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A Banach space is a complete normed vector space: A vector space equipped with a norm such that every Cauchy sequence converges.

11 votes
1 answer
465 views

Is the spectrum of a "self adjoint" operator real on $\ell^p$?

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 …
an_ordinary_mathematician's user avatar
4 votes
Accepted

Is the spectrum of a "self adjoint" operator real on $\ell^p$?

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 …
an_ordinary_mathematician's user avatar
4 votes
Accepted

Are these two norms on localized versions of $L^p_q$ equivalent?

The opposite inequality cannot be true. If that were true, then consider a positive function $g$ with the property such that for all $s\in \mathbb{T}$ it holds that $g(s,x) \leq C g(s,y)$ whenever $|x …
an_ordinary_mathematician's user avatar
4 votes

Integral means vs infinite convex combinations

I don't think so. Consider the functions $f(x,y)=e^{ixy}, -1<x<1, y\in \mathbb{R}$. Then, $$ \int_{-1}^1 f(x,y) \frac{dx}{2} = \frac{\sin(y)}{y}. $$ The question is if this is representable as $$ \su …
an_ordinary_mathematician's user avatar
1 vote
Accepted

A counterexample showing $BV_p \neq AC_p$

So first of all, what is claimed in Love's paper is slightly different, It says that for a sufficient large choice of the parameter $c>0$ the function $$ g(x): = \sum_{n=0}^\infty c^{-n/p}\cos(c^n \pi …
an_ordinary_mathematician's user avatar
0 votes
0 answers
67 views

Closed graph theorem for cones?

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 …
an_ordinary_mathematician's user avatar
0 votes
0 answers
73 views

Sufficient condition for weak convergence in Banach spaces

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 …
an_ordinary_mathematician's user avatar