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### Integral representation of joint projection valued measures

Given two positive $\sigma$-finite measures $\mu_{1/2}$ on the spaces $X_{1/2}$ one can define the product measure $\mu_1\otimes\mu_2$ on the product space $X_1\times X_2$. It can be proved that the ...
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### Reference Request: Generalization of spectral theory to symmetric KL divergence type metrics?

Spectral theory(Courant Fischer Theorem) provides a definition of the spectrum in term of the minima/maxima of the rayleigh coefficient of a matrix. So I can say that kth eigenvector and associated ...
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### Why is this operator compact?

Let $D$ be the Dirac-Operator on $\mathbb{R}^n$ or more generally the Dirac spinor bundle $\mathcal{S}\to M$ of a (semi-)Riemannian spin manifold $M$. Then we consider $D$ as an unbouded Operator on ...
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I encountered an inequality when reading a paper. Can someone help to show how to prove it? Let be the spectral radius of matrix $A$ or $\rho(A)=\max\{|\lambda|, \lambda \text{ are eigenvalues of ... 1answer 97 views ### Spectrum of composition of graphs( lexicographic product) I was wondering how to relate the spectra of the composition of two graphs in term of the factors...someone can help me? 0answers 87 views ### Global solution for spectral clustering I used spectral clustering for directed graphs suggested by Dengyong Zhou paper to partition the graph.I selected the eigen vectors corresponding to k largest eigen values and then I use kmeans or FCM ... 1answer 473 views ### Eigenvalues of directed Laplacian matrix$L$and$DL$, where$D$is a diagonal matrix with positive entries I have a weighted Laplacian matrix$L$of a strongly connected directed graph and a diagonal matrix$D$with positive entries. Since the graph is directed,$L$is non-symmetric real. Further, since ... 1answer 230 views ### A doubt about the parts of the spectrum of tensor products Let$\mathcal{H}$be any complex Hilbert space of infinite dimensional. By an operator$T$I mean a linear bounded transformation from$\mathcal{H}$into$\mathcal{H}$, i.e, ... 0answers 224 views ### Eigen-decomposition perturbation Let$A$,$B$and$A_k + B$be symmetric matrices with eigenvalues$\sigma_1 \geq \sigma_2 \ldots \geq \sigma_n$,$\rho_1 \geq \rho_2 \ldots \geq \rho_n$and$\lambda_1 \geq \lambda_2 \ldots \geq ...
Intuitively one understands that if one is solving the Schroedinger's equation for energies $E$ such that $\{ x \vert U(x)\leq E \}$ is compact (..is there a weaker criteria?..) then the spectrum ...