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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 ):

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

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 . (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.)