On Some Algebraic and Operator-Theoretic Properties of λ-Toeplitz Operators

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 H 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, Calgebras,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, andmany others. Also, there exists a vast literature on the theory of Toeplitz andHankel operators; see, for example, [3–8]. Maybe a näıve reason also for their importance is the fact that Toeplitz and Hankel operators are compressions of (bounded) multiplication operators and their flipped, respectively, toH. Indeed, any essentially bounded function φ on ∂U induces, in a natural way, three bounded operators, one on L(∂U) and the two others onH, as follows.


Introduction
All of the work I am about to describe takes place in the Hardy-Hilbert space of the unit circle U, denoted by 2 ( U)(= 2 ), and consists of all square-integrable (with respect to the normalized arc-length measure /2 ) functions, on U, whose negative Fourier coefficients all vanish; that is, 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 2 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, *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, [3][4][5][6][7][8].
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 2 . Indeed, any essentially bounded function on U induces, in a natural way, three bounded operators, one on 2 ( U) and the two others on 2 , as follows.
(i) The Multiplication operator is given by = , for ∈ 2 ( U).
(ii) The Toeplitz operator is defined, in terms of the orthogonal projection from 2 ( U) onto 2 ( U), as the compression of to 2 ; = , for ∈ 2 ( U).
In each case, 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 Generalizations of such operator-equations have been studied and explored for some time. For instance, in [9] the operator-equation * = , for arbitrary contractions and acting on different Hilbert spaces, has been studied. Pták in [10] studied the solutions to the operator-equation *

=
, where and are contractions. Here we study an operator-equation, on B( 2 ), which is a slight modification to the Toeplitz equation; namely, , for an arbitrary complex number . This operator-equation appeared in [11] 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 = 1 (Toeplitz operators). Fortunately, this problem is a spectral one; that is, its solutions are the eigen-operators of a bounded operatorvalued linear transformation on B( 2 ), which have been found and characterized by Sun in [12].
In this paper we study and develop some algebraic and operator-theoretic properties of -Toeplitz operators as the bounded operator-solutions to the operator-equation , for an arbitrary complex number . In most cases, it is shown that -Toeplitz operators behave the same as the classical Toeplitz operators, on 2 . We also introduce the classes of analytic and coanalytic -Toeplitz operators (Definition 9), which generalize the most commonly considered classes of Toeplitz operators, and apply them to study the multiplicative properties of -Toeplitz operators. In Theorem 14, we show that a product of two -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) -Toeplitz operators need not commute, (which violates [13,Theorem 9] in the -Toeplitz operators' context). Though, we can still obtain some necessary and sufficient conditions for pairs of (co-)analytic -Toeplitz operators to commute (Theorem 17). We also obtain an interesting result (Corollary 18) on the problem of invertibility for -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 -Toeplitz operators and the classical Hankel operators (Theorem 19). This work justifies Barría and Halmos' suggestion, in [11], that the notion of -Toeplitzness may be worthy of study.
Finally, it should be mentioned that this work has its roots in [13] and been inspired by [12,14].
We close this section by setting up the notations required for what is to follow.

Notations
is the * -algebra of all bounded linear operators on 2 .
(ii) K stands for the two-sided ideal of all compact operators in B( 2 ).
(iv) The standard tensor notation will be used for operators of rank one: for and vectors in 2 , the operator ⊗ is defined by ( ⊗ )ℎ = ⟨ℎ, ⟩ 2 .

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

For a fixed complex number , a typical example of a singly infinite -Toeplitz matrix is
] . (2) In [13], it is shown that Toeplitz operators, on 2 , are the solutions of the operator-equation − = , on B( 2 ). Equivalently, this means that they are the eigen-operators of the following operator-valued linear transformation (let us call it Toeplitz mapping on B( 2 )): 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( 2 ), a -Toeplitz operator if it is an eigen-operator of the Toeplitz mapping Γ corresponding to one of its eigenvalues.
More precisely, for ∈ (Γ), the set T := ker( − Γ) consists of -Toeplitz operators corresponding to , or, equivalently, It can be easily checked that the Toeplitz mapping is a contraction; moreover, since − = , we should have . Therefore, (Γ) = (Γ) = U; that is, the only eigen-operators for Γ are the ones corresponding to the eigenvalues living in U.
, exist if and only if ∈ U. For more details on the form of a -Toeplitz operator, see [12].  (ii) If | | < 1, all solutions are compact operators of the form for some and ∈ 2 .
For convenience, let us divide -Toeplitz operators into two main classes: unimodular -Toeplitz operators and nonunimodular -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 -Toeplitz operators are compact (so are not invertible) and unimodular -Toeplitz operators are not. Indeed, in the latter case, the only compact unimodular -Toeplitz operator is the zero operator.
Hence, the entries of its matrix representation ( , ) ∞ , =0 , with respect to the monomial basis for 2 , satisfy This yields where

Basic Properties of -Toeplitz Operators
Recall that T 1 consists of all (classical) Toeplitz operators, and it turns out, as we will see later, that other -Toeplitz operators behave like them. Also, notice that, for each ∈ U, T forms a complex vector subspace of B( 2 ).
Here we look at the following straightforward properties of the -Toeplitz operators. First, from Definition 2, we observe that, for each ∈ U, T is topologically well behaved. Indeed, for ∈ T , since Γ( ) = − is weakly continuous in its middle factor, T 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 -Toeplitz operators; that is, -Toeplitz operators correspond to real eigenvalues for Γ.

Proposition 6.
For ∈ U and ∈ T , one has the following.
(ii) If is a nonzero self-adjoint element of T , then ∈ T ∩ T . But this means that which implies = , or, equivalently, ∈ R.
Remark 7. As a consequence of Sun's Theorem, every nonunimodular -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 ∈ U, letting ∈ T , and + be nonnegative integers, we have Now, if is a compact operator, then ‖ ‖ 2 ( U) → 0, as → ∞; it follows that ⟨ 0 , ⟩ 2 ( U) = 0, for all nonnegative .

Analyticity and Coanalyticity of -Toeplitz Operators
In [13, p. 96], analyticity and coanalyticity of a (classical) Toeplitz operator are defined and characterized in terms of its commutativity with and − , respectively [13,Theorem 7]. Here, we define and give an analogous characterization of these two properties in the -Toeplitz operators' setting. But let us first assign them a "symbol" (similar to one in the classical case) as a generating function.
and is denoted by sym( ), in which 0 is called the analytic symbol and * 0 is called the coanalytic symbol of .
Observation 2. For a -Toeplitz operator , sym( ) 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 2 , and whose nonpositive Fourier coefficients are the terms of the 0th-row of that matrix.
Definition 8 determines two 2 -functions, namely, 0 and * 0 , by which we may characterize the properties of analyticity and coanalyticity, for such operators, as follows.
This observation provides us with two classes of typical examples of analytic and coanalytic -Toeplitz operators.
Before stating the first result of this section, we need to introduce some terms. For a Hilbert space bounded operator , consider the operator-equation for some complex number . If there is a nonzero (bounded) operator and a scalar as above that satisfy (16), according to [15], it is said that -commutes with and that is an extended eigenvalue and is an extended eigen-operator of .
Equation (16) has been studied in [16] and, independently, in [17]. These works provided extensions of Lomonosov's classic result [18]. Now, we use these terms to state our next result which characterizes analyticity and coanalyticity of -Toeplitz operators in terms of their -commutativity with and − .

Theorem 11. Let ∈ U \ {0} and ∈ B( 2 ). A necessary and sufficient condition that is an analytic (coanalytic) -Toeplitz operator in T is that itcommutes with
( −1 -commutes with − ); that is, Proof. (i) Let be an analytic -Toeplitz operator in T ; that is, − * 0 = 0. Hence, Journal of Operators 5 Now, let ∈ B( 2 ) be such that = . Just by multiplying both sides by − from the left, one can easily see that ∈ T . To show it is analytic, we need to prove ⟨ * 0 , ⟩ = 0, for = 1, 2, . . .. So, which means is an analytic -Toeplitz operator.
(ii) If is a coanalytic -Toeplitz operator in T , that is, Now, let ∈ B( 2 ) −1 -commute with − ; that is, − = − . Multiplying both sides by from the right shows that ∈ T . To prove coanalyticity, we need to show − 0 = 0. So, we have Remark 12. Using the terms aforementioned, Theorem 11 can also be restated as follows.

Theorem 11
(i) Let ∈ U \ {0}. A necessary and sufficient condition in which ∈ T is analytic is that is an extended eigen-operator of corresponding to the extended eigenvalue .
(ii) Let ∈ U \ {0}. A necessary and sufficient condition in which ∈ T is coanalytic is that − is an extended eigen-operator of corresponding to the extended eigenvalue −1 .
Remark 13. Notice that if ∈ T 0 , it can be represented by the finite-rank operator is analytic if and only if * 0 is the constant function ⟨ 0 , 0 ⟩ 2 and, in this case, = 0 ⊗ 0 .

Multiplicative Properties of -Toeplitz Operators
Although, for a fixed ∈ U, T 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 -Toeplitzness is preserved under multiplication, on the right, by analytic -Toeplitz operators and, on the left, by coanalytic ones.
A necessary and sufficient condition that the product 1 2 is a -Toeplitz operator in T 1 2 is that either 1 is coanalytic or 2 is analytic.
Let us suppose, for now, 1 is a coanalytic -Toeplitz operator in T 1 and 2 ∈ T 2 . To show that 1 2 ∈ T 1 2 , we apply Theorem 11 to write − 1 2 which proves 1 2 is a -Toeplitz operator in T 1 2 . And if 2 is an analytic -Toeplitz operator in T 2 and 1 ∈ T 1 , again using Theorem 11 results in which proves the same thing.
From the proof of Theorem 14, along with considering Remark 7, one may deduce the following property for unimodular -Toeplitz operators.
Proof. Considering the assumptions, we have which implies Since ̸ = 0 and 1 2 is a nonzero -Toeplitz operator in T , 1 2 cannot be of finite rank (see Remark 7). Hence, both sides in (25) should be zero. Therefore, = 1 2 , and this in turn implies that either 1 should be coanalytic or 2 is analytic.

Journal of Operators
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 -Toeplitz operators; for, let us look at an example.
(1) let 1 , 2 ∈ T be analytic such that 2 is arbitrary and 1 is given by where , +1 is the Kronecker delta. Note that 2 can also be represented as Hence, we have Therefore, as analytic -Toeplitz operators, 1 and 2 do not commute.
(2) In the other direction, consider two coanalytic -Toeplitz operators 3 , 4 ∈ T , such that 4 is arbitrary and 3 = * 1 , the Hilbert space adjoint of 1 in the previous case; that is, Again, note that 4 can also be represented as Hence, we have Therefore, as coanalytic -Toeplitz operators, 3 and 4 do not commute.
(ii) Assume that 1 ∈ T 1 and 2 ∈ T 2 are both coanalytic -Toeplitz operators such that for almost all ∈ U. Their coanalyticity implies that * 1 ∈ T 1 and * 2 ∈ T 2 are analytic, which satisfy (39). Therefore, by the sufficiency condition in (37), they commute. This in turn implies that 1 and 2 commute. This proves the necessity condition in (37). Now, suppose coanalytic -Toeplitz operators 1 and 2 commute, which means analytic -Toeplitz operators * 1 ∈ T 1 and * 2 ∈ T 2 commute. Hence, by the necessity condition in (37), we should have which proves the sufficiency condition in (37).
Another consequence of Theorem 11 characterizes the -Toeplitz operators having -Toeplitz operator inverses.

Corollary 18.
For ∈ U, let ∈ T . If is invertible, then a necessary and sufficient condition that −1 is a -Toeplitz operator is that is either analytic or coanalytic.
Proof. Suppose that is invertible. If is analytic, then, by Theorem 11, -commutes with ; that is, from which follows But, on one hand, (42) implies that −1 is a -Toeplitz operator in T and, on the other hand, that it is an analytic -Toeplitz operator, using Theorem 11. The case for coanalyticity walks through the same steps as the latter case.
Suppose now that −1 is known to be a -Toeplitz operator in T , for some ∈ U. Having the following operator-equations, we obtain each of which implies = ; that is, −1 ∈ T , and, in this case, from the first equation follows either is coanalytic or −1 is analytic. And the second one also implies either −1 is coanalytic or is analytic.
If is not coanalytic, then −1 is analytic and not constant; this implies that −1 is not coanalytic and hence that is analytic. The same reasoning also works when it is assumed that is not analytic.

-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 , ∈ B( 2 ) such that is a Hankel operator and is an analytic Toeplitz operator, then which states that satisfies the Hankel equation, so is a Hankel operator. A similar way shows that is a Hankel operator, where is a coanalytic Toeplitz operator.
It turns out that Hankel operators behave in a similar manner when they meet analytic/coanalytic -Toeplitz operators.

Theorem 19. Let
∈ B( 2 ) be a Hankel operator and ∈ U. If 1 is an analytic and 2 is a coanalytic -Toeplitz operator in T , then 1  Proof. As it is well known, is a Hankel operator if and only if − = . Then, simply, we have proving the assertion.

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