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Compressed resolvents of selfadjoint contractive exit space extensions and holomorphic operator-valued functions associated with them
Methods Funct. Anal. Topology 21 (2015), no. 3, 199-224
Contractive selfadjoint extensions of a Hermitian contraction $B$ in a Hilbert space $\mathfrak H$ with an exit in some larger Hilbert space $\mathfrak H\oplus\mathcal H$ are investigated. This leads to a new geometric approach for characterizing analytic properties of holomorphic operator-valued functions of Krein-Ovcharenko type, a class of functions whose study has been recently initiated by the authors. Compressed resolvents of such exit space extensions are also investigated leading to some new connections to transfer functions of passive discrete-time systems and related classes of holomorphic operator-valued functions.
Methods Funct. Anal. Topology 12 (2006), no. 2, 131-150
Let $q$ be a scalar generalized Nevanlinna function, $q\in\mathcal N_\kappa$. Its gene alized zeros and poles (including their orders) are defined in terms of the function's operator representation. In this paper analytic properties associated with the underlying root subspaces and their geometric structures are investigated in terms of the local behaviour of the function. The main results and various characterizations are expressed by means of (local) moments, asymptotic expansions, and via the basic factorization of $q$. Also an inverse problem for recovering the geometric structure of the root subspace from an appropriate asymptotic expansion is solved.
Methods Funct. Anal. Topology 11 (2005), no. 2, 170-187
Methods Funct. Anal. Topology 7 (2001), no. 3, 1-21
Methods Funct. Anal. Topology 6 (2000), no. 3, 24-55
Let A be a symmetric linear operator (or relation) with equal, possibly infinite, defect numbers. It is well know that one can associate with A a boundary value space and the Weyl function M(λ). The authors show that certain fractional-linear transforms of M(λ) are identified as Weyl functions of extensions of A, and vice versa. This connection is applied to various problems arising in the extension theory of symmetric operators. Some new criteria for a linear operator to be selfadjoint are established. When the defect numbers of A are finite the structure of all selfadjoint extensions with an exit space is completely characterized via a pair of boundary value spaces and their respective Weyl functions. New admissibility criteria are given which guarantee that a generalized resolvent of a nondensely defined symmetric operator corresponds to a selfadjoint operator extension.
Methods Funct. Anal. Topology 5 (1999), no. 1, 65-87