Tikhonov filter norm bound ($\Vert (K^{\ast }K+\alpha I{)}^{-1}{K}^{\ast }\Vert \le 1/(2\sqrt{\alpha })$)

For a bounded linear operator between Hilbert spaces, the spectral regularization method for quadratic Tikhonov regularization has scalar coefficient $s/(s^2+\alpha )$. Since $(s-\sqrt{\alpha }{)}^2\ge 0$, this coefficient is at most $1/(2\sqrt{\alpha })$. The spectral theorem for normal operators transfers this scalar bound to the operator norm, providing the noise term in the noise-bias decomposition for linear regularization.