Oneparameter versions of the Heinz inequality
The celebrated Heinz inequality ([6]) states
where are Hilbert space operators with
.
This estimate actually remains valid for arbitrary unitarily
invariant norms
, and in the recent years such
norm inequalities for various operator means have been
investigated by several authors. For example, the following
arithmeticlogarithmicgeometric mean inequality ([7])
is known:
Firstly, the general theory on operator means worked out in [8,9])
is reviewed, which is built upon the theory of Stieltjes double integral
transformations ([4]). It enables us to associate operator means to
certain scalar means and moreover to establish norm comparison results as
above in a unified fashion.
Secondly, as applications several norm inequalities involving quantities
such as
and
are reported.
 1
 T. Ando,
Matrix Young inequalities,
Oper. Theory Adv. Appl., 75 (1995), 3338.
 2
 R. Bhatia and C. Davis,
More matrix forms of the arithmetic geometric mean inequality,
SIAM J. Matrix Anal. Appl., 14 (1993), 132136.
 3
 R. Bhatia and K. R. Parthasarathy,
Positive definite functions and operator inequalities,
Bull. London Math. Soc., 32 (2000), No 2, 214228.
 4

M. Sh. Birman and M. Z. Solomyak,
Stieltjes double operator integrals,
Dokl. Akad. Nauk SSSR, 165 (1965), 12231226 (Russian);
Soviet Math. Dokl., 6 (1965), 15671571.
 5

U. Haagerup,
On Schur multipliers in
,
unpublished handwritten notes (1980).
 6
 E. Heinz,
Beiträge zur Störungstheorie der Spektralzerlegung,
Math. Ann., 123 (1951), 415438.
 7
 F. Hiai and H. Kosaki,
Comparison of various means for operators,
J. Funct. Anal., 163 (1999), 300323.
 8
 F. Hiai and H. Kosaki, Means for matrices and comparison
of their norms,
Indiana Univ. Math. J., 48 (1999), 899936.
 9
 F. Hiai and H. Kosaki, Means of Hilbert space operators,
to appear as LNM, Springer.
 10
 H. Kosaki,
Arithmeticgeometric mean and related inequalities for operators,
J. Funct. Anal., 156 (1998), 429451.
 11
 A. McIntosh,
Heinz inequalities and perturbation of spectral families,
Macqaurie Mathematical Reports, 790006, 1979.
 12

V. V. Peller,
Hankel operators and differentiability properties of functions
of selfadjoint (unitary) operators,
LOMI Preprints E184, USSR Academy of Sciences Steklov
Mathematical Institute Leningrad Department, 1984.
 13
 X. Zhan,
Inequalities for unitarily invariant norms,
SIAM J. Matrix Anal. Appl., 20 (1998), 466470.