UPPER AND LOWER BOUNDS FOR AN INTEGRAL TRANSFORM OF POSITIVE OPERATORS IN HILBERT SPACES WITH APPLICATIONS

For a continuous and positive function w ( ) ; > 0 and a positive measure on (0;1) we consider the following integral transform D (w; ) (T ) := Z 1 0 w ( ) ( + T ) 1 d ( ) ; where the integral is assumed to exist for T a positive operator on a complex Hilbert space H. In this paper we show, among others that, if the positive operators A; B satisfy the separation condition 0 < A < B for some positive constants ; ; ; ; then 0 [D (w; ) ( ) D (w; ) ( )] D (w; ) (A) D (w; ) (B) [D (w; ) ( ) D (w; ) ( )] : If A; B > 0 with kAk B 1 < 1; then 0 1 kAk B 1 (kBk kAk) kB 1k [D (w; ) (kAk) D (w; ) (kBk)] D (w; ) (A) D (w; ) (B) kBk A 1 1 kA 1k kB 1k B 1 hD (w; ) A 1 1 D (w; ) B 1 1 i : Some natural applications for operator monotone and operator convex functions are also given. 1. Introduction Consider a complex Hilbert space (H; h ; i). An operator T is said to be positive (denoted by T 0) if hTx; xi 0 for all x 2 H and also an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. A real valued continuous function f on (0;1) is said to be operator monotone if f(A) f(B) holds for any A B > 0: We have the following representation of operator monotone functions [10], see for instance [1, p. 144-145]: 1991 Mathematics Subject Classication. 47A63, 47A60.

where the integral is assumed to exist for T a positive operator on a complex Hilbert space H.
In this paper we show, among others that, if the positive operators A; B satisfy the separation condition i : Some natural applications for operator monotone and operator convex functions are also given.

Introduction
Consider a complex Hilbert space (H; h ; i). An operator T is said to be positive (denoted by T 0) if hT x; xi 0 for all x 2 H and also an operator T is said to be strictly positive (denoted by T > 0) if T is positive and invertible. A real valued continuous function f on (0; 1) is said to be operator monotone if f (A) f (B) holds for any A B > 0: We have the following representation of operator monotone functions [10], see for instance [1, p. 144-145]: Theorem 1. A function f : (0; 1) ! R is operator monotone in (0; 1) if and only if it has the representation where a 2 R; b 0 and a positive measure on (0; 1) such that A real valued continuous function f on an interval I is said to be operator convex (operator concave) on I if where a; b 2 R; c 0 and a positive measure on (0; 1) such that (1.2) holds. If f is operator convex in [0; 1); then a = f (0) and b = f 0 + (0) ; the right derivative, in (1.1).
We have the following integral representation for the power function when t > 0; r 2 (0; 1]; see for instance [1, p. 145] t r 1 = sin (r ) Motivated by these representations, we introduce, for a continuous and positive function w ( ) ; > 0; the following integral transform where is a positive measure on (0; 1) and the integral (1.4) exists for all t > 0: For the Lebesgue usual measure, we put Now, assume that T > 0, then by the continuous functional calculus for selfadjoint operators, we can de…ne the positive operator where w and are as above. Also, when is the usual Lebesgue measure, then for T > 0: If we take to be the usual Lebesgue measure and the kernel w r ( ) = r 1 ; r 2 (0; 1]; then We de…ne the upper incomplete Gamma function as [8] (a; z) := Z 1 z t a 1 e t dt; which for z = 0 gives Gamma function We have the integral representation [9] (1.9) (a; z) = z a e z (1 a) for Re a < 1 and jph zj < : Now, we consider the weight w a e ( ) := a e for > 0: Then by (1.9) we have Let a = 1 n; with n a natural number with n 0; then by (1.10) we have If we de…ne the generalized exponential integral [6] by for n 1 and t > 0: Using the identity [6, Eq 8.19.7], for n 2 for n 2 and t > 0: for a < 1.
In particular, and, for n 2 where T > 0: For n = 2; we also get for T > 0: We consider the weight w ( +a) 1 ( ) := 1 +a for > 0 and a > 0: Then, by simple calculations, we get for all a > 0 and t > 0 with t 6 = a: From this, we get for all t; a > 0: If T > 0; then Let a > 0: Assume that either 0 < T < a or T > a; then by (1.21) we get We can also consider the weight w ( 2 +a 2 ) 1 ( ) := 1 2 +a 2 for > 0 and a > 0: Then, by simple calculations, we get for t > 0 and a > 0: If T > 0 and a > 0; then and, in particular, Assume that 0 < A < B: We say that these operators are separated if there exists 0 < < such that 0 < A < B: For a positive operator T > 0; we have the operator inequalities T 1 1 T kT k : Therefore, if A; B > 0 with kAk B 1 < 1; then The class of two separated positive operators play an important role in establishing various re…nements and reverses of operator Young inequalities as pointed out in numerous recent papers from which we only mention [3], [13] and the references therein.
In this paper we show, among others that, if the positive operators A; B satisfy the separation condition Some natural applications for operator monotone and operator convex functions are also given.

Main Results
In the following, whenever we write D (w; ) we mean that the integral from (2.3) exists and is …nite for all t > 0: Proof. Observe that, for all A; B > 0 If we write this equality for the function f (t) = t 1 and C; D > 0; then we get the representation Now, if we take in (2.3) C = + B; D = + A; then and by (2.2) we derive which, by the change of variable t = 1 s; gives (2.1).
We have the following double inequality for two positive separated operators: If we multiply this inequality by w ( ) 0 and integrate over the measure ( ) ; we get and, by (2.1) we derive the inequality of interest for all s 2 [0; 1] and 0: Therefore Finally, on making use of (2.6), (2.9) and (2.10), we derive (2.5).

Some Examples
Consider the operator monotonic function f (t) = t r ; r 2 (0;  i : In particular, The interested author may state other similar inequalities by using the examples of operator monotone functions from [2], [4] and the references therein.