Compact Operators

Let \(T : X \to X\) be a compact linear operator on a normed space \(X\). Show that for every \(\lambda \neq 0\) the null space \(N(T - \lambda I)\) is finite dimensional.

Let \(X\) be a Banach space and \(A\), \(T\) be bounded linear operators from \(X\) to itself. Assume further that \(A\) is compact. Show that \(TA\) and \(AT\) are also compact operators.

Let \(H\) be a Hilbert space and \(T : H \to H\) be a compact operator. Show that the Hilbert adjoint \(T^*\) is also compact.

Let \(T:X\to X\) be a compact linear operator and \(S:X\to X\) be a bounded linear operator, defined on a Banach space \(X\).

1. Show that \(TS\) is a compact linear operator.
2. Show that \(ST\) is a compact linear operator.

Let \(T:X\to X\) be a compact linear operator on a normed space \(X\). Show that for every \(\lambda \ne 0\), the null space of \(T_\lambda =T-\lambda I\) is finite dimensional.

Let \(X\) be an inner product space with inner product \(\langle \cdot ,\cdot \rangle \). Let \((a_j)\) be a sequence of complex numbers and let \((\psi _j)\) and \((\phi _j)\) be two orthonormal sequences in \(X\). Let \(T:X\to X\) be an operator defined by \[ Tx=\sum _{j=1}^{\infty } a_j\langle \psi _j,x\rangle \phi _j. \] State and prove a (nontrivial) sufficient condition on \((a_j)\) so that \(T\) is a compact operator.

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 \(T_n\).

Let \(T:\ell ^2\longrightarrow \ell ^2\) be defined by \(y=Tx\) with \(y_j=x_je^{-j}\) for \(j=1,2,3,\ldots \), where \(x=(x_1,x_2,\ldots )\) and \(y=(y_1,y_2,\ldots )\). Prove that \(T\) is a compact operator on \(\ell ^2\).

1. Define what is meant by a precompact (or sequentially compact) set and what is meant by a compact linear operator, assuming linear operators and bounded sets have been already been defined.
2. If \(K\) is a compact linear operator and if \(\{\phi _n\}\) is an infinite set of orthonormal functions then (using Bessel’s inequality, if needed) show that \[ \lim _{n\to \infty }K\phi _n=0. \]

Prove that if a linear operator \(K\) can be approximated (in the operator norm) by a sequence of compact operators \(K_n\), with, \[ \lim _{n\to \infty }\lVert K_n-K\rVert =0, \] then \(K\) is a compact operator.

Let \(K:H\to H\) be a compact linear operator on the infinite-dimensional Hilbert space \(H\).

1. Prove that \(K^{-1}\), if it exists, is not bounded.
2. Given \(T>0\), let \(H=L^2(0,T)\). For \(u\in H\), define \[ (Ku)(t)=\int _0^T e^{-st}u(s)\,ds,\qquad 0\le t\le T. \] Prove that \(K:H\to H\) is a compact operator. What are the consequences of this fact for the utility of the Laplace transform as a numerical method?