Guelfen H, Kittaneh F.
On Numerical Radius Inequalities for Operator Matrices. Numerical Functional Analysis and Optimization. 2019;40 (11) :1231-1241.
Ameur S.
SELFADJOINT OPERATORS, NORMAL OPERATORS, AND CHARACTERIZATIONS. Operators and Matrices. 2019;13 (3) :835–842.
Abstract
Let B(H) be the C∗ -algebra of all bounded linear operators acting on a complex separable Hilbert space H . We shall show that:
1. The class of all selfadjoint operators in B(H) multiplied by scalars is characterized by ∀X ∈ B(H), S2X +XS2 =>2||SXS||, (S ∈ B(H)).
2. The class of all normal operators in B(H) is characterized by each of the three following properties (where DS = S∗S−SS∗ , for S ∈ B(H)),
(i) ∀X ∈ B(H), S2X + XS2 =>2||SXS||,(S ∈ B(H)),
(ii) S∗DSS = 0 = SDSS∗,(S ∈ B(H)),
(iii) S∗DSS=> 0 =>SDSS∗,(S ∈ B(H)).
Lombarkia F, Boussaid A.
Operator equations and inner inverses of elementary operators. Linear and Multilinear Algebra . 2019.
Abstract
Let E,F,G,D be infinite complex Banach spaces and B(F,E) the Banach space of all bounded linear operators from F into E. Consider A1,A2∈B(F,E), B1,B2∈B(D,G)B1,B2∈B(D,G). Let MA1,B1:X→A1XB1 be the multiplication operator on B(G,F) induced by A1,B1. In particular, LA1=MA1,I and RB1=MI,B1, where I is the identity operator are the left and the right multiplication operators, respectively. The elementary operator Ψ defined on B(G,F)B(G,F) is the sum of two multiplication operators Ψ=MA1,B1+MA2,B2. This paper gives necessary and sufficient conditions for the existence of a common solution of the operator equations MA1,B1(X)=C1 and MA2,B2(X)=C2 and derive a new representation of the general common solution via the inner inverse of the elementary operator Ψ; we apply this result to determine new necessary and sufficient conditions for the existence of a Hermitian solution and a representation of the general Hermitian solution to the operator equation MA,B(X)=C. As a consequence, we obtain well-known results of Dajic´ and Koliha.
