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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 …

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 $\mathbb H$ 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 …

Too long for a comment. Why don't use your third criterion: if $K_L$ is the kernel of the operator $L$, that gives you $$ K_A=K_B\circ K_C, $$ i.e. $ K_A(x,y)=\int K_B(x,z) K_C(z,y) dz. $

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 $\mathbb H$ be a Hilbert space and let $\mathcal B(\mathbb H)$ be the Banach algebra of bounded operators on $\mathbb H$. Let $(A_k)_{k\ge 1}$ be a sequence in $\mathcal B(\mathbb H)$. $\bullet$ I …

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 …

The standard $n$-dimensional harmonic oscillator is the operator $ \mathcal H=\frac{1}{2}\sum_{1\le j\le n}(D_j^2+x_j^2), \text{ $D_j=-i\partial_{x_j}$}, $ and its spectral decomposition is $$ \mathca …

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 $ …

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 …

Your question must be reformulated to take into account the comments. Let us just say here that Hilbert-Schmidt operators (operators whose kernels are in $L^2((0,1)^2)$) make an ideal of the bounded o …

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 $u,v\in C^\infty_c(\mathbb R)$, …

Using Fourier transformation, your question is about the existence of a subspace $X$ of $L^2(\mathbb R^d)$ such that $$ \lim_{t\rightarrow 0_+}\left\{\sup_{v\in X, \Vert v\Vert=1}\int(1-e^{-t\vert \xi …

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 …

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. …