The Schwarz inequality and Jensen’s one are fundamental in a Hilbert space. Regarding a sesquilinear map B(X, Y ) = Y*X as an OPERATORvalued inner product, we discuss OPERATOR versions for the above inequalities and give simple conditions that the equalities hold.

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In the present paper, taking some advantages offered by the context of finite dimensional Hilbert spaces, we shall give a complete characterizations of certain distinguished classes of OPERATORs (self-adjoint, unitary reflection, normal) in terms of OPERATOR inequalities. These results extend previous characterizations obtained by the second author.

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Any notion of purity is normally defined in terms of solvability of some set of equations. To study mathematical notions, such as injectivity, tensor products, flatness, one needs to have some categorical and algebraic information about the pair (A, M), for a category A and a class M of monomorphisms in a category A. In this paper we take A to be the category Act-S of S-acts, for a semigroup S, and Msp to be the class of CspI-pure monomorphisms and study some categorical and algebraic properties of this class concerning the closure OPERATOR CspI.

In this paper by using a nice criterion, we show that the perturbation of identity OPERATORs by some multiples of the standard backward shift is hypercyclic. This gives a new proof for Salas Theorem in ( [10 ], Theorem 3.3).