If $V$ is a vector space over $\mathbb{C}$ , the spectrum of a linear mapping $T:V\rightarrow V$ is the set$$\sigma(T) = \{\lambda\in \mathbb{C}: T-\lambda I \mbox{is not invertible}\}$$ 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 $\mathcal{A}$ is a Banach algebra over $\mathbb{C}$ with identity element $e$ , the spectrum of an element $a \in \mathcal{A}$ is the set $$\sigma(a) = \{ \lambda \in \mathbb{C} : a - \lambda e \mbox{is not invertible in} \mathcal{A} \}$$

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