# Inequalities for exponentials in Banach algebras

Studia Mathematica (1991)

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

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topPryde, 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

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

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