# Spectral radius

*of an element of a Banach algebra*

The radius of the smallest closed disc in the plane that contains the spectrum of this element (cf. Spectrum of an element). The spectral radius of an element is connected with the norms of its powers by the formula

which, in particular, implies that . The spectral radius of a bounded linear operator on a Banach space is the spectral radius of it regarded as an element of the Banach algebra of all operators. In a Hilbert space, the spectral radius of an operator is equal to the greatest lower bound of the norms of the operators similar to it (see [2]):

If the operator is normal, then (cf. Normal operator).

As a function of the elements of a Banach algebra, the spectral radius is upper semi-continuous (but not, in general, continuous). The subharmonicity of the spectral radius has been proved [3]. (This means that if is a holomorphic mapping of some domain into a Banach algebra , then is a subharmonic function.)

#### References

[1] | M.A. Naimark, "Normed rings" , Reidel (1959) (Translated from Russian) |

[2] | P.R. Halmos, "A Hilbert space problem book" , Springer (1980) |

[3] | E. Vesentini, "On the subharmonicity of the spectral radius" Boll. Union. Mat. Ital. , 1 (1968) pp. 427–429 |

[4] | V. Ptak, "On the spectral radius in Banach algebras with involution" Bull. London Math. Soc. , 2 (1970) pp. 327–334 |

#### Comments

#### References

[a1] | N. Dunford, J.T. Schwartz, "Linear operators. General theory" , 1 , Interscience (1958) |

**How to Cite This Entry:**

Spectral radius.

*Encyclopedia of Mathematics.*URL: http://encyclopediaofmath.org/index.php?title=Spectral_radius&oldid=15490