Namespaces
Variants
Actions

Spectral radius

From Encyclopedia of Mathematics
Revision as of 17:12, 7 February 2011 by 127.0.0.1 (talk) (Importing text file)
(diff) ← Older revision | Latest revision (diff) | Newer revision → (diff)
Jump to: navigation, search

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
This article was adapted from an original article by V.S. Shul'man (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article