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We study some properties of -normal operators and we present various inequalities between the operator norm and the numerical radius of -normal operators on Banach algebraℬ() of all bounded linear operators , where is Hilbert space.

Abstract. Let A, B and C be adjointable operators on a Hilbert C * -module E . Giving a suitable version of the celebrated Douglas theorem in the context of Hilbert C * -modules, we present the general solution of the equation AX + Y B = C when the ranges of A, B and C are not necessarily closed. We examine a result of Fillmore and Williams in the setting of Hilbert C * -modules. Moreover, we obtain some necessary and sufficient conditions for existence of a solution for AXA * + BY B * = C. Finally, we deduce that there exist nonzero operators X, Y ≥ 0 and Z such that AXA * + BY B * = CZ, when A, B and C are given subject to some conditions. Introduction and PreliminariesRecently several operator equations have been extended from matrices to infinite dimensional spaces, i.e., Hilbert spaces and Hilbert C * -modules; see [11] and references therein. Recall that the notion of Hilbert C * -module is a natural generalization of that of Hilbert space arising by replacing the field of scalars C by a C * -algebra. Generalized inverses are useful tools for investigation of solutions of operator equations in the setting of Hilbert C * -modules but these inverses need the strong condition of closedness of ranges of considered operators. Fang et al. [6,7] have studied the solvability of operator equations without the closedness condition on ranges of operators by employing a generalization of a known theorem of Douglas [4, Theorem 1] in the framework of Hilbert C * -modules. In their results, concentration is based on the idea of using more general (orthogonal) projections instead of projections such as AA † . They investigated the equations AX = B Inspired by Fang et al., we investigate the solution of equations AX + Y B = C and AXA * +BY B * = C without the condition of closedness of ranges. This paper is organized as follows. First, we recall some basic information about Hilbert C * -modules. In Section 2, we present an example that shows that the conditions CC * ≤ λAA * for some λ > 0 and R(C) ⊆ R(A) are not equivalent in the setting 2010 Mathematics Subject Classification. 15A24, 46L08, 47A05, 47A62.

In the first part of the paper, we use states on $C^{*}$ -algebras in order to establish some equivalent statements to equality in the triangle inequality, as well as to the parallelogram identity for elements of a pre-Hilbert $C^{*}$ -module. We also characterize the equality case in the triangle inequality for adjointable operators on a Hilbert $C^{*}$ -module. Then we give certain necessary and sufficient conditions to the Pythagoras identity for two vectors in a pre-Hilbert $C^{*}$ -module under the assumption that their inner product has a negative real part. We introduce the concept of Pythagoras orthogonality and discuss its properties. We describe this notion for Hilbert space operators in terms of the parallelogram law and some limit conditions. We present several examples in order to illustrate the relationship between the Birkhoff–James, Roberts, and Pythagoras orthogonalities, and the usual orthogonality in the framework of Hilbert $C^{*}$ -modules.

Necessary and sufficient conditions are given for the operator system $A_1X=C_1$, $XA_2=C_2$, $A_3XA^*_3=C_3$, and $A_4XA^*_4=C_4$ to have a common positive solution, where $A_i$'s and $C_i$'s are adjointable operators on Hilbert $C^*$-modules. This corrects a published result by removing some gaps in its proof. Finally, a technical example is given to show that the proposed investigation in the setting of Hilbert $C^*$-modules is different from that of Hilbert spaces.