**2. Preliminaries**

In this paper, we consider complex Hilbert spaces H and K. The symbol B(X) stands to the Banach space of bounded linear operators on a Hilbert space X. The cone of positive operators on X is denoted by <sup>B</sup>(X)+. For Hermitian operators *A* and *B* in <sup>B</sup>(X), the situation *A* ≥ *B* means that *A* − *B* ∈ <sup>B</sup>(X)+. Denote the set of all positive invertible operators on X by <sup>B</sup>(X)++.

We fix the following orthogonal decompositions:

$$\mathbb{H} = \bigoplus\_{i=1}^{m} \mathbb{H}\_i \quad \mathbb{K} = \bigoplus\_{k=1}^{n} \mathbb{K}\_k$$

where all H*i* and K*j* are Hilbert spaces. Such decompositions lead to a unique representation for each operator *A* ∈ B(H) and *B* ∈ B(K) as a block-matrix form:

$$A = \begin{bmatrix} A\_{ij} \end{bmatrix}\_{i,j=1}^{m,m} \quad \text{and} \quad B = \begin{bmatrix} B\_{kl} \end{bmatrix}\_{k,l=1}^{n,u}$$

where *Aij* ∈ <sup>B</sup>(<sup>H</sup>*j*, <sup>H</sup>*i*) and *Bkl* ∈ <sup>B</sup>(<sup>K</sup>*l*, <sup>K</sup>*k*) for each *i*, *j*, *k*, *l*.
