6710-2021-01

Let \(H\) be a Hilbert space endowed with the inner product \(\langle \cdot , \cdot \rangle \). Let \(T\) be a bounded linear operator satisfying \(\langle T x, x\rangle \ge 0\) for all \(x \in H\). Show that \(I + T\) is injective.

Let \(T : H \to H\) be a bounded linear operator on a Hilbert space \(H\). Let \(A\) and \(B\) be subsets of \(H\) such that \(T(A) \subset B\). Show that \(A^\perp \supset T^*(B^\perp )\). Note: We used the notation \(T(A) = \{T x \mid x \in A\}\).

Let \(X\) be a Banach space and \(T : X \to X\) be a compact linear operator. Use the Fredholm alternative to show that if \(\lambda \notin \sigma _p(T)\) and \(\lambda \neq 0\) then \(\lambda \in \rho (T)\), or in other words: that all non-zero spectral values of a compact operator are eigenvalues.

Let \(X\) be a Banach space and consider the non-linear mapping \(F : X \to X\) defined by

where \(y \in X\) is fixed and \(\alpha \) is a scalar. Show that there is a constant \(C > 0\) such that for any \(|\alpha | < C, F\) is a contraction on the open ball of radius 1 centered at \(y\). Note: You may use that for any \(a, b \in \mathbb {R}\) we have \(|\sin (a) - \sin (b)| \le |a - b|\).

Let \(T : X \to X\) be a bounded linear operator on a complex Banach space \(X\). Prove that \(\sigma (T)\) lies in the complex plane disk: \(\{\lambda \in \mathbb {C} \mid |\lambda | \le \|T\|\}\).