator, then XenH2(U)0, as n; it follows that Xe0,ekH2(U)=0, for all nonnegative k.

And, if we apply the same procedure for X*, we obtain X*e0,ekH2(U)=0, for all nonnegative k. Therefore, X=0.

4. Analyticity and Coanalyticity of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M214"><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:math></inline-formula>-Toeplitz Operators

In [13, p. 96], analyticity and coanalyticity of a (classical) Toeplitz operator are defined and characterized in terms of its commutativity with Teıθ and Te-ıθ, respectively [13, Theorem 7]. Here, we define and give an analogous characterization of these two properties in the λ-Toeplitz operators’ setting. But let us first assign them a “symbol” (similar to one in the classical case) as a generating function.

Definition 8.

For λU¯, let XTλ. The symbol of X is defined to be (11)Xe0+X*e0¯-Xe0,e0H2 and is denoted by sym(X), in which Xe0 is called the analytic symbol and X*e0¯ is called the coanalytic symbol of X.

Observation 2.

For a λ-Toeplitz operator X, sym(X) is the function whose nonnegative Fourier coefficients are the terms of the 0th-column of its matrix representation, with respect to the monomial basis for H2, and whose nonpositive Fourier coefficients are the terms of the 0th-row of that matrix.

Definition 8 determines two H2-functions, namely, Xe0 and X*e0, by which we may characterize the properties of analyticity and coanalyticity, for such operators, as follows.

Definition 9.

A λ-Toeplitz operator, X, is called

analytic if sym(X) is an analytic function (i.e., X*e0 is the constant function e0,Xe0H2);

coanalytic if sym(X) is a coanalytic function (i.e., Xe0 is the constant function Xe0,e0H2).

This definition makes the following remark obvious.

Remark 10.

For λU¯, letting XTλ

X is analytic if and only if X*e0,enH2=0, for all n>0 (i.e., Te-ıθX*e0=0),

X is coanalytic if and only if Xe0,enH2=0, for all n>0 (i.e., Te-ıθXe0=0).

Hence, X is analytic if and only if X* is coanalytic.

Observation 3.

Notice that for, λU¯ and ϕL(U), DλTϕTλ, where Dλ is the diagonal operator with diagonal (λn)n=0. Indeed, (12)Te-ıθDλTϕTeıθ=Te-ıθDλ(TeıθTe-ıθ+e0e0)TϕTeıθ=(Te-ıθDλTeıθ)(Te-ıθTϕTeıθ)+Te-ıθDλe0Te-ıθϕ¯e0=λDλTϕ+Te-ıθe0=0Te-ıθϕ¯e0=λDλTϕ. This observation provides us with two classes of typical examples of analytic and coanalytic λ-Toeplitz operators.

For λU¯ and ϕH(U), DλTϕ is an analytic λ-Toeplitz operator. Indeed for n=1,2,, (13)DλTϕ*e0,enH2=Dλ¯e0,ϕen=e0,ϕen=ϕ¯^(n)=0.

For λU¯ and ϕ¯H(U), DλTϕ is a coanalytic λ-Toeplitz operator. Indeed for n=1,2,, (14)DλTϕe0,enH2=Tϕe0,Dλ¯en=Tϕe0,λ¯nen=λnϕ^(n)=0.

Note that if XTλ is analytic, its matrix representation with respect to the monomial basis for H2 is lower triangular. Indeed, for m,n=0,1,2, with m<n(15)Xen,emH2=XTeıθmen-m,Teıθme0=Te-ıθmXTeıθmen-m,e0=λmXen-m,e0=λmen-m,X*e0=0. With the same reasoning one can show that coanalytic λ-Toeplitz operators correspond to upper triangular λ-Toeplitz matrices.

Before stating the first result of this section, we need to introduce some terms. For a Hilbert space bounded operator A, consider the operator-equation (16)AX=λXA for some complex number λ. If there is a nonzero (bounded) operator X and a scalar λ as above that satisfy (16), according to , it is said that A  λ-commutes with X and that λ is an extended eigenvalue and X is an extended eigen-operator of A.

Equation (16) has been studied in  and, independently, in . These works provided extensions of Lomonosov’s classic result .

Now, we use these terms to state our next result which characterizes analyticity and coanalyticity of λ-Toeplitz operators in terms of their λ-commutativity with Teıθ and Te-ıθ.

Theorem 11.

Let λU¯{0} and XB(H2). A necessary and sufficient condition that X is an analytic (coanalytic)  λ-Toeplitz operator in Tλ is that it λ-commutes with Teıθ(λ-1-commutes with Te-ıθ); that is, XTeıθ=λTeıθX  (Te-ıθX=λXTe-ıθ).

Proof.

(i) Let X be an analytic λ-Toeplitz operator in Tλ; that is, Te-ıθX*e0=0. Hence, (17)λTeıθX=Teıθ(λX)=Teıθ(Te-ıθXTeıθ)=(I-e0e0)XTeıθ=XTeıθ-(e0Te-iθX*e0)=XTeıθ. Now, let XB(H2) be such that XTeıθ=λTeıθX. Just by multiplying both sides by Te-ıθ from the left, one can easily see that XTλ. To show it is analytic, we need to prove X*e0,en=0, for n=1,2,. So, (18)X*e0,enH2=e0,XTeıθen-1H2=e0,λTeıθXen-1H2=λ¯X*Te-ıθe0,en-1H2=0, which means X is an analytic λ-Toeplitz operator.

(ii) If X is a coanalytic λ-Toeplitz operator in Tλ, that is, Te-iθXe0=0, then (19)λXTe-ıθ=(Te-ıθXTeıθ)Te-ıθ=Te-ıθX(I-e0e0)=Te-ıθX-(Te-ıθXe0e0)=Te-ıθX. Now, let XB(H2)  λ-1-commute with Te-ıθ; that is, Te-ıθX=λXTe-ıθ. Multiplying both sides by Teıθ from the right shows that XTλ. To prove coanalyticity, we need to show Te-ıθXe0=0. So, we have (20)(Te-ıθX)e0=(λXTe-ıθ)e0=0.

Remark 12.

Using the terms aforementioned, Theorem 11 can also be restated as follows.

Theorem 11

Let λU¯{0}. A necessary and sufficient condition in which XTλ is analytic is that Teıθ is an extended eigen-operator of A corresponding to the extended eigenvalue λ.

Let λU¯{0}. A necessary and sufficient condition in which XTλ is coanalytic is that Te-ıθ is an extended eigen-operator of A corresponding to the extended eigenvalue λ-1.

Remark 13.

Notice that if XT0, it can be represented by the finite-rank operator [(Xe0-Xe0,e0H2e0)e0]+e0X*e0. Thus,

X is analytic if and only if X*e0 is the constant function e0,Xe0H2 and, in this case, X=Xe0e0.

And X is coanalytic if and only if Xe0 is the constant function Xe0,e0H2 and, in this case, X=e0X*e0.

5. Multiplicative Properties of <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M352"><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:math></inline-formula>-Toeplitz Operators

Although, for a fixed λU¯, Tλ is closed under finite summation of its elements, the corresponding result rarely holds for products. As an application of Theorem 11, we will see that λ-Toeplitzness is preserved under multiplication, on the right, by analytic λ-Toeplitz operators and, on the left, by coanalytic ones.

Theorem 14.

For λ1,λ2U¯, let X1Tλ1 and X2Tλ2.

A necessary and sufficient condition that the product X1X2 is a λ-Toeplitz operator in Tλ1λ2 is that either X1 is coanalytic or X2 is analytic.

Proof.

Let us first assume X1X2Tλ1λ2. Hence, (21)λ1λ2X1X2=Te-ıθX1X2Teıθ=Te-ıθX1(TeıθTe-ıθ+e0e0)X2Teıθ=(Te-ıθX1Teıθ)(Te-ıθX2Teıθ)+(Te-ıθX1e0Te-ıθX2*e0)=λ1λ2X1X2+(Te-ıθX1e0Te-ıθX2*e0). So, for holding the equality, we should have either Te-ıθX1e0=0 or Te-ıθX2*e0=0; that is, either X1 is coanalytic or X2 is analytic.

Let us suppose, for now, X1 is a coanalytic λ-Toeplitz operator in Tλ1 and X2Tλ2. To show that X1X2Tλ1λ2, we apply Theorem 11 to write (22)Te-ıθX1X2Teıθ=λ1X1Te-ıθX2Teıθ=(λ1X1)(λ2X2)=λ1λ2X1X2, which proves X1X2 is a λ-Toeplitz operator in Tλ1λ2.

And if X2 is an analytic λ-Toeplitz operator in Tλ2 and X1Tλ1, again using Theorem 11 results in (23)Te-ıθX1X2Teıθ=Te-ıθX1(λ2TeıθX2)=λ2(Te-ıθX1Teıθ)X2=λ1λ2X1X2, which proves the same thing.

From the proof of Theorem 14, along with considering Remark 7, one may deduce the following property for unimodular λ-Toeplitz operators.

Proposition 15.

For λ1,λ2U, let X1Tλ1 and X2Tλ2. If X1X2Tμ{0} for some μU, then μ=λ1λ2. Moreover, either X1 is coanalytic or X2 is analytic.

Proof.

Considering the assumptions, we have (24)μX1X2=Te-ıθX1X2Teıθ=Te-ıθX1(TeıθTe-ıθ+e0e0)X2Teıθ=(Te-ıθX1Teıθ)(Te-ıθX2Teıθ)+(Te-ıθX1e0Te-ıθX2*e0)=λ1λ2X1X2+(Te-ıθX1e0Te-ıθX2*e0), which implies (25)(μ-λ1λ2)X1X2=Te-ıθX1e0Te-ıθX2*e0. Since μ0 and X1X2 is a nonzero λ-Toeplitz operator in Tμ, X1X2 cannot be of finite rank (see Remark 7). Hence, both sides in (25) should be zero. Therefore, μ=λ1λ2, and this in turn implies that either X1 should be coanalytic or X2 is analytic.

Recall that if the symbols of two (classical) Toeplitz operators are either analytic or coanalytic, they necessarily commute [13, Theorem 9]. But, surprisingly, this is not the case among λ-Toeplitz operators; for, let us look at an example.

Example 16.

For some λU¯{1},

let X1,X2Tλ be analytic such that X2 is arbitrary and X1 is given by (26)X1ej,eiH2=λjδi,j+1i,j=0,1,2,,

where δi,j+1 is the Kronecker delta. Note that X2 can also be represented as (27)X2ej,eiH2=0ifi<jλjX2e0,ei-jH2ifij.

Hence, we have (28)X1X2ej,eiH2=0ifijλi+j-1X2e0,ei-j-1H2ifi>j,X2X1ej,eiH2=0ifijλ2j+1X2e0,ei-j-1H2ifi>j.

Therefore, as analytic λ-Toeplitz operators, X1 and X2 do not commute.

In the other direction, consider two coanalytic λ-Toeplitz operators X3,X4Tλ, such that X4 is arbitrary and X3=X1*, the Hilbert space adjoint of X1 in the previous case; that is, (29)X3ej,eiH2=λ¯iδj,i+1i,j=0,1,2,.

Again, note that X4 can also be represented as (30)X4ej,eiH2=0ifi>jλiX4ej-i,e0H2ifij.

Hence, we have (31)X3X4ej,eiH2=0ifjiλ2iλX4ej-i-1,e0H2ifj>i,X4X3ej,eiH2=0ifjiλ¯j-1λiX4ej-i-1,e0H2ifj>i.

Therefore, as coanalytic λ-Toeplitz operators, X3 and X4 do not commute.

Though, we can still obtain some necessary and sufficient conditions for pairs of (co-)analytic λ-Toeplitz operators to commute.

Theorem 17.

For λ1,λ2U¯, let X1Tλ1 and X2Tλ2.

If both X1 and X2 are analytic, then X1X2=X2X1 if and only if (X2e0)(λ1ζ)(X1e0)(ζ)=(X1e0)(λ2ζ)(X2e0)(ζ), for almost all ζU.

If both X1 and X2 are coanalytic, then X1X2=X2X1 if and only if (X2*e0)(λ1¯ζ)(X1*e0)(ζ)=(X1*e0)(λ2¯ζ)(X2*e0)(ζ), for almost all ζU.

Proof.

(i) Assume that X1Tλ1 and X2Tλ2 are both analytic λ-Toeplitz operators such that X2e0(λ1ζ)X1e0(ζ)=X1e0(λ2ζ)X2e0(ζ), for almost all ζU. So, by Theorem 14, X1X2,X2X1Tλ1λ2. Also, analyticity of X1 and X2 reveals that (32)X1ej,eiH2=0i<jλ1jX1e0,ei-jH2ij,X2ej,eiH2=0i<jλ2jX2e0,ei-jH2ij, which in turn implies (33)X1X2ej,ei=0i<jk=jiλ1kλ2jX1e0,ei-kX2e0,ek-jij. But since X1X2 is an analytic λ-Toeplitz operator, we just need to consider the Fourier coefficients of X1X2e0, which can be obtained from the finite sum in (33) by letting j=0, which gives us the ith-Fourier coefficient of X1X2e0; that is, (34)k=0iλ1kX2e0,ekH2X1e0,ei-kH2, which is nothing but the ith-Fourier coefficient of (X2e0)(λ1ζ)(X1e0)(ζ), for almost all ζU, since (35)(X2e0)(λ1ζ)(X1e0)(ζ)=i=0X2e0^iλ1iζii=0X1e0^iζi=i=0k=0iλ1kX2e0^kX1e0^i-kζi, where Xje0^(i)=Xje0,eiH2, for j=1,2 and i=0,1,2,.

By the assumption (X2e0)(λ1ζ)(X1e0)(ζ)=(X1e0)(λ2ζ)(X2e0)(ζ), hence the finite sum in (34) is also equal to the ith-Fourier coefficient of (X1e0)(λ2ζ)(X2e0)(ζ); that is, (36)k=0iλ2kX1e0,ekH2X2e0,ei-kH2, which is the ith-Fourier coefficient of X2X1e0. What we already showed is that the ith-Fourier coefficients of sym(X1X2) and sym(X2X1) are equal. Therefore, X1X2=X2X1. This proves the sufficiency condition in (37).

Let us assume that X1X2=X2X1. This assumption, along with analyticity of X1 and X2, implies (37)k=jiλ1kλ2jX1e0,ei-kX2e0,ek-j=k=jiλ2kλ1jX2e0,ei-kX2e0,ek-j, for i,j=0,1,2,, such that ij. Now, letting j=0 in (37), we obtain (38)(X2e0)(λ1ζ)(X1e0)(ζ)=(X1e0)(λ2ζ)(X2e0)(ζ) for almost all ζU. This proves the necessity condition in (37).

(ii) Assume that X1Tλ1 and X2Tλ2 are both coanalytic λ-Toeplitz operators such that (39)(X2*e0)(λ1¯ζ)(X1*e0)(ζ)=(X1*e0)(λ2¯ζ)(X2*e0)(ζ), for almost all ζU. Their coanalyticity implies that X1*Tλ1¯ and X2*Tλ2¯ are analytic, which satisfy (39). Therefore, by the sufficiency condition in (37), they commute. This in turn implies that X1 and X2 commute. This proves the necessity condition in (37).

Now, suppose coanalytic λ-Toeplitz operators X1 and X2 commute, which means analytic λ-Toeplitz operators X1*Tλ1¯ and X2*Tλ2¯ commute. Hence, by the necessity condition in (37), we should have (40)(X2*e0)(λ1¯ζ)(X1*e0)(ζ)=(X1*e0)(λ2¯ζ)(X2*e0)(ζ), which proves the sufficiency condition in (37).

Another consequence of Theorem 11 characterizes the λ-Toeplitz operators having λ-Toeplitz operator inverses.

Corollary 18.

For λU, let XTλ. If X is invertible, then a necessary and sufficient condition that X-1 is a λ-Toeplitz operator is that X is either analytic or coanalytic.

Proof.

Suppose that X is invertible. If X is analytic, then, by Theorem 11, X  λ-commutes with Teıθ; that is, (41)XTeıθ=λTeıθX, from which follows (42)X-1Teıθ=λ¯TeıθX-1. But, on one hand, (42) implies that X-1 is a λ-Toeplitz operator in Tλ¯ and, on the other hand, that it is an analytic λ-Toeplitz operator, using Theorem 11.

The case for coanalyticity walks through the same steps as the latter case.

Suppose now that X-1 is known to be a λ-Toeplitz operator in Tμ, for some μU. Having the following operator-equations, (43)Te-ıθXTeıθ=λX,Te-ıθX-1Teıθ=μX-1, we obtain (44)Te-ıθXe0Te-ıθX-1*e0=1-λμI,Te-ıθX-1e0Te-ıθX*e0=(1-λμ)I, each of which implies μ=λ¯; that is, X-1Tλ¯, and, in this case, from the first equation follows either X is coanalytic or X-1 is analytic. And the second one also implies either X-1 is coanalytic or X is analytic.

If X is not coanalytic, then X-1 is analytic and not constant; this implies that X-1 is not coanalytic and hence that X is analytic. The same reasoning also works when it is assumed that X is not analytic.

6. <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M537"><mml:mrow><mml:mi>λ</mml:mi></mml:mrow></mml:math></inline-formula>-Toeplitzness versus Hankelness

One of the properties of Hankelness is that it is preserved under multiplication, on the right, by analytic Toeplitz operators or, on the left, by coanalytic Toeplitz operators. Indeed, if H,TB(H2) such that H is a Hankel operator and T is an analytic Toeplitz operator, then (45)Te-ıθHT=HTeıθT=HTTeıθ, which states that HT satisfies the Hankel equation, so is a Hankel operator. A similar way shows that TH is a Hankel operator, where T is a coanalytic Toeplitz operator.

It turns out that Hankel operators behave in a similar manner when they meet analytic/coanalytic λ-Toeplitz operators.

Theorem 19.

Let HB(H2) be a Hankel operator and λU¯. If X1 is an analytic and X2 is a coanalytic λ-Toeplitz operator in Tλ, then HX1 and X2H satisfy the Hankel equation in the sense that λTe-ıθHX1=HX1Teıθ and Te-ıθX2H=λXHTeıθ.

Proof.

As it is well known, H is a Hankel operator if and only if Te-ıθH=HTeıθ. Then, simply, we have (46)λTe-ıθHX1=λHTeıθX1=H(λTeıθX1)=HX1Teıθ,Te-ıθX2H=λX2Te-ıθH=λX2Te-ıθH=λX2HTeıθ, proving the assertion.

Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The author would like to express his sincere gratitude to the anonymous referee for his/her helpful comments that will help improve the quality of the paper.

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