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Let $A:D_A \subseteq H \to K$ a linear closed surjctive operator between two Hilbert spaces $H$ and $K$. One would expect that in such a situation there must exist a bounded right inverse of $A$, name …

In the paper "A strong open mapping theorem for surjections from cones onto Banach spaces, Marcel de Jeu and Miek Masserschmidt, Adv. Math." it is proved (among other things) that if $X, Y$ are comp …

Let $\{ g_n \} $ be a frame in a separable Hilbert space $H$. Then the frame operator $S:H\to H$ defined as \begin{equation} S f := \sum_{n=1}^\infty (f,g_n)g_n \end{equation} is a Hilbert space isomo …

The so called integral Minkowski inequality claims that for suitable positive functions $f$ defined on the product of two measure spaces $ (X\times Y, \mu \times \nu) $ and $p\geq 1$ we have that $$ \ …

The question is quite elementary but nonetheless no proof or counter example comes to mind immediately. Suppose that $X$ is a Banach space and $\{x_n\}$ is a sequence in $X$ such that $(x_n,y)$ conver …

I have been reading the paper "Carleson Measures in Hardy and Weighted Bergman Spaces of Polydiscs" by F. Jafari and there are a few things that going on that I am not entirely convinced of. The paper …

Beurling's theorem characterize the closed subspaces $M\subset H^2$ of the Hardy space, which are invariant under the shift operator $Sf(z):=zf(z)$, as spaces of the form $\varphi H^2 $ where $\varphi …

No, I don't think so. For an explicit example, for $v,u\in L^2(\mathbb{R})$, let me use the notation $v\otimes w$ for the rank $1$ operator $(v\otimes w) (f) = (w,f) v$. Then consider any fixed $g\in …

It is not neccesary in general that $\varphi \in L^\infty(\mathbb{D})$, but it is necessary and sufficient that in a certain sense $\varphi$ must be bounded ``on average in the hyperbolic sense''. The …

In the paper "Hilbert's inequality", Montgomery and Vaughan proved that a generalization of the discrete Hilbert transform is bounded in $\ell^2$. The inequality reads as follows $$ \Big| \sum_{k\neq …

As stated this property cannot be true. Consider $f(z)=z$. Clearly $f \in X_\frac12$. Let any other $g\in \mathcal{B}$ such that $\Vert g \Vert_\mathcal{B} < \varepsilon$. Then we have that $|f'(0)+g' …

It seems that I have found a counter example myself. For the Hilbert matrix $$ H_\lambda:= \big( \frac{1}{1-\lambda+k+n} \big)_{k,n\geq 0}, \lambda < 1 $$ Rosenblum in "On the Hilbert Matrix I, Pro …

I will try to prove that such functions do not exist. Suppose that $f,g$ are positive (pluri)harmonic functions in the pluri harmonic Drury Arveson space $\mathcal{H}DA_d$ such that $ \langle f ,g \ra …

If a sequence is weakly separated, i.e. there exists a multiplier $\varphi_{ij}$ of multiplier norm at most one such that $\varphi_{ij}(\lambda_i)=\varepsilon, \varphi_{ij}(\lambda_j)=0$, then necessa …

There might be an obvious answer to the question, but it doesn't come to mind. Suppose we have an infinite matrix $A=(a_{ij})$, which defines a bounded linear operator on $\ell^p$, i.e. for all seque …