Abstract. The problem considered is the determination of``lower bounds'' of matrix operators on the spaces p w or d wY p . Under fairly general conditions, the solution is the same for both spaces and is given by the in®mum of a certain sequence. Speci®c cases are considered, with the weighting sequence de®ned by w n 1an . The exact solution is found for the Hilbert operator. For the averaging operator, two di erent upper bounds are given, and for certain values of p and it is shown that the smaller of these two bounds is the exact solution.1991 Mathematics Subject Classi®cation. Primary 47B37; secondary 15A60, 26D15, 46B45.

In this paper, we generalize several Berezin number inequalities involving product of operators, which acting on a Hilbert space H (Ω). Among other inequalities, it is shown that if A, B are positive operators and X is any operator, then ber r (H α (A, B)2010 Mathematics Subject Classification. Primary 47A30, Secondary 15A60, 30E20, 47A12 .

We generalize several inequalities involving powers of the numerical radius for off-diagonal part of 2 × 2 operator matrices of the form T = 0 B C 0 , where B, C are two operators. In particular, if T = 0 BC 0 , then we get

Abstract. In this paper, we considered the problem of finding the upper bound Hausdorff matrix operator from sequence spaces l p (v) (or d(v, p)) into l p (w) (or d(w, p)

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