Lebesgue type decompositions for nonnegative forms

Abstract

A nonnegative form t on a complex linear space is decomposed with respect to another nonnegative form w: it has a Lebesgue decomposition into an almost dominated form and a singular form. The part which is almost dominated is the largest form majorized by t which is almost dominated by w. The construction of the Lebesgue decomposition only involves notions from the complex linear space. An important ingredient in the construction is the new concept of the parallel sum of forms. By means of Hilbert space techniques the almost dominated and the singular parts are identified with the regular and a singular parts of the form. This decomposition addresses a problem posed by B. Simon. The Lebesgue decomposition of a pair of finite measures corresponds to the present decomposition of the forms which are induced by the measures. T. Ando's decomposition of a nonnegative bounded linear operator in a Hilbert space with respect to another nonnegative bounded linear operator is a consequence. It is shown that the decomposition of positive definite kernels involving families of forms also belongs to the present context. The Lebesgue decomposition is an example of a Lebesgue type decomposition, i.e., any decomposition into an almost dominated and a singular part. There is a necessary and sufficient condition for a Lebesgue type decomposition to be unique. This condition is inspired by the work of Ando concerning uniqueness questions.

Parallel Sum of Unbounded Positive Operators

“…However, these authors dealt with positive quadratic forms (in the algebraic sense) with a common domain. The study in [5] and subsequent related works (see [13] for instance) might be related to ours, but the exact relationship is not clear to the author. When a, b are bounded, the following expression (for the quadratic form corresponding to the parallel sum a : b) is well-known:…”

“…In fact, by including unbounded operators, we can make the whole theory much more transparent. In [5] parallel sums for (unbounded) positive quadratic forms were studied. However, these authors dealt with positive quadratic forms (in the algebraic sense) with a common domain.…”

Domain and Range Descriptions for Adjoint Relations, and Parallel Sums and Differences of Forms

“…The underlying idea behind the present approach is that the pair of forms induces a linear relation between the Hilbert spaces generated by the forms, which makes it possible to apply the range characterization appearing in the first part of the paper. As was shown in [13] the parallel sum and difference of forms play an essential role in the Lebesgue type decomposition of one form with respect to another form.…”

“…The parallel sum t : w of nonnegative forms t and w was introduced in [13], where the main properties of t : w can be found. From Theorem 5.2 one obtains the following result for parallel sums.…”

“…[13]. This theory was developed parallel to the potential theoretic treatment for operators as developed by Eriksson and Leutwiler [8].…”