Finite Rank Operators and Approximation

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence of finite rank operators \((T_k)\) that converges to \(T\) in an appropriate sense.

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.

Let \(H\) be a separable Hilbert space with \((e_n)\) being a total orthonormal sequence of \(H\). Let \(T : H \to H\) be a bounded linear operator satisfying

Show that \(T\) is compact by approximating \(T\) by a sequence \((T_k)\) of finite rank operators that converges to \(T\) in an appropriate sense.

Let \((\lambda _n)\in \ell ^2\) and consider the operator \(T:\ell ^2\to \ell ^2\) defined by \(y=Tx\), where \(x=(\xi _j)\in \ell ^2\), \(y=(\eta _j)\in \ell ^2\) and \[ \eta _j=\sum _{k=1}^{\infty }\alpha _{jk}\xi _k,\qquad j=1,2,\ldots , \] where the \(\alpha _{jk}\) satisfy \[ \sum _{j=1}^{\infty }\sum _{k=1}^{\infty }\lvert \alpha _{jk}\rvert ^2<\infty . \] Show that \(T\) is a compact operator. Hint: Approximate \(T\) with finite rank operators \(T_n\).

Dependence on Initial Conditions. Consider the scalar initial value problem \[ \begin {aligned} y'' &= y-y^3,\\ y(0) &= a,\\ y'(0) &= b. \end {aligned} \] Denote the dependence of \(y(t)\) on its initial conditions by \(y(t;a,b)\). Without explicitly solving for \(y(t;a,b)\), find \(\frac {\partial y}{\partial a}(t;0,0)\) and \(\frac {\partial y}{\partial b}(t;0,0)\), then write down a first-order approximation for \(y(t;a,b)\) which is valid for small \(\lvert a\rvert ,\lvert b\rvert \).