Normal operator norm equals spectral radius

$\mathcal{B}(H)$ is a unital complex C*-algebra, hence in particular a nonzero unital complex Banach algebra; an operator $T$ is normal when $T^{\ast }T=T{T}^{\ast }$ (Bounded Hilbert operators form a C star algebra, Self-adjoint, positive, unitary and normal operators).

For every normal element $a$ of a unital complex C*-algebra $A$ the spectral radius satisfies $r(a)=\mathrm{\parallel }a\mathrm{\parallel }$ (C star spectral radius equals norm for normal elements).

For a bounded operator $T$ on a nonzero complex Banach space the spectral radius is $r(T)=\max \{\mathrm{\mid }z\mathrm{\mid }:z\in \sigma (T)\}$, computed in $\mathcal{B}(X)$, and $\lambda \in \rho (T)$ exactly when $\lambda I-T$ is bijective with bounded inverse (Spectral radius, Spectrum and resolvent of a bounded operator).

In a nonzero unital complex Banach algebra the spectrum of every element is nonempty and compact (Spectrum is nonempty compact and norm bounded).

A bounded bijective linear map between Banach spaces has a bounded inverse (Bounded inverse theorem), so for $A=\mathcal{B}(H)$ invertibility in the algebra and bijectivity with bounded inverse coincide (C star algebra generated by a normal operator for the generated-algebra convention used on this page).