PROCEEDINGS OF THE
AMERICAN MATHEMATICAL SOCIETY
Volume 36, Number 1, November 1972
SOME OPERATOR MONOTONE FUNCTIONS1
GERT K. PEDERSEN
Abstract.
A short proof is given based on C*-algebra theory
for the well-known theorem that if S and Tare bounded selfadjoint
operators on a Hubert space such that 0^5^T
then Sai|7"a for
each0^a<n.
Theorem.
If S and T are bounded selfadjoint operators on a Hilbert
space $ such that O^S^T then S°<T*for
each a in the interval [0, 1].
Remark.
The theorem says that each function t—t", with 0^oc<l, is
operator monotone on the set of positive operators in B(9f). This was
first proved by K. Löwner, who gave a complete description of operator
monotone functions. Later T. Ogasawara gave a short proof of the operator monotonicity for the square root function. We present here a simple
proof based on C*-algebra theory.
Proof.
If 0<S^Fthen
S+el^T+el
for each £>0; and S+e/and
T+el are both invertible. Since (S+eI)x converges to Sx in norm
e-0 for each a>0, and since the positive operators in B(%>)form a
closed set, it suffices to prove the theorem assuming that S and
invertible. (The case a=0 can be verified directly, since S° is the
when
norm
T are
range
projection of 5.)
Let E denote the set of exponents a in [0, 1] for which the function
t—t" is operator monotone. Trivially 0e£ and 1 eE. Since the function
a-S" is continuous from [0, 1] to F(§) in the norm topology we see that
Fis a closed set. The proof will be complete when we show that Fis convex.
Take a and ß in E. Then S°^TX; hence r^^F""72^/.
It follows that
\\S"/2T-"'*\\£l. Similarly \\Sß'2T-ß/2\\^\. With P(A) the spectral radius
of an operator A we have p(AB) = p(BA). Therefore
/j*-U+ß)/lMa+ß)/2j--<"+ß1li)
_
D('r'.a-ß)li'j—iix+ß'iHc{a+ß)lij-l.!i+ß)H-Y-t.!ii-ß)li\
_ „(T—f/!c(»+j!)/2T-a/2\
<;
II j—0/2ç(«t+/»/2<T--«/2t|
<; \\T-pl2Sßl2\\ ||S°,/2T_,I/21| ^ 1.
Received by the editors April 27, 1972.
AMS 1970 subject classifications. Primary 47B15; Secondary 46L05.
Key words and phrases. Operator monotone functions, C*-algebras, positive operators.
1 The preparation of this paper was supported in part by NSF Grant 28976X.
© American Mathematical Society 1972
309
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310
G. K. PEDERSEN
It follows that t-(«+'î,/4S(i+'',/2F-<0I+î)/4^/,
shows that (x+ß)ß
so that S^+^/2^Tfx+ß)/2.
This
e E which completes the proof.
References
1. J. Dixmier, Les C*-algèbres et leurs représentations,
Cahiers Scientifiques,
fase.
29, Gauthier-Villars, Paris, 1964. MR 30 #1404.
2. K. Löwner, Über monotone matrixfunctionen, Math. Z. 38 (1934), 177-216.
3. T. Ogasawara, A theorem on operator algebras, J. Sei. Hiroshima Univ. Ser. A 18
(1955), 307-309. MR 17, 514.
Department
of Mathematics,
Current address:
Denmark
Matematisk
University
Institut,
of Copenhagen, Copenhagen,
Universitetsparken
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Denmark
5, 2100 Copenhagen,