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We will show that $T:L^{2}[1,\infty)\rightarrow L^{2}[1,\infty)$ is compact if and only if $-\infty<\alpha<0$. Observe that the integral kernel $K$ of $T$ is given by \begin{align*} K(x,y)=\dfrac{x^{\alpha}}{y}\chi_{[1\leq x\leq y<\infty)}, \tag{1} \end{align*} where $\chi$ is the characteristic function of the set $\left\{(x,y)\in [1,\infty)^{2} : ... 1 By Lax-Milgram there exists only one$\theta\in W_0^{1,2}$such that $$\int_\Omega\nabla\theta\cdot\nabla\phi dx=\int_\Omega f\phi dx$$ for all$\phi\in W_0^{1,2}$. So in fact$T$is well defined, linear and continuous from$W_0^{1,2}$to$W_0^{1,2}$with norm$|\theta|_{W_0^{1,2}}$. Obviously it is compact, since the range of$T$... 0 No. Since$\lambda_k\to 0$you have a norm-convergent expansion. The$*$-operation is norm-continuous. 2 To show noncompactness, let$a> 0$and set$f_a(x)= (a/2)^{-1/2}\chi_{(a/2,a)}.$Then$f$is a unit vector in$L^2.$It is straightforward to see $$T(f_a)(x) = \begin{cases} 0,\, 0 < x < a/2 \\ (a/2)^{1/2}/x,\, x> a.\end{cases}.$$ Suppose$a<b/2.$Because$T(f_b) = 0$on$(0,b/2),$$$T(f_a)(x)-T(f_b)(x) = (a/2)^{1/2}/x, a<x<b/2.$$ ... 6 I thought I would just offer an alternate solution using Minkowski's integral inequality applied to: $$\| Tf \|_2 = \left\{ \int_0^{\infty} \left( \int_0^x x^{-1} f(t)\,dt \right)^{2}\,dx \right\}^{1/2}$$ But first we do a variable substitution$t \rightarrow xt\$ so the inner integral is integrating over a fixed space: \| Tf \|_2 = \left\{ ... 6 For the boundedness, observe that \begin{align*} \| Tf \|_2^2 &= \int_{0}^{\infty} |Tf(x)|^2 \, dx \\ &\leq \int_{0}^{\infty} \frac{1}{x^2} \int_{0}^{x} \int_{0}^{x} |f(u)||f(v)| \, dudvdx \\ &= \int_{0}^{\infty}\int_{0}^{\infty} |f(u)||f(v)| \left( \int_{u \vee v}^{\infty} \frac{dx}{x^2} \right) \, dudv \\ &= ... 3 Doing only 3). For each n \in \mathbb{N} we define the operators T_n: \ell^2 \to \ell^2 as follows, for each x=\{ x_j\}_j \in \ell^2 put T_n(x) = \left( \sum_{k=1}^\infty a_{1k} x_k, \cdots, \sum_{k=1}^\infty a_{nk} x_k, 0 ,\cdots \right) $$By Hölder, for any j\in \mathbb{N}$$ \left|\sum_{k=1}^\infty a_{jk} x_k\right| \leq \left( ...