New classes of generalized Nevanlinna functions, which under multiplication with an arbitrary fixed symmetric rational function remain generalized Nevanlinna functions, are introduced. Characterizations for these classes of functions are established
by connecting the canonical factorizations of the product function and the original generalized Nevanlinna function in a constructive manner. Also a detailed functional analytic treatment of these classes of functions is carried out by investigating the connection between the realizations of the product function and the original function. The operator theoretic treatment of these realizations is based on the notions of rigged spaces, boundary triplets, and associated Weyl functions.
Let $A$ be a, not necessarily closed, linear relation in a Hilbert space $sH$ with a multivalued part $mul A$. An operator $B$ in $sH$ with $ran Bperpmul A^{**}$ is said to be an operator part of $A$ when $A=B hplus ({0}times mul A)$, where the sum i
s componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of $A$. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation $A$ is said to have a Cartesian decomposition if $A=U+I V$, where $U$ and $V$ are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of $A$ and the real and imaginary parts of $A$ is investigated.
