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Let me define the Fourier transform of a function $u$, say in the Schwartz space $\mathscr S(\mathbb R)$ as $ \hat u(\xi)=\int_{\mathbb R} e^{-2iπ x\cdot \xi} u(x) dx. $ The Hilbert transform $\maths …

Let $\mathbb H$ be a Hilbert space and let $A\in \mathcal B(\mathbb H)$ such that the spectrum of $A$ does not meet a closed half-line issued from 0 in the complex plane. Then I guess that $ A=\exp L …

Your operator $K$ is a Hilbert-Schmidt operator since its kernel belongs to $L^2$. As a result this is a compact operator whose spectrum contains a sequence of eigenvalues $\\{\lambda_k\not=0\\}$ with …

Too long for an additional comment. I guess that you can keep the assumption $\hat P, \hat Q$ Hermitian and require $$ [\hat P, \hat Q]=1/(2πi), $$ as it is the case with the prototypical example $ \h …

I may assume that $H=Z=L^2(\mathbb R)$ and the mapping $K$ to be given by a distribution kernel $k(s,t)$ via a formula $$ Ku(s)=\int k(s,t) u(t) dt, $$ meaning that for $u,v\in C^\infty_c(\mathbb R)$, …

Let $\mathbb H$ be a Hilbert space, and let $\mathcal B(\mathbb H)$ be the space of bounded operators on $\mathbb H$, equipped with the operator-norm topology. Let $\mathbb R\ni t\mapsto A(t)\in \math …

Let me follow your notations with $\Delta=\sum_{1\le j\le 3}\partial_{x_j}^2$. You have with $r=\Vert x\Vert$ (the Euclidean norm) $$ r^2\Delta=(r\partial_r)^2+r\partial_r+\Delta_{\mathbb S^2},\quad\t …

Let us consider the classical Harmonic Oscillator (a selfadjoint operator) $$ \mathcal H=\frac12\left(-\frac{d^2}{dx^2}+x^2\right),\quad\text{with spectrum $\frac12+\mathbb N$.} $$ This one-dimensiona …

Here are some very classical references: M. Cwikel. Weak type estimates for singular values and the number of bound states of Schrödinger operators. Ann. of Math. (2), 106(1):93–100, 1977. E. Lieb. …

If $B$ is positive self-adjoint then $B=A^2$ with $A$ positive self-adjoint. If $X$ is bounded non-negative and commutes with $B$, it commutes as well with a function of $B$ such as $A=\sqrt B$. Then …

Yes, it is a classical pseudo-differential operator of order $-1$ with principal symbol $\vert p_D(x,\xi)\vert^{-1}$ where $p_D$ is the principal symbol of $D$; it is also possible to prove that you h …

Let $k:[0,1]^2\rightarrow (0,+\infty)$ be a continuous function and let $f,g:[0,1]\rightarrow (0,+\infty)$ be measurable functions. We assume that $$\forall x\in [0,1],\quad f(x)=\int_0^1 k(x,y) g(y) …

Too long for a comment. Your requirement is too stringent and it is quite likely that to get continuity from $L^\infty$ into itself, it is indeed necessary to have $$ \text{esssup}_x\int\vert k(x,y)\v …

Let $A$ be a selfadjoint bounded operator on a Hilbert space. Let $M$ be another bounded selfadjoint operator. Let me assume the commutation property $$ [A, e^{iM}]=0. \tag 1$$ Does (1) imply that $ …

Consider the most classical example, $D=\frac{d}{dx}-x$, which is the creation operator. Note that $D$ is injective since, with $L^2$ norms and dot-products and say $u$ in the Schwartz class, $$ \Vert …