A simplified proof of Heinz inequality and scrutiny of its equality

Author:
Takayuki Furuta

Journal:
Proc. Amer. Math. Soc. **97** (1986), 751-753

MSC:
Primary 47A30

DOI:
https://doi.org/10.1090/S0002-9939-1986-0846001-3

MathSciNet review:
846001

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Abstract | References | Similar Articles | Additional Information

Abstract: An operator means a bounded linear operator on a Hilbert space . We give a simplified proof of the following inequality:

*In case*,

*the equality in*

*holds iff*

*and*

*are linearly dependent iff*

*and*

*are linearly dependent*. is equivalent to

**[1]**J. Dixmier,*Sur une inégalité de E. Heinz*, Math. Ann.**126**(1953), 75-78. MR**0056200 (15:39e)****[2]**E. Heinz,*On an inequality for linear operators in Hilbert space*, Report on Operator Theory and Group Representations, Publ. No. 387, Nat. Acad. Sci.-Nat. Res. Council, Washington, D. C., 1955, pp. 27-29. MR**0079139 (18:35b)****[3]**T. Kato,*Notes on some inequalities for linear operators*, Math. Ann.**125**(1952), 208-212. MR**0053390 (14:766e)**

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Additional Information

DOI:
https://doi.org/10.1090/S0002-9939-1986-0846001-3

Keywords:
Heinz inequality,
polar decomposition

Article copyright:
© Copyright 1986
American Mathematical Society