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.

All of the work I am about to describe takes place in the Hardy-Hilbert space of the unit circle

Two of the most intensely studied classes of bounded operators on

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

The

The

The

These classes of operators can also be considered as solutions to some linear operator-equations involving the Toeplitz operator

Generalizations of such operator-equations have been studied and explored for some time. For instance, in [

Here we study an operator-equation, on

In this paper we study and develop some algebraic and operator-theoretic properties of

Finally, it should be mentioned that this work has its roots in [

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

For

The standard tensor notation will be used for operators of rank one: for

The standard orthonormal basis

For

Here we give two approaches for defining the concept of “

One calls a singly infinite matrix a

For a fixed complex number

One calls an operator, in

More precisely, for

Bounded operator-solutions to

Let

If

If

for some

For convenience, let us divide

Some immediate consequences of Sun’s Theorem are that the nonunimodular

By Definition

Recall that

Here we look at the following straightforward properties of the

The next result, inspired by [

For

If

(i) Since

(ii) If

As a consequence of Sun’s Theorem, every nonunimodular

And, if we apply the same procedure for

In [

For

For a

Definition

A

This definition makes the following remark obvious.

For

Notice that for,

For

For

Note that if

Before stating the first result of this section, we need to introduce some terms. For a Hilbert space bounded operator

Equation (

Now, we use these terms to state our next result which characterizes analyticity and coanalyticity of

Let

(i) Let

(ii) If

Using the terms aforementioned, Theorem

Theorem

Notice that if

And

Although, for a fixed

For

A necessary and sufficient condition that the product

Let us first assume

Let us suppose, for now,

And if

From the proof of Theorem

For

Considering the assumptions, we have

Recall that if the symbols of two (classical) Toeplitz operators are either analytic or coanalytic, they necessarily commute [

For some

let

where

Hence, we have

Therefore, as analytic

In the other direction, consider two coanalytic

Again, note that

Hence, we have

Therefore, as coanalytic

Though, we can still obtain some necessary and sufficient conditions for pairs of (co-)analytic

For

If both

If both

(i) Assume that

By the assumption

Let us assume that

(ii) Assume that

Now, suppose coanalytic

Another consequence of Theorem

For

Suppose that

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

Suppose now that

If

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

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

Let

As it is well known,

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

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.