Recent results have shown that any closed operator A commuting with the backwards shift S∗ restricted to K u := H 2 ⊖ uH, where u is an inner function, can be realized as a Nevanlinna function of S∗ u := S ∗| K2 u , A = φ(S∗ u), where φ belongs to a certain class of Nevanlinna functions which depend on u. In this paper this result is generalized to show that given any contraction T of class C0(...

A Nevanlinna function is a function which is analytic in the open upper half plane and has a non-negative imaginary part there. In this paper we study a fractional linear transformation for a Nevanlinna function n with a suitable asymptotic expansion at ∞, that is an analogue of the Schur transformation for contractive analytic functions in the unit disc. Applying the transformation p times we ...

We consider symmetric second-order differential operators with real coefficients such that the corresponding equation is in limit circle case at infinity. Our goal to construct theory of self-adjoint realizations by an analogy Jacobi operators. introduce a new object, quasiresolvent maximal operator, and use it obtain very explicit formula for resolvents all realizations. In particular, this yi...

The classical scalar Nevanlinna-Pick interpolation problem has a long and distinguished history, appearing in a variety of applications in mathematics and electrical engineering. There is a vast literature on this problem and on its various far reaching generalizations; for a quick historical survey see [1] and [38]. It is widely known that the now classical algorithm for solving this problem p...

This paper establishes a version of Nevanlinna theory based on Jackson difference operator $D_{q}f(z)=\frac{f(qz)-f(z)}{qz-z}$ for meromorphic functions zero order in the complex plane $\mathbb{C}$. We give logarithmic lemma, second fundamental theorem, defect relation, Picard theorem and five-value sense $q$-difference operator. By using this theory, we investigate growth entire solutions line...

In this paper the scattering matrix of a scattering system consisting of two selfadjoint operators with finite dimensional resolvent difference is expressed in terms of a matrix Nevanlinna function. The problem is embedded into an extension theoretic framework and the theory of boundary triplets and associated Weyl functions for (in general nondensely defined) symmetric operators is applied. Th...

Abstract We prove several statements about arithmetic hyperbolicity of certain blow-up varieties. As a corollary we obtain multiple examples simply connected quasi-projective varieties that are pseudo-arithmetically hyperbolic. This generalizes results Corvaja and Zannier obtained in dimension 2 to arbitrary dimension. The key input is an application the Ru–Vojta’s strategy. also analogue for f...

This article treats Nevanlinna-Pick interpolation in the setting of a special class of algebraic curves called distinguished varieties. An interpolation theorem, along with additional operator theoretic results, is given using a family of reproducing kernels naturally associated to the variety. The examples of the Neil parabola and doubly connected domains are discussed.