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Let us say that a bounded smooth function $f:\mathbb{R}\rightarrow\mathbb{R}$ has vanishing variation at infinity (or satisfies "property $A$" for short) if, for any $r\neq 0$, we have $$\lim_{x\righ …

Suppose $D$ is a Dirac operator acting on sections of a bundle $E$ over a manifold $M$, and define the Sobolev spaces $H^i(E)$ via the inner products $$\langle e_1,e_2\rangle_{H^i}:=\sum_{k=0}^i\langl …

Let $G$ be a Lie group and $C_r^*(G)$ and $C^*(G)$ be its reduced and maximal group $C^*$-algebras respectively. The left-regular representation of a group $G$ induces a surjective map $$\lambda_G:C …

Suppose $X$ is a non-compact manifold. Let $P$ be an order-$0$ pseudodifferential operator on $X$ and $f:L^2(X)\rightarrow L^2(X)$ a compact operator. I'm wondering: 1) Is $P + f$ always a pseudodiff …

If $M$ is a non-compact smooth manifold, then an analogue of Rellich's lemma states that the operator of multiplication by a compactly supported function $f:M\rightarrow\mathbb{C}$ is a compact operat …

Let $f:\mathbb{R}^n\rightarrow\mathbb{R}$ be a non-negative, smooth, uniformly bounded function with uniformly bounded first derivative. Then $f$ defines a bounded operator on $L^2(\mathbb{R}^n)$ as w …

Let $(M,g)$ be a closed Riemannian manifold, $D$ an essentially self-adjoint differential operator on $L^2(M)$. Then the operator group $\{e^{itD}\}_{t\in\mathbb{R}}$ formed via functional calculus so …

In a paper I was reading, it was mentioned that if $M$ is a closed Riemannian manifold, then by fixing a basis for $L^2(M)$ consisting of eigenfunctions of the Laplacian, the space of smoothing operat …

Let $X$ be a non-compact complete Riemannian manifold and $P$ a first-order elliptic pseudodifferential operator on $X$. Let $Q$ be a parametrix for $P$, so that $PQ - 1 = T$ and $QP - 1 = R$ are lowe …

Let $X\rightarrow Y$ be a smooth principal $G$-bundle for some Lie group $G$. Then $L^2(X)$ has a natural $G$-action determined by fibrewise action of $G$ on $X$. We have an abstract isomorphism of H …

I'm looking for references about the following aspect of cyclic vectors for regular representations. Let $K$ be a compact Lie group. Let $K$ act on $L^2(K)$ by the left regular representation. Then $ …

For $n\geq 1$, let $D_n$ be the Dirac operator on the spinor bundle on the $n$-dimensional sphere $S^n$. For example, $D_1$ acts on the trivial bundle $S^1\times\mathbb{C}\to S^1$, and can be explicit …

Let $E$ be a Hilbert module over a $C^*$-algebra $A$. Let $T\colon E\to E$ be a densely defined, unbounded $A$-linear operator. (In particular, the initial domain of $T$ is an $A$-submodule of $E$.) I …

Let $(H,||\cdot||_H)$ be a Banach space and $K$ a (not necessarily closed) subspace. Suppose that $K$ is a Banach space under another norm $||\cdot||_K$, which satisfies $$||x||_H\leq ||x||_K$$ for …

Let $M$ be a smooth manifold on which a Lie group $G$ acts properly, such that the orbit space $M/G$ is compact. Suppose $c:M\rightarrow [0,\infty)$ is a compactly supported smooth function with the p …