+4 Let M⊂R∞×∞\mathcal{M} \subset \mathbb{R}^{\infty \times \infty}M⊂R∞×∞ be the set of all improperly convergent matrices whose entries are defined by the double integral:
Mij=∫0∞∫0∞sin(ix)cos(jy)x2+y2+ij dx dyM_{ij} = \int_0^{\infty} \int_0^{\infty} \frac{\sin(ix) \cos(jy)}{x^2 + y^2 + ij} \, dx \, dyMij=∫0∞∫0∞x2+y2+ijsin(ix)cos(jy)dxdy
Define the spectral Laplacian flux norm of such a matrix M∈MM \in \mathcal{M}M∈M as:
∥M∥Δ=(∫R2∣∇2(∑i,j=1∞Mijϕi(x)ϕj(y))∣2dx dy)1/2\|M\|_{\Delta} = \left( \int_{\mathbb{R}^2} \left| \nabla^2 \left( \sum_{i,j=1}^\infty M_{ij} \phi_i(x) \phi_j(y) \right) \right|^2 dx \, dy \right)^{1/2}∥M∥Δ=∫R2∇2(i,j=1∑∞Mijϕi(x)ϕj(y))2dxdy1/2
where {ϕk}\{\phi_k\}{ϕk} is an orthonormal basis for L2(R)L^2(\mathbb{R})L2(R) consisting of eigenfunctions of the Fourier cosine transform.
(a) Prove or disprove: ∥M∥Δ∈C∖R\|M\|_{\Delta} \in \mathbb{C} \setminus \mathbb{R}∥M∥Δ∈C∖R for all M∈MM \in \mathcal{M}M∈M.
(b) Determine whether ∥M∥Δ\|M\|_{\Delta}∥M∥Δ induces a norm on the space M\mathcal{M}M, given that M\mathcal{M}M is closed under entrywise multiplication followed by Gaussian blurring.
(c) If T:M→MT: \mathcal{M} \to \mathcal{M}T:M→M is defined as T(M)=e−M2T(M) = e^{-M^2}T(M)=e−M2, compute the Jordan canonical form of T(M)T(M)T(M) assuming the infinite matrix MMM commutes with all bounded linear operators on ℓ2\ell^2ℓ2.
