This might be a dumb question, but I thought the Laplacian (classical) is defined for $C^2$ functions. How do we extend that to be a self-adjoint operator on all of $L_2$? Is it t …

Let $X_0, X_1, \dots, X_k$ be smooth vector fields over ${\mathbb R}^n$, and let us consider the operator $$ L = \sum_{i=1}^k X_i^2 + X_0~. $$ Here, I assume that Hörmander's brack …

Let $H$ be a Hilbert space, and let $A_t$ be a family of unbounded positive (self-adjoint) operators on $H$ parametrized by $\mathbb t\in R_{\ge 0}$. Consider the ordinary differen …

In the article below, it is shown that the proposition "Every linear operator defined on a whole Hilbert space is bounded" is consistent with the axioms of ZF + a weakened version …

Hi Everyone, I was recently reading Reed & Simon's functional analysis textbook (the first volume), and it mentions casually on page 294 that weak graph limits of a sequence o …

Suppose $A \in \mathcal{L}(E_1, E_0)$ is a bounded linear operator between Banach spaces $E_1$ and $E_0$, and we also have that $E_1$ is densely, continuously embedded in $E_0$ (i. …

Let $X$ and $Y$ be Banach spaces, and let $L : X \rightarrow Y$ be a unbounded operator with dense domain $\operatorname{dom}(L)$. We can then talk about the transposed operator $ …

Assume, $M$ is a smooth manifold with a measure $\mu$ and let $L^2(M, \mu)$ be a space of all square-integrable functions on $M$. Recall that $L$ is a hypoelliptic differential o …

How do you define unbounded measurable functions for a general von Neumann algebra? For the commutative algebra $L^\infty(X,\mu)$, we can consider the space of all measurable func …

Is it true that every closed operator on a separable Hilbert H space only has countably many eigenvalues? Or put the other way around, if I want to ensure that a (not necessarily …

I would like to know if there are theorems that state under which circumstances spectra of operator families depend smoothly on the parameter. To clarify, suppose I have a 1-param …

Let $L$ be a linear differential operator (with smooth coefficients) on a compact differentiable manifold $M$ (without boundary). Suppose $M$ is endowed with a smooth volume form ( …

Consider the unbounded operator $L$ on $L^2(\mathbb{R^d})$ to be the self-adjoint extension of $$Lf := \nabla \cdot \left(a(x) \nabla f(x) \right)$$ on $C^2_c(\mathbb{R^d})$. I a …

Assume we have a linear operator $P_0: D->H$ where $D$ is the domain and $H$ some Hilbert space it is acting on. Assume moreover that the spectrum of $P_0$ is purely essential. If …

Let $V$ be a normed infinite dimensional vector space. Let $L: V \longrightarrow V$ be a bounded linear operator. Moreover assume that $L$ is 'locally nilpotent' that is: $$ \foral …