Solutions of the system of operator equations $BXA=B=AXB$ via the *-order

Abstract

Abstract. In this paper, some necessary and sufficient conditions are established for the existence of solutions to the system of operator equations BXA = B = AXB in the setting of bounded linear operators on a Hilbert space, where the unknown operator X is called the inverse of A along B. After that, under some mild conditions, it is proved that an operator

The Sum Two of Hermitian Operators Ai=Ti+Mi for Solving the Equations{Ai X=Ui },i=1,2

“…Researchers have increased interest in studying matrix and operator equation and the system of it, can you see [8][9][10][11][12][13]. [28], [29] Τ𝒳 = 𝒰 and Τ 𝑖 𝒳 = 𝒰 𝑖 , 𝑖 = 1,2, 1 In [14], Zhang studied the Hermitian positive semidefinite solution of Τ𝒳𝒱 = 𝒰, 2 The Banach or Hilbert spaces was used for matrices and bounded linear operators (see, [13,[15][16][17][18][19][20][21][22]), and finding necessary conditions and sufficient conditions (N-SCs) for an existence of a combined solutions, and generalization of two equations Τ 𝑖 𝒳𝒱 𝑖 = 𝒰 𝑖 , 𝑖 = 1,2, 3 Also, the equation's solvability, as follows Τ 1 𝒳𝒱 1 + Τ 2 𝒳𝒱 2 = 𝒰, 4 Vosough and Moslehian [23] gave characterizations of the existence and representations of the solutions to the system and restricted the case of operator systems ℬ𝒳𝒜 = ℬ = 𝒜𝒳, 5…”

Parametrized solutions $X$ of the system $AXA = AY A$ and $A^k Y AX = XAY A^k$

“…For instance, the problem of finding all the solutions of the classical Yang-Baxter matrix equation AXA = XAX is a representative case [14,17,21]. On the other hand, some recent results finding algebraic solutions for operator equations such as AXB = B = BXA can be found in [22,26].…”