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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 $ …

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

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. $

Interesting question: obviously a sufficient condition for your operator $T$ to be a bounded endomorphism of $L^\infty$ is that $$ \sup_x\int\vert K(x,y)\vert dy=C<+\infty \ (\sharp).\quad\text{This …

The following result is classical: let $\mathbb H$ be a Hilbert space, and let $A,B\in \mathcal B(\mathbb H)$, then $ [A,B]\not=I. $ In finite dimension, just take the trace, and if the dimension is i …

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 $k:[0,1]^2\rightarrow (0,+\infty)$ 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) …

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 …

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 …

The Kadison-Singer problem is the following statement: for any $\epsilon >0$, there exists $r\in \mathbb N$ such that for any bounded operator $A$ on $\ell^2(\mathbb Z)$, there exists a partition $(\m …

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 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 …

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 …