Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Mixed Hadamard's theorems


Author: Takayuki Furuta
Journal: Proc. Amer. Math. Soc. 96 (1986), 217-220
MSC: Primary 47A05; Secondary 47A30
MathSciNet review: 818447
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: An operator $ T$ means a bounded linear operator on a complex Hilbert space $ H$. We give two types of mixed Hadamard's theorems containing the terms $ T,\left\vert T \right\vert$ and $ \left\vert {{T^ * }} \right\vert$ as extensions of Hadamard's theorem and mixed Schwarz's inequality $ {\left\vert {(Tx,y)} \right\vert^2} \leq (\left\vert T \right\vert x,x)(\left\vert {{T^ * }} \right\vert y,y)$ for any $ T$ and for any $ x$ and $ y$ in $ H$. Also we scrutinize the cases when the equalities in these mixed Hadamard's theorems hold.


References [Enhancements On Off] (What's this?)

  • [1] Takayuki Furuta, An elementary proof of Hadamard’s theorem, Mat. Vesnik 8(23) (1971), 267–269. MR 0294371
  • [2] F. G. Gantmacher, The theory of matrices, vol. 1, Chelsea, New York, 1960.
  • [3] Paul Richard Halmos, A Hilbert space problem book, 2nd ed., Graduate Texts in Mathematics, vol. 19, Springer-Verlag, New York-Berlin, 1982. Encyclopedia of Mathematics and its Applications, 17. MR 675952
  • [4] G. H. Hardy, J. E. Littlewood and G. Pólya, Inequalities, Cambridge Univ. Press, Cambridge, 1934.
  • [5] Harold Widom, Lectures on integral equations, Notes by David Drazin and Anthony J. Tromba. Van Nostrand Mathematical Studies, No. 17, Van Nostrand; Reinhold Co., New York-Toronto, Ont.-London, 1969. MR 0243299

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 47A05, 47A30

Retrieve articles in all journals with MSC: 47A05, 47A30


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1986-0818447-0
Keywords: Hadamard's theorem, Schwarz's inequality, polar decomposition
Article copyright: © Copyright 1986 American Mathematical Society