Tagged Questions

A compact operator is an operator from normed space $X$ to a normed space $Y$, such that image of every bounded subset of $X$ is relatively compact in $Y$. It's used with (functional-analysis) and (operator-theory) tags.

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Show that the following is a bounded linear operator on $L^2(R_+)$ Calculate the adjoint operator.

The following is a question I have been working on for some time with help from my teacher. Unfortunately we have a solution but are not 100% confident with it. Some guidance on if part(s) of our ...
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I want to calulate the range of an operator $S$ which maps $L^{2}$ into $H^{1}(\Omega)$?

I have an equation $s(\lambda,\mu)=l(\mu)$, where s(.,.) is a symetric positive definite bilinear form in $L^{2}(\Omega)$, and $l(.)$ is in $H^{1}(\Omega)$. I want to show that the range of the ...
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Is $Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$ a compact operator?

Is the operator $A$ defined by $$Ax = (\alpha_1 x_1, \alpha_2 x_2, \alpha_3 x_3, \dots, \alpha_k x_k, 0, 0, \dots)$$ a compact operator? It only has finitely non-zero dimensions, so does this mean ...
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How to show $\sigma(T_q) = \overline{\{q(t) : t \in [0,1]\}}$ where $T_q$ is the multiplication operator?

Let $B$ be the Banach space of bounded complex functions on $[0,1]$ with sup-norm. For $q \in B$, define the (multiplication) operator $T_q : B\rightarrow B$ by $(M_q f)(t) = q(t)f(t)$. How do you ...
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Integral operator with sub-exponential eigenvalue decay

I am looking for a positive-definite symmetric function $K: \Omega \times \Omega \to \mathbb{R}$ ($\Omega \subset \mathbb{R}$), such that  K(x,y) = \sum_{k=0}^\infty \exp(-\sqrt{k}) \phi_k(x) \...
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Show that there is an operator on $H^{2}$ and it's compact.
Let $H^{2}=W_{0}^{2,2}(\Omega)$. Define $(u,v)=\int_{\Omega} (\triangle u\triangle v+2v\triangle u)\mathrm{d}S$ as an inner product on $H^{2}$. Define \$a(u;v)=\int_{\Omega} (\nabla u\cdot \nabla v-...