On the Spectrum of the Product of Closed Operators

Adjoints and formal adjoints of matrices of unbounded operators

“…So as to mention some partial results, let us refer to [6] and [14] where the only nonzero entries are those off the diagonal. On the other hand, in [3] and [13] examples are given showing that the adjoint of a matrix operator A and its formal adjoint A × , that is, the matrix of adjoints of all the particular entries, may be quite different.…”

Local Definitizability of $${{T^{[\ast]}T}}$$ and $${{TT^{{[\ast]}}}}$$

“…In the first one we consider the spectra of the products AB and BA of two arbitrary linear operators A and B acting between Banach spaces. We give a simple proof of a theorem from [5], saying that the non-zero spectra of AB and BA are equal, provided the resolvent sets of AB and BA are non-empty. As a by-product we establish some estimates on the norms of the resolvents of AB and BA.…”

Right (Or Left) Invertibility of Bounded and Unbounded Operators and Applications to the Spectrum of Products

“…For instance, see [3], [4], [8] and [10] We conclude this introduction with an application of Corollary 1.5 (cf. [9]).…”