Search Results

Let $\Omega$ be a convex, bounded open subset of $\mathbb{R}^d$, and let $C^1(\bar \Omega)$ be usual space of continuous functions on $\bar \Omega$ which are $C^1$ in $\Omega$ and whose partials in $\ …

I think that in general $L(.)$ and $\Vert.\Vert_\mathcal{H}$ measure quite different things. Writing $L(f)$ for $$ \inf\{ M>0:|f(x)-f(x')| \leq Md(x,y) \;\forall \;x,x'\in \mathcal{X}\} $$ let $\mathc …

Take $N=1$, $T=1$, and let $u=1/2$. Let $v(x) =1/2 - x$ for $x\in \mathbb{Z}+[h,1-h]$ for some small positive number $h$. By taking $h$ small enough, you can obviously make sure $$ \int_{[0,1]} v'(u- …

If $F(f) = G(f,f')$ for some $G\in C^\infty(\mathbb{R}^2)$ then I think you can establish that $$ F'(f)u = (\partial_1 G)(f,f')u + (\partial_2 G)(f,f')u' $$ holds in much the same way as if you were d …

The reason I'm asking is because characteristic\indicator functions have no smooth derivatives and plus I don't understand in which function space of test functions the authors define the weak form …

This isn't an answer, but it's a bit long for a comment. I am going to write $\varphi$ for your $T$, just for consistency with the notation in a book I'm going to mention. Let $X = \{ g \hspace{.2pc} …

The fact that the authors of https://arxiv.org/pdf/math/0602479.pdf refer to using "representation (4) for the distance" makes me think that the estimate you want is supposed to come from something al …

Let $X$ be a complex Banach space, and let $T:X\longrightarrow X$ be a bounded linear operator. Let $\sigma_1$ and $\sigma_2$ be disjoint compact subsets of $\mathbb{C}$ for which $\sigma_1\cup \sigma …

You may want to check the details, but I think that the following argument (or something like it) is enough to give you compactness. I am assuming throughout that your smaller space is contained in $C …

I know it's a while since this question was asked, but I think the book "Linear Operators and their Spectra" by E. Brian Davies might contain some information about what's being asked here. In particu …

Just to add another answer to the pile... I suspect that the 'magic' observed here may be related to so-called local spectral theory. It is possible to do quite a lot (at least in the Banach space se …