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of an element of a Banach algebra

The radius $\rho$ 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 $a$ is connected with the norms of its powers by the formula

$$\rho(a)=\lim_{n\to\infty}\|a^n\|^{1/n}=\inf\|a^n\|^{1/n},$$

which, in particular, implies that $\rho(a)\leq\|a\|$. 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]):

$$\rho(A)=\inf_X\|XAX^{-1}\|.$$

If the operator is normal, then $\rho(A)=\|A\|$ (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 $z\mapsto h(z)$ is a holomorphic mapping of some domain $D\subset\mathbf C$ into a Banach algebra $\mathfrak A$, then $z\mapsto\rho(h(z))$ 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