Let A be a positive bounded operator on a Hilbert space H, ·, · . The semi-inner product x, y A := Ax, y , x, y ∈ H, induces a seminorm · A on H. Let T A , w A (T ), and c A (T ) denote the A-operator seminorm, the A-numerical radius, and the A-Crawford number of an operator T in the semi-Hilbertian space H, · A , respectively. In this paper, we present some seminorm inequalities and equalities for semi-Hilbertian space operators. More precisely, we give some necessary and sufficient conditions for two orthogonal semi-Hilbertian operators satisfy Pythagoras' equality. In addition, we derive new upper and lower bounds for the numerical radius of operators in semi-Hilbertian spaces. In particular, we show that

We review some significant generalizations and applications of the celebrated Douglas theorem on equivalence of factorization, range inclusion, and majorization of operators. We then apply it to find a characterization of the positivity of 2 × 2 block matrices of operators on Hilbert spaces and finally describe the nature of such block matrices and provide several ways for showing their positivity.

Let n be any natural number. The n-centered operator is introduced for adjointable operators on Hilbert C * -modules. Based on the characterizations of the polar decomposition for the product of two adjointable operators, n-centered operators, centered operators as well as binormal operators are clarified, and some results known for the Hilbert space operators are improved. It is proved that for an adjointable operator T , if T is Moore-Penrose invertible and is n-centered, then its Moore-Penrose inverse is also n-centered. A Hilbert space operator T is constructed such that T is n-centered, whereas it fails to be (n + 1)-centered.2010 Mathematics Subject Classification. Primary 46L08; Secondary 47A05.

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