Bounds and positivity conditions for operator valued functions in a Hilbert lattice

In this paper we investigate regular functions of a bounded operator A acting in a Hilbert lattice and having the form A=D + T, where T is a positive operator and D is a selfadjoint operator whose resolution of the identity P(t) $(ale s le b)$ has the property $P(s_2)-P(s_1);;(s_1 are non-negative in the sense of the order. Upper and lower bounds and positivity conditions for the considered operator valued functions are derived. Applications of the obtained estimates to functions of integral operators, partial integral operators, infinite matrices and differential equations are also discussed.