Nihonkai Mathematical Journal

Simultaneous extensions of Selberg and Buzano inequalities

Masatoshi Fujii, Akemi Matsumoto, and Masaru Tominaga

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

We give a simultaneous extension of Selberg and Buzano inequalities: If $y1, y2$ and nonzero vectors $\{ z_i; i = 1, 2, \dots , n \}$ in a Hilbert space $\mathscr{H}$ satisfy the orthogonality condition $\langle y_k; z_i \rangle = 0$ for $i = 1, 2, \dots , n$ and $k = 1, 2,$ then \[ | \langle x, y_1 \rangle \langle x, y_2 \rangle | + \mathit{B} (y_1, y_2) \sum_i \frac{| \langle x, z_i \rangle |^2}{\sum_j | \langle z_i, Z-j \rangle |} \leq \mathit{B} (y_1, y_2) \|x\|^2 \] holds for all $x \in \mathscr{H}$, where $\mathit{B} (y1; y2) = \frac{1}{2} (\|y_1\| \|y_2\| + | \langle y_1, y_2 \rangle |)$. As an application, we discuss some refinements of the Heinz-Kato-Furuta inequality and the Bernstein inequality.

Article information

Source
Nihonkai Math. J., Volume 25, Number 1 (2014), 45-63.

Dates
First available in Project Euclid: 17 October 2014

Permanent link to this document
https://projecteuclid.org/euclid.nihmj/1413555412

Mathematical Reviews number (MathSciNet)
MR3270971

Zentralblatt MATH identifier
1311.47020

Subjects
Primary: 47A63: Operator inequalities

Keywords
Selberg inequality Buzano inequality Heinz-Kato-Furuta inequality Furuta inequality Bernstein inequality

Citation

Fujii, Masatoshi; Matsumoto, Akemi; Tominaga, Masaru. Simultaneous extensions of Selberg and Buzano inequalities. Nihonkai Math. J. 25 (2014), no. 1, 45--63. https://projecteuclid.org/euclid.nihmj/1413555412


Export citation

References

  • H.J. Bernstein, An inequality for selfadjoint operators on a Hilbert space, Proc. Amer. Math. Soc., 100 (1987), 319–321.
  • M.L. Buzano, Generalizzazione della diseguaglianza di Cauchy-Schwarz, Rend. Sem. Mat. Univ. e Politech. Torino, 31 (1971-73). (1974), 405–409.
  • M. Fujii, Furuta's inequality and its mean theoretic approach, J. Operator Theory, 23 (1990), 67–72.
  • M. Fujii, T. Furuta and E. Kamei, Furuta's inequality and its application to Ando's theorem, Linear Algebra Appl., 179 (1993), 161–169.
  • M. Fujii, T. Furuta and Y. Seo, An inequality for some nonnormal operators –Extension to normal approximate eigenvalues, Proc. Amer. Math. Soc., 118 (1993), 899–902.
  • M. Fujii, J.-F. Jiang and E. Kamei, Characterization of chaotic order and its application to Furuta inequality, Proc. Amer. Math. Soc., 125 (1997), 3655–3658.
  • M. Fujii, J.-F. Jiang, E. Kamei and K. Takahashi, A characterization of chaotic order and a problem, J. Inequal. Appl., 2 (1998), 149–156.
  • M. Fujii and E. Kamei, Furuta's inequality and a generalization of Ando's theorem, Proc. Amer. Math. Soc., 115 (1992), 409–413.
  • M. Fujii, K. Kubo and S. Otani, A graph theoretic observation on the Selberg inequality, Math. Japon., 35 (1990), 381–385.
  • M. Fujii, C.-S. Lin and R. Nakamoto, Bessel type extension of the Bernstein inequality, Sci. Math., 3 (2000), 95–98.
  • M. Fujii and R. Nakamoto, Simultaneous extensions of Selberg inequality and Heinz-Kato-Furuta inequality, Nihonkai Math. J., 9 (1998), 219–225.
  • M. Fujii and R. Nakamoto, Extensions of Heinz-Kato-Furuta inequality, Proc. Amer. Math. Soc., 128 (2000), 223–228.
  • M. Fujii and R. Nakamoto, Extensions of Heinz-Kato-Furuta inequality, II, J. Inequal. Appl., 3 (1999), 293–302.
  • M. Fujii, R. Nakamoto and Y. Seo, Covariance in Bernstein's inequality for operators, Nihonkai Math. J., 8 (1997), 1–6.
  • T. Furuta, $A \ge B \ge 0$ assures $(B^rA^pB^r)^{1/q} \ge B^{(p+2r)/q}$ for $r \ge 0, p \ge 0, q \ge 1$ with $(1+2r)q \ge p+2r$, Proc. Amer. Math. Soc., 101 (1987), 85–88.
  • T. Furuta, An inequality for some nonnormal operators, Proc. Amer. Math. Soc., 104 (1988), 1216–1217.
  • T. Furuta, Elementary proof of an order preserving inequality, Proc. Japan Acad., 65 (1989), 126.
  • T. Furuta, When does the equality of a generalized Selberg inequality hold?, Nihonkai Math. J., 2 (1991), 25–29.
  • T. Furuta, Determinant type generalizations of the Heinz-Kato theorem via the Furuta inequality, Proc. Amer. Math. Soc., 120 (1994), 223–231.
  • E. Kamei, A satellite to Furuta's inequality, Math. Japon., 33 (1988), 883–886.
  • K. Kubo and F. Kubo, Diagonal matrix dominates a positive semidefinite matrix and Selberg's inequality, preprint.
  • C.-S. Lin, Heinz's inequality and Bernstein's inequality, Proc. Amer. Math. Soc., 125 (1997), 2319–2325.
  • G.K. Pedersen, Some operator monotone functions, Proc. Amer. Math. Soc., 36 (1972), 309–310.
  • K. Tanahashi, Best possibility of the Furuta inequality, Proc. Amer. Math. Soc., 124 (1996), 141–146.
  • M.Uchiyama, Some exponential operator inequalities, Math. Inequal. Appl., 2 (1999), 469–471.