Approximating n-time differentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor type expansion

On utilizing the spectral representation of selfadjoint operators in Hilbert spaces, some approximations for the n-time di¤erentiable functions of selfadjoint operators in Hilbert spaces by two point Taylor’s type expansions are given. 1. Introduction Let U be a selfadjoint operator on the complex Hilbert space (H; h:; :i) with the spectrum Sp (U) included in the interval [m;M ] for some real numbers m < M and let fE g be its spectral family. Then for any continuous function f : [m;M ]! C, it is well known that we have the following spectral representation in terms of the Riemann-Stieltjes integral : (1.1) f (U) = Z M m 0 f ( ) dE ; which in terms of vectors can be written as (1.2) hf (U)x; yi = Z M m 0 f ( ) d hE x; yi ; for any x; y 2 H: The function gx;y ( ) := hE x; yi is of bounded variation on the interval [m;M ] and gx;y (m 0) = 0 and gx;y (M) = hx; yi for any x; y 2 H: It is also well known that gx ( ) := hE x; xi is monotonic nondecreasing and right continuous on [m;M ]. For a recent monograph devoted to various inequalities for continuous functions of selfadjoint operators, see [11] and the references therein. For other recent results see [1], [2], [5]-[9], [13], [14], [15] and [16]. The following result provides a Taylor’s type representation for a function of selfadjoint operators in Hilbert spaces with integral remainder: Theorem 1 (Dragomir, 2010, [10]). Let A be a selfadjoint operator in the Hilbert space H with the spectrum Sp (A) [m;M ] for some real numbers m < M , fE g be its spectral family, I be a closed subinterval on R with [m;M ] I (the interior of I) and let n be an integer with n 1: If f : I ! C is such that the n-th derivative 1991 Mathematics Subject Classi…cation. 47A63; 47A99.