Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: Lebesgue type decompositions for linear relations and Ando's uniqueness
  criterion   A linear relation, i.e., a multivalued operator $T$ from a Hilbert space
${\mathfrak H}$ to a Hilbert space ${\mathfrak K}$ has Lebesgue type
decompositions $T=T_{1}+T_{2}$, where $T_{1}$ is a closable operator and
$T_{2}$ is an operator or relation which is singular. There is one canonical
decomposition, called the Lebesgue decomposition of $T$, whose closable part is
characterized by its maximality among all closable parts in the sense of
domination. All Lebesgue type decompositions are parametrized, which also leads
to necessary and sufficient conditions for the uniqueness of such
decompositions. Similar results are given for weak Lebesgue type
decompositions, where $T_1$ is just an operator without being necessarily
closable. Moreover, closability is characterized in different useful ways. In
the special case of range space relations the above decompositions may be
applied when dealing with pairs of (nonnegative) bounded operators and
nonnegative forms as well as in the classical framework of positive measures.

12
The categories of this paper are: nucl-th
30
The categories of this paper are: q-alg
4
The categories of this paper are: cond-mat
32
The categories of this paper are: atom-ph
24
The categories of this paper are: dg-ga
33
The categories of this paper are: plasm-ph
26
The categories of this paper are: bayes-an
7
The categories of this paper are: hep-th
11
The categories of this paper are: hep-lat
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The categories of this paper are: acc-phys
17
The categories of this paper are: q-fin
6
The categories of this paper are: astro-ph
1
The categories of this paper are: cs
5
The categories of this paper are: gr-qc
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The categories of this paper are: eess
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The categories of this paper are: alg-geom