Heinz-Kato-Furuta 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 $.

The following Heinz–Kato–Furuta inequality can be considered as an extension of the Heinz–Kato inequality, since for $ \alpha + \beta = 1 $ the Heinz–Kato inequality is obtained from the Heinz–Kato–Furuta inequality.

The Heinz–Kato–Furuta inequality (1994; cf. [a2]): If $ A $ and $ B $ are positive operators such that $ \| {Tx } \| \leq \| {Ax } \| $ and $ \| {T ^ {*} y } \| \leq \| {By } \| $ for all $ x, y \in H $, then for all $ x,y \in H $:

$$ \tag{a1 } \left | {( T \left | T \right | ^ {\alpha + \beta - 1 } x,y ) } \right | \leq \left \| {A ^ \alpha x } \right \| \left \| {B ^ \beta y } \right \| $$

for all $ \alpha, \beta \in [ 0,1 ] $ such that $ \alpha + \beta \geq 1 $.

As generalizations of the Heinz–Kato–Furuta inequality, two determinant-type generalizations, expressed in terms of $ T $, $ | T | $ and $ | {T ^ {*} } | $, can be obtained by using the Furuta inequality. It turns out that each of these two generalizations is equivalent to the Furuta inequality. Results similar to these determinant-type generalizations but under the conditions $ { \mathop{\rm log} } | T | \leq { \mathop{\rm log} } A $ and $ { \mathop{\rm log} } | {T ^ {*} } | \leq { \mathop{\rm log} } B $, which are weaker than the original conditions $ T ^ {*} T \leq A ^ {2} $ and $ TT ^ {*} \leq B ^ {2} $ in the Heinz–Kato inequality, have also been obtained. A nice application of the Heinz–Kato–Furuta inequality is given in [a1].

Additional references can be found in Heinz inequality.

References

[a1]	M. Fujii, S. Izumino, R. Nakamoto, "Classes of operators determined by the Heinz–Kato–Furuta inequality and the Hölder–MacCarthy inequality" Nihonkai Math. J. , 5 (1994) pp. 61–67
[a2]	T. Furuta, "An extension of the Heinz–Kato theorem" Proc. Amer. Math. Soc. , 120 (1994) pp. 785–787