Löwner-Heinz inequality

In the sequel, a capital letter denotes a bounded linear operator on a Hilbert space $ H $. An operator $ T $ is said to be positive (denoted by $ T \geq 0 $) if $ ( {Tx } , x ) \geq 0 $ for all $ x \in H $.

A real-valued continuous function $ f $ is called an operator-monotone function if

$$ A \geq B \geq 0 \Rightarrow f ( A ) \geq f ( B ) . $$

K. Löwner [a3] has completely characterized all operator-monotone functions as the class of Pick functions. These functions play an essential and very important role in the theory of analytic functions. He has also proved that a real-valued continuous function $ f $ on $ ( 0, \infty ) $ is operator monotone if and only if it has a representation

$$ f ( t ) = at + b + \int\limits _ { 0 } ^ \infty { \frac{t}{t + s } } d \mu ( s ) , $$

where $ a \geq 0 $ and $ b $ are two arbitrary real constants. This representation shows that $ f ( t ) = t ^ \alpha $ is operator monotone if $ \alpha \in [ 0,1 ] $ and it is not operator monotone if $ \alpha > 1 $. This fact can be expressed as follows (the Löwner–Heinz inequality, 1934): $ A \geq B \geq 0 $ implies $ A ^ \alpha \geq B ^ \alpha $ for all $ \alpha \in [ 0,1 ] $.

The Löwner–Heinz inequality is not only the most famous and profound one in operator theory, but also a useful and fundamental tool for treating operator inequalities. The original proof in [a3] used the above integral representation for operator-monotone functions; an alternative proof was been discovered in 1951 (cf. [a1]), while a short proof can be found in [a2]. Various different proofs of the Löwner–Heinz inequality have been given by many authors. There are proofs based on an analytical method in the theory of differential equations as well as simple algebraic proofs. A breathtakingly elegant proof can be found in [a4]. (This is a motivation of the Furuta inequality.) It is known that the Löwner–Heinz inequality, the Heinz–Kato inequality and the Cordes inequality are mutually equivalent, although the first is an operator inequality while the latter two are norm inequalities. Moreover, they are also equivalent to the Jensen inequality:

$$ ( X ^ {*} A X ) ^ \alpha \geq X ^ {*} A X $$

$$ \textrm{ for } A \geq 0, \textrm{ contractions } X, \alpha \in [ 0,1 ] . $$

See also Furuta inequality.

References

[a1]	E. Heinz, "Beiträge zur Störungstheorie der Spektralzerlegung" Math. Ann. , 123 (1951) pp. 415–438
[a2]	T. Kato, "Notes on some inequalities for linear operators" Math. Ann. , 125 (1952) pp. 208–212
[a3]	K. Löwner, "Über monotone Matrixfunktionen" Math. Z. , 38 (1934) pp. 177–216
[a4]	G.K. Pedersen, "Some operator monotone functions" Proc. Amer. Math. Soc. , 36 (1972) pp. 309–310