SOME APPLICATIONS OF SCHWARZ LEMMA FOR OPERATORS

A generalized Schwarz lemma and some Harnack type inequalities for operators have been obtained in this paper.


INTRODUCTION.
Let A be a bounded linear operator on a complex Hilbert space H.For a complex valued function f analytic on a domain E of the complex plane containing the spectrum o(A) of A, let f(A) denote the operator on H defined by the Riesz Dunford integral ([2, p.568]).where C is a postively oriented simple closed rectifiable contour containing o(A) in its inside domain R and satisfying CURe E.Fan [3] has obtained Schwarz 1emma for f(A) and has given several applications of his results including the Harnack's inequalities for operators in [3,4].
In this paper, we obtain a generalized Schwarz lemma and some further Harnack type inequalities for operators.

SOME PRELIMINARY LEMMAS.
We need the following lemmas.
LEMMA i.Let a,b,c,d be complex numbers such that ad bc 0, c # 0 and let T be a bounded linear operator on a Hilbert space H such that -d/c is not in o(T).The operator inside the square brackets can be written as PROOF.The inequality (2.3) is equivalent to rladbc ll{(aT+bl)( Idl 2 r21cl 2 (b r 2 aE) (cT + dl)} (cT + dl)-lll .<After simplication the above can be written as 2 (dT + r El) (cT + dl)-lll _<-r Now an application of Lemma 1 shows that (2.4) is equivalent to T l{r 3. A GENERALIZED SCHWARZ LEMMA.
Let D denote the open unit disc {z: z I<i} n the complex plane and let H(D) be the class of complex valued functions analytic in D. PROOF.Since f is in B(D) and A is a proper contraction, by a result of F and ([3, Theorem i, p.276]), T f(A) is also a proper contraction.Now, if we define the complex valued function g by g(z) Using triangle inequality we get both the inequalities in (3.1).CORALLARY I. Let f in B (D) be given by the series f(z) bzn+ o b 0, and let A be a proper contraction on a Hilbert space H. Then (f(z)/zn), z 0 and g(0)=b, is in B(D) and f(A) Ang(A).Hence the result follows from Theorem I. REMARK.The author learned from Professor R. Finn that Theorem follows independently from some results of K. Fan that are now in press.
THEOREM 2. Let p in P(a,6) be given by the series p(z) 1+2b(1-a) [3za+ 0 < [b[ _<-1, z in D and let A be a proper contraction on a Hilbert space H.Then, ACKNOWLEDGEMENT.The author wishes to thank Professor R. Finn for his useful suggestions.
Further, let B(D) {f H(D): If(z)l <i z D} and let B (D) {f B (D): f(0) 0} O THEOREM I. Let f be in B(O) and let A be a proper contraction on a Hilbert space H.Then, function g, defined by g(z)
1 is also a proper contraction.Further by