spectral radius

If $V$ is a vector space over $ℂ$, the spectrum of a linear mapping $T:V\to V$ is the set

$$\sigma (T)=\{\lambda \in ℂ:T-\lambda I\text{is not invertible}\},$$

where $I$ denotes the identity mapping.If $V$ is finite dimensional, the spectrum of $T$ is precisely the set of its eigenvalues. For infinite dimensional spaces this is not generally true,although it is true that each eigenvalue of $T$ belongs to $\sigma (T)$. The spectral radius of $T$ is

$$\rho (T)=sup\{|\lambda |:\lambda \in \sigma (T)\}.$$

More generally, the spectrum and spectral radius can be defined for Banach algebras with identity element: If $𝒜$ is a Banach algebra over $ℂ$ with identity element $e$, the spectrum of an element $a\in 𝒜$ is the set

$$\sigma (a)=\{\lambda \in ℂ:a-\lambda e\text{is not invertible in}𝒜\}$$

The spectral radius of $a$ is $\rho (a)=sup\{|\lambda |:\lambda \in \sigma (a)\}$.