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Let $H$ be a separable Hilbert space over $\mathbb{R}$. Let ${A} = \{a\colon H\to H\,|\,a\text{ is a positive, compact linear operator}\}$. By the spectral theorem, given $a \in A$, there are scalars $...

Let $H$ be a separable complex Hilbert space and let $A\in B(H)$ be positive with $||A||=1$ and have eigenvalue 1 with multiplicity 1. Suppose $A=T^*T$ for some $T\in B(H)$. Denote the spectrum of $A$ ...

Suppose $k:\mathcal{X} \times \mathcal{X} \to \mathbb{R}$ be a symmetric positive (semi-)definite kernel. The Moore-Aronszajn Theorem indicates that there is a reproducing kernel Hilbert Space $\...

Consider the Hilbert space $H=L^2_w(I)$ as the weighted $L^2$ space, where $I\subseteq\mathbb{R}$: $$ L_w^2(I)=\{\phi:I\rightarrow\mathbb{R}:\,\|\phi\|^2=\int_I \phi(x)^2w(x)\,dx<\infty\}. $$ In ...

Let $T$ be a closed operator on a Hilbert space with discrete spectrum. Then for $\{\lambda_1,...\lambda_n\}\in\sigma(T)$ one can define the spectral projections $$P_{\{\lambda_1,...\lambda_n\}}=\frac{...