Lebesgue decomposition theorems

A Lebesgue-type decomposition for non-positive sesquilinear forms

“…Now, we prove the main theorem of this paper, whose proof is based on Theorem 4.3 of [2]. To do this we will use the following construction of [15] of the w-absolutely continuous s a and w-singular s s parts of a non-negative form s. Let J be the embedding operator π s+w (φ) → π w (φ), from D/ ker (s + w) ⊆ H s+w into H w . In particular, J is a densely defined contraction and J * * is the closure of J.…”

“…Their framework involves the notion of parallel sum of forms, which is inspired by the one for non-negative operators used by Ando [1]. A proof with a different approach was developed by Sebestyén, Tarcsay and Titkos [15].…”

“…Moreover, these notions are related. Indeed, a non-negative measure induces a non-negative form and the absolutely continuous parts are in correspondence, as well as the singular parts (see [8,Theorem 5.5] and also [15,Theorems 3.2 and 3.4]).…”

On the parallel sum of positive operators, forms, and functionals

“…Throughout this section D is a complex vector space and t, w are nonnegative Hermitian forms on it. The parallel sum of t and w was introduced by Hassi, Sebestyén and de Snoo [5] who employed it successfully by developing the Lebesgue decomposition theory of nonnegative sesquilinear forms (for an operator theoretic approach see also [14]). The main idea in their considerations was in going to the quadratic forms and proving that the mapping ( Let us consider the product Hilbert space H t × H w and define three bounded operators J T , J W and J from H t × H w into H t + w , as follows:…”

Lebesgue type decompositions and Radon-Nikodym derivatives for pairs of bounded linear operators

“…In particular, there will be an explicit connection to the recent papers by Z. Sebestyén, Zs. Tarcsay, and T. Titkos; see for instance [27,28,30]. Moreover, further work will also be connected with the situation where at least one of the operators A and B is not bounded.…”