Since the Frobenius norm on matrices is generated by an inner product, solving the optimization/approximation problem of approximating an operator $X$ with a scalar multiple of another operator $Y$ $$\text{minimize:}\,\|X-\alpha Y\|_\mathsf{F}$$ has a closed form solution $\alpha = \frac{\langle X, Y \rangle}{\langle Y,Y \rangle}$ via an orthogonal projection.

What if I want to consider a non-Frobenius norm, specifically the standard operator norm / spectral norm / 2-norm? In other words, is there a closed form and/or efficient solution to the optimization problem $$\text{minimize:}\,\|X-\alpha Y\|_2$$ Since $\|A\|_2\leq \|A\|_\mathsf{F}$ for all $A$, solving the Frobenius version would provide an upper bound for the operator norm approximation. Is this approximation decent?

It feels like you could frame this problem as a semidefinite program under the self-adjoint assumption, but a closed form solution would be more optimal.

exactexplicit formula for the minimizer is certainly out of question. The rest depends on how much you are willing to settle with. $\endgroup$ – fedja Jun 10 '17 at 10:10