Khatri-Rao Products for Operator Matrices Acting on the Direct Sum of Hilbert Spaces

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 andHadamard product of operators and theKhatri-Rao product ofmatrices.We investigate algebraic properties, positivity, and monotonicity of the Khatri-Rao product. Moreover, there is a unital positive linear map taking TracySingh products to Khatri-Rao products via an isometry.


Introduction
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, [1][2][3].Denote by  , (C) the set of -by- complex matrices and abbreviate  , (C) to   (C).Recall that the Kronecker product of  = [  ] ∈  , (C) and  ∈  , (C) is given by ⊗ = [  ]  ∈  , (C) . ( The Hadamard product of ,  ∈  , (C) is defined by the entrywise product Now, let  and  be complex matrices partitioned into blocks   and   for each ,  (the sizes of   and   may be different).Then, the Khatri-Rao product [4] of  and  is defined by When  and  are nonpartitioned (i.e., each has only one block), their Khatri-Rao product is just their Kronecker product.If  and  are entrywise partitioned (i.e., each block is a 1 × 1 matrix), then their Khatri-Rao product is their Hadamard product.Interesting algebraic, order, and analytic properties of this product were studied in the literature; see, for example, [5][6][7][8][9][10][11][12].Their applications in statistics, computer science, and related fields can be seen, for example, in [13,14].The tensor product of Hilbert space operators is a natural extension of the Kronecker product to the infinitedimensional setting.Let H, H  , K, and K  be Hilbert spaces.Recall that the tensor product of two operators  : H → H  and  : K → K  is the unique bounded linear operator from H ⊗ K into H  ⊗ K  such that, for all  ∈ H and  ∈ K, 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  ⊠  to the Khatri-Rao product  ⊡ .Hence, the Khatri-Rao product can be viewed as a generalization of the Hadamard product of operators.Moreover, positivity, strict positivity, and operator orderings are preserved under the Khatri-Rao product.Our result extends well-known results for Khatri-Rao products of complex matrices (see [4,9,15,16]).
This paper is organized as follows.In Section 2, we provide some preliminaries about Tracy-Singh products for operators.These facts will be used in Sections 4 and 5.In Section 3, we introduce the Khatri-Rao product for operator matrices and deduce its algebraic properties.Section 4 explains how the Khatri-Rao product can be regarded as a generalization of the Hadamard product.Section 5 discusses positivity and monotonicity of Khatri-Rao products.

Preliminaries on Tracy-Singh Products for Operators
where all H  , H Basic properties of the Tracy-Singh product are listed below.
Lemma 1.The Tracy-Singh product (, )  →  ⊠  is a bilinear map for operators.It is positive in the sense that if  ⩾ 0 and  ⩾ 0, then  ⊠  ⩾ 0.

Compatibility of Khatri-Rao Products with Algebraic Operations
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: That is, we fix how to partition any operator matrix in B(H, H  ) and B(K, K  ).We now extend the Khatri-Rao product of matrices [4] to that of operators on a Hilbert space.
,=1 ∈ B(K, K  ) be operators partitioned into matrices according to decomposition (8).We define the Khatri-Rao product of  and  to be the bounded linear operator from If both  and  are 1 × 1 block operator matrices, then  ⊡  is  ⊗ .When H  = K  = C and H   = K   = C for all , , the Khatri-Rao product is the Hadamard product of complex matrices.
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  ∈  , (C) and  ∈  , (C), the induced maps, are bounded linear operators.We identify C  ⊗ C  with C  together with the canonical bilinear map (, )  → ⊗ for each (, ) ∈ C  × C  .Lemma 3.For any  ∈  , (C) and  ∈  , (C), one has Proof.Recall that the Kronecker product of matrices has the following property (see, e.g., [3]): (⊗) (⊗) = ⊗ (12) provided that all matrix products are well defined.It follows that, for any  ∈ C  and  ∈ C  , The uniqueness of tensor products implies that   ⊗   =  ⊗ .Proposition 4. For any complex matrices  = [  ] and  = [  ] partitioned in block-matrix form, one has Proof.Recall that the (, )th block of the matrix representation of   is    .By Lemma 3, we obtain The next result states that the Khatri-Rao product is bilinear and compatible with the adjoint operation.Proposition 5. Let  ∈ B(H, H  ) and ,  ∈ B(K, K  ) be operator matrices, and let  ∈ C.Then, The fact that ( + )  =   +   for all ,  together with the left distributivity of the tensor product over the addition implies Similarly, we obtain property (17).Since ()  =   for all , , we get Similarly,  ⊡ () = ( ⊡ ).
By property (15), the self-adjointness of operators is closed under taking Khatri-Rao products; that is, if  and  are self-adjoint, then so is  ⊡ .The next proposition shows that, in order to compute the Khatri-Rao product of operator matrices, we can freely merge the partition of each operator.
,=1 ∈ B(K, K  ) be operator matrices represented according to decomposition (8).We merge the partition of  to be  = [  ] , ,=1 , where ,  are given natural numbers such that  ⩽  and  ⩽ .Here, each operator   is of   ×   block in which the (, ℎ)th block of   is the (, V)th block of , where Similarly, we repartition  = [  ] , ,=1 , where each operator   is of   ×   block in which the (, ℎ)th block of   is the (, V)th block of .Then, That is, each (, )th block of  ⊡  is just   ⊡   .
Recall that the direct sum of   ∈ B(H  , H   ),  = 1, . . ., , is defined to be the operator matrix The next result shows that the Khatri-Rao product is compatible with the direct sum of operators.
In summary, the Khatri-Rao product is compatible with fundamental algebraic operations for operators.

The Khatri-Rao Product as a Generalization of the Hadamard Product
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 E for H. Recall that the Hadamard product of  and  in B(H) is defined to be the operator  ⊙  in B(H) such that ⟨( ⊙ ) , ⟩ = ⟨, ⟩ ⟨, ⟩ for all  ∈ E.More explicitly, it was shown in [15] that where  : H → H ⊗ H is the isometry defined by  =  ⊗  for all  ∈ E. When H = C  and E is the standard ordered basis of C  , the Hadamard product of two matrices reduces to the entrywise product (2).We now extend selection matrices in [9] to selection operators.Fix an ordered 4-tuple (H, H  , K, K  ) of Hilbert spaces endowed with decomposition (8).For each  = 1, . . ., , consider the operator matrix where  ()  ℎ is the identity operator if  = ℎ =  and the others are zero operators.Similarly, for  = 1, . . ., , we define the operator matrix where  () ℎ is the identity operator if  = ℎ =  and the others are zero operators.Now, construct We call  1 and  2 selection operators associated with the ordered tuple (H, H  , K, K  ).Notice that  1 depends only on the ordered tuple (H  , K  ) and how we decomposed H  and K  .The operator  2 depends on (H, K) and how we decomposed H and K.For instance, an ordered tuple (H, H  , K, K  ) with decompositions has the following selection operators: In the case of H = H  and K = K  , construction (31) gives is the ordered pair of selection operators associated with the ordered tuple (H, H  , K, K  ) with decompositions given by ( 8), then ( 2 ,  1 ) is the ordered pair of selection operators associated with the ordered collection (H  , H, K  , K) with the same decompositions.
Remark 10.If we partition  and  into row operator matrices, we have If both  and  are column operator matrices, then Comparing ( 28) and (37), Theorem 9 shows that the Khatri-Rao product can be regarded as a generalization of the Hadamard product.
Recall that a map Φ between two  * -algebras is said to be positive if Φ preserves positive elements.The map Φ is unital if Φ preserves the multiplicative identity.
Corollary 11.There is a unital positive linear map such that Φ( ⊠ ) =  ⊡  for any  ∈ B(H) and  ∈ B(K).
Proof.Define Φ() =  * , where  is the selection operator defined by (37) in Theorem 9.The map Φ is clearly linear and positive.The map Φ is unital since  is an isometry (Lemma 8).
Corollary 11 provides a natural way to derive operator inequalities concerning Khatri-Rao products from existing inequalities for Tracy-Singh products.
The next result extends [16,Corollary 3] to the case of Khatri-Rao and Tracy-Singh products of operators.

Positivity and Monotonicity of Khatri-Rao Products
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.
Proof.It follows from the positivity of the Tracy-Singh product (Lemma 1) and Theorem 9.
The next result provides the monotonicity of Khatri-Rao product which is an extension of [9,Theorem 5] to the case of operators.Proof.The strict positivity of  and the spectral theorem imply the existence of an increasing sequence (H  ) Proof.The proof is similar to that of Corollary 14.Instead of Theorem 13, we apply Theorem 15.
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.
, K  , and K   are Hilbert spaces.For each , , let   : H  → H and   : K  → K be the canonical embeddings.For each  and , let   : H  → H   and   : K  → K   be the orthogonal projections.Two operators  ∈ B(H, H  ) and  ∈ B(K, K  ) can thus be represented uniquely as operator matrices  = [  ]  =     ∈ B(H  , H   ) and   =     ∈ B(K  , K   ) for each , , , and .We define the Tracy-Singh product of  and  to be the bounded linear operator from ⨁ ∞ =1 of closed subspaces of H such that, for each  ∈ N, H  .Let   be the orthogonal projection onto H  for each  ∈ N.There are similar subspaces K  and orthogonal projections   for the operator .Then, for each  ∈ N, we have  ⩾ (1/)  and  ⩾ (1/)  and hence ⊡  ⩾ 1  2   ⊡  (47)by Corollary 14.Since the union of the subspaces H  in H and of the subspaces K  in K is dense, it follows that, for any  ∈ H ⊗ K, there is  ∈ N for which ⟨(  ⊡   ), ⟩ > 0. Hence,⟨( ⊡ ) , ⟩ ⩾ 1  2 ⟨(  ⊡   ) , ⟩ > 0. Let  1 ,  2 ∈ B(H) and  1 ,  2 ∈ B(K).If  1 >  2 > 0 and  1 >  2 > 0, then  1 ⊡  1 >  2 ⊡  2 .