Inequalities for exponentials in Banach algebras

A. Pryde

Studia Mathematica (1991)

  • Volume: 100, Issue: 1, page 87-94
  • ISSN: 0039-3223

Abstract

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For commuting elements x, y of a unital Banach algebra ℬ it is clear that ∥ e x + y ∥ ≤ ∥ e x ∥ ∥ e y ∥ . On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form ∥ e ' ∥ ≤ c ( 1 + | ξ | ⟩ s for all ξ ∈ R m , where x = ( x 1 , . . . , x m ) ∈ ℬ m and c, s are constants.

How to cite

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Pryde, A.. "Inequalities for exponentials in Banach algebras." Studia Mathematica 100.1 (1991): 87-94. <http://eudml.org/doc/215876>.

@article{Pryde1991,
abstract = {For commuting elements x, y of a unital Banach algebra ℬ it is clear that $∥e^\{x+y\}∥ ≤ ∥e^x∥ ∥e^y∥$. On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form $∥e^\{\prime \}∥ ≤ c(1 + |ξ|⟩^s$ for all $ξ ∈ R^m$, where $x = (x_1,..., x_m) ∈ ℬ^m$ and c, s are constants.},
author = {Pryde, A.},
journal = {Studia Mathematica},
keywords = {inequalities for exponentials in Banach spaces; self-adjoint operator; skew-adjoint operator; growth estimates},
language = {eng},
number = {1},
pages = {87-94},
title = {Inequalities for exponentials in Banach algebras},
url = {http://eudml.org/doc/215876},
volume = {100},
year = {1991},
}

TY - JOUR
AU - Pryde, A.
TI - Inequalities for exponentials in Banach algebras
JO - Studia Mathematica
PY - 1991
VL - 100
IS - 1
SP - 87
EP - 94
AB - For commuting elements x, y of a unital Banach algebra ℬ it is clear that $∥e^{x+y}∥ ≤ ∥e^x∥ ∥e^y∥$. On the order hand, M. Taylor has shown that this inequality remains valid for a self-adjoint operator x and a skew-adjoint operator y, without the assumption that they commute. In this paper we obtain similar inequalities under conditions that lie between these extremes. The inequalities are used to deduce growth estimates of the form $∥e^{\prime }∥ ≤ c(1 + |ξ|⟩^s$ for all $ξ ∈ R^m$, where $x = (x_1,..., x_m) ∈ ℬ^m$ and c, s are constants.
LA - eng
KW - inequalities for exponentials in Banach spaces; self-adjoint operator; skew-adjoint operator; growth estimates
UR - http://eudml.org/doc/215876
ER -

References

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  1. [1] A. McIntosh and A. Pryde, A functional calculus for several commuting operators, Indiana Univ. Math. J. 36 (1987), 421-439. Zbl0694.47015
  2. [2] A. McIntosh, A. Pryde and W. Ricker, Estimates for solutions of the operator equation ∑ j = 1 m A j Q B j = U , in: Operator Theory: Adv. Appl. 28, Birkhäuser, Basel 1988, 43-65. 
  3. [3] A. McIntosh, A. Pryde and W. Ricker, Systems of operator equations and perturbation of spectral subspaces of commuting operators, Michigan Math. J. 35 (1988), 43-65. Zbl0657.47021
  4. [4] A. Pryde, A non-commutative joint spectral theory, Proc. Centre Math. Anal. Canberra 20 (1988), 153-161. 
  5. [5] H. Radjavi, A trace condition equivalent to simultaneous triangularizability, Canad. J. Math. 38 (1986), 376-386. Zbl0577.47018
  6. [6] M. Reed and B. Simon, Methods of Modern Mathematical Physic. I, II, Academic Press, New York 1980. 
  7. [7] M. E. Taylor, Functions of several self-adjoint operators, Proc. Amer. Math. Soc. 19 (1968), 91-98. Zbl0164.16604
  8. [8] C. Thompson, Inequalities and partial orders on matrix spaces, Indiana Univ. Math. J. 21 (1971), 469-480. Zbl0227.15005

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