We introduce the notion of Khatri-Rao product for operator matrices acting on the direct sum of Hilbert spaces. This notion generalizes the tensor product and Hadamard product of operators and the Khatri-Rao product of matrices. We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking Tracy-Singh products to Khatri-Rao products via an isometry.

In matrix theory, there are various matrix products which are of interest in both theory and applications, such as the Kronecker product, Hadamard product, and Khatri-Rao product; see, for example, [

The tensor product of Hilbert space operators is a natural extension of the Kronecker product to the infinite-dimensional setting. Let

In this paper, we generalize the tensor product of operators to the Khatri-Rao product of operator matrices acting on a direct sum of Hilbert spaces. We investigate fundamental properties of this operator product. Algebraically, this product is compatible with the addition, the scalar multiplication, the adjoint operation, and the direct sum of operators. By introducing suitable operator matrices, we can prove that there is a unital positive linear map taking the Tracy-Singh product

This paper is organized as follows. In Section

Throughout, let

Decompose

The Tracy-Singh product

In this section, we define the Khatri-Rao product for operator matrices and show that this product is compatible with certain algebraic operations of operators.

From now on, fix the following orthogonal decompositions of Hilbert spaces:

Let

If both

Next, we shall show that the Khatri-Rao product of two linear maps induced by matrices is just the linear map induced by the Khatri-Rao product of these matrices. Recall that, for each

For any

Recall that the Kronecker product of matrices has the following property (see, e.g., [

For any complex matrices

Recall that the

The next result states that the Khatri-Rao product is bilinear and compatible with the adjoint operation.

Let

Since

By property (

Let

Write

Recall that the direct sum of

For each

It follows directly from Proposition

In summary, the Khatri-Rao product is compatible with fundamental algebraic operations for operators.

In this section, we explain how the Khatri-Rao product can be viewed as a generalization of the Hadamard product. To do this, we construct two isometries which identify which blocks of the Tracy-Singh product we need to get the Khatri-Rao product.

Fix a countable orthonormal basis

We now extend selection matrices in [

Let

A direct computation shows that

Next, we relate the Khatri-Rao and the Tracy-Singh product of operators.

For any operator matrices

Let

We mention that Theorem

If we partition

Comparing (

Recall that a map

There is a unital positive linear map

Define

Corollary

The next result extends [

Let

Using the fact that

In this section, we show that the Khatri-Rao product preserves positivity and strict positivity. It follows that operator orderings are preserved under Khatri-Rao products.

Let

It follows from the positivity of the Tracy-Singh product (Lemma

The next result provides the monotonicity of Khatri-Rao product which is an extension of [

Let

Applying Proposition

Now, we will develop the result of [

Let

The strict positivity of

Let

The proof is similar to that of Corollary

Finally, we mention that, by using the results in this paper, we can develop further operator identities/inequalities parallel to matrix results for Khatri-Rao products.

The authors declare that they have no competing interests.

This work was supported by the Thailand Research Fund. The second author would like to thank the Thailand Research Fund for the financial support.