Products of orthogonal projections as Carleman operators
Let \(X\neq\emptyset\) and let \(\mu\) be a \(\sigma\)-finite measure on \(X\) such that the Hilbert space \(L^2(X,\mu)\) is infinite-dimensional and separable. Let \(H\) be an infinite-dimensional separable Hilbert space. An operator \(A: H\to L_q(X,\mu)\) is said to be a Carleman operator if there is a measurable function \(k: X\to H\) such that (ii) \(D(A)\subseteq \{f\in H:\langle f,k(\cdot)\rangle_H\in L_q(X,\mu)\}\) and (ii) \(Af= \langle f,k(\cdot)\rangle_H\) for all \(f\in D(A)\). Let \(Q\), \(P\) be two orthogonal projections on \(H\). Then \(QP\) is either a Carleman operator or a semi-Fredholm operator with finite defect.
scientific article; zbMATH DE number 1346933
scientific article